SO(3) MONOPOLES, LEVEL-ONE SEIBERG-WITTEN MODULI SPACES, AND WITTEN’S CONJECTURE IN LOW DEGREES

PAUL M. N. FEEHAN AND THOMAS G. LENESS

Abstract. We prove Witten’s formula relating the Donaldson and Seiberg-Witten series modulo powers of degree c + 2, with c = − \frac{1}{4} (7χ + 11σ), for four-manifolds obeying some mild conditions, where χ and σ are their Euler characteristic and signature. We use the moduli space of SO(3) monopoles as a cobordism between a link of the Donaldson moduli space of anti-self-dual SO(3) connections and links of the moduli spaces of Seiberg-Witten monopoles. Gluing techniques allow us to compute contributions from Seiberg-Witten moduli spaces lying in the first (or ‘one-bubble’) level of the Uhlenbeck compactification of the moduli space of SO(3) monopoles.

1. Introduction

1.1. Main results. In the present article we extend our results in [17, 18], showing that Witten’s conjecture [53] relating the Donaldson and Seiberg-Witten series holds in ‘low degrees’ for a broad class of four-manifolds. We apply our work on gluing SO(3) monopoles [19, 21] — restricting to the case of one ‘instanton bubble’ in this article — to prove that these two series are equivalent through a higher range of degrees than was possible in [17, 18]. We assume throughout that X is a closed, connected, smooth four-manifold with an orientation for which b_{2}^{+}(X) > 0. The Seiberg-Witten (SW) invariants (see §2.3.1) comprise a function, SW_{X} : Spin^{c}(X) → ℤ, where Spin^{c}(X) is the set of isomorphism classes of spin^{c} structures on X. For w ∈ H^{2}(X; ℤ), define the Seiberg-Witten series by

SW_{X}^{w}(h) = ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·w)}SW_{X}(𝔰)e^{⟨c_{1}(𝔰),h⟩}, h ∈ H_{2}(X; ℝ), (1.1)

and let D_{X}^{w}(h) denote the Donaldson series (see [31, Theorem 1.7] or §2.5 here). There is a map c_{1} : Spin^{c}(X) → H^{2}(X; ℤ) and the image of the support of SW_{X} is the set B of SW-basic classes [53]. A four-manifold X has SW-simple type, when b_{1}(X) = 0, if c_{1}(𝔰)^{2} = 2χ + 3σ for all c_{1}(𝔰) ∈ B, where χ and σ are the Euler characteristic and signature of X. Let B^{⊥} ⊂ H^{2}(X; ℤ) denote the orthogonal complement of B with respect to the intersection form Q_{X} on H^{2}(X; ℤ). Denote c(X) = −\frac{1}{4}(7χ + 11σ). Our main result is

Theorem 1.1. Let X be four-manifold with b_{1}(X) = 0 and odd b_{2}^{+}(X) ≥ 3. Assume X is abundant, SW-simple type, and effective. Then there exist Λ ∈ B^{⊥} and w ∈ H^{2}(X; ℤ) for which Λ^{2} = 4 − (χ + σ) and w − Λ ≡ w_{2}(X) (mod 2). For any such Λ and w, and any h ∈ H_{2}(X; ℝ), one has

D_{X}^{w}(h) ≡ 0 ≡ SW_{X}^{w}(h) (mod h^{c(X)−2}),
D_{X}^{w}(h) ≡ 2^{2−c(X)}e^{\frac{1}{2} h·h}SW_{X}^{w}(h) (mod h^{c(X)+2}).

(1.2)

Date: This version: August 6, 2001. math.DG/0106238. Topology and its Applications, to appear.

PMNF was supported in part by NSF grants DMS 9704174 and, through the Institute for Advanced Study, by DMS 9729992. TGL was supported in part by a Florida International University Provost’s Office Summer Research Grant.

Noting that c(X) = χ_{h}(X) − c_{1}^{2}(X) (see §1.2), Theorem 1.1 can be used to compute Donaldson invariants for four-manifolds in the region c_{1}^{2} ≤ χ_{h} of the (χ_{h}, c_{1}^{2}) plane; all simply connected, compact, complex algebraic surfaces with odd b_{2}^{+} ≥ 3 are abundant (see below) and SW-simple type.

We shall explain below the terminology and notation in the statement of Theorem 1.1. Witten’s conjecture [53] asserts that a four-manifold X with b_{1}(X) = 0 and odd b_{2}^{+}(X) ≥ 3 has SW-simple type if and only if it has KM-simple type, that is, simple type in the sense of Kronheimer and Mrowka (see Definition 1.4 in [31]), and that the SW-basic and KM-basic classes (see Theorem 1.7 in [31]) coincide; if X has simple type, then

D_{X}^{w}(h) = 2^{2−c(X)}e^{\frac{1}{2} Q_{X}(h,h)}SW_{X}^{w}(h), h ∈ H_{2}(X; ℝ). (1.3)

The quantum field theory argument giving equation (1.3) when b_{2}^{+}(X) ≥ 3 has been extended by Moore and Witten [38] to allow b_{2}^{+}(X) ≥ 1, b_{1}(X) ≥ 0, and four-manifolds X of non-simple type. Recall that b_{2}^{+}(X) is the dimension of a maximal positive-definite linear subspace H^{2,+}(X; ℝ) for the intersection pairing Q_{X} on H^{2}(X; ℝ).

In [17, 18] we proved that the second equation (1.2) holds modulo h^{c(X)}, with Λ^{2} = 2−(χ+σ) but otherwise identical hypotheses. If one desires a mod h^{δ} relation such as (1.2) for larger values of δ relative to c(X), one must allow more bubbles and the difficulty of the calculations rapidly increases: see [18, §1] for a more detailed discussion. In particular, to prove Witten’s conjecture (1.3) in full, we would need to prove that equation (1.2) holds modulo h^{δ} for all δ ∈ ℤ_{≥0}: this is the goal of the remaining papers [19, 20, 21] in our series.

If the hypotheses concerning a class Λ are omitted, one still obtains a formula for Donaldson invariants in terms of Seiberg-Witten invariants (see Theorem 1.5), but it has a more complicated structure and one cannot immediately see if it has the shape (1.2) consistent with Witten’s conjecture.

Definition 1.2. [14, p. 169] We say that a closed, oriented four-manifold X is abundant if the restriction of the intersection form to B^{⊥} contains a hyperbolic sublattice.

The abundance condition is just a convenient way of formulating the weaker, but more technical condition that one can find (for example) classes Λ_{j} ∈ B^{⊥} such that Λ_{j}^{2} = 2j − (χ + σ), for j = 1, 2: this is the only property of Q_{X}|B^{⊥} which we use to prove Theorem 1.1. All compact, complex algebraic, simply-connected surfaces with b_{2}^{+} ≥ 3 are abundant [17, Theorem A.1].

We expect the second equation (1.2) to hold modulo h^{δ}, for any δ, without reference to whether a class Λ exists with the properties stated and without the constraint on w. For an explanation of why the limitations of the present article prevent us from relaxing the constraints on w in Theorem 1.1, see Remark 6.8.

In the present article and its companion [18] we prove Theorem 1.1 using the moduli space M_{𝔱} of SO(3) monopoles [45] to provide a cobordism between the link of the moduli space M_{κ}^{w} of anti-self-dual connections and links of moduli spaces of Seiberg-Witten moduli spaces, M_{𝔰}, these moduli spaces being (topologically) embedded in M_{𝔱}. Pairing certain cohomology classes with the links of these moduli spaces gives multiples of the Donaldson and Seiberg-Witten invariants, respectively, and thus provides a relation between the two types of invariants. Let M̄_{𝔱} denote the Uhlenbeck compactification (see Theorem 2.2) of M_{𝔱} in the space of ideal SO(3) monopoles, ∪_{ℓ=0}^{∞}(M_{𝔱_{ℓ}} × Sym^{ℓ}(X)).

Definition 1.3. [17, Definition 1.3] We say that a closed, oriented, smooth four-manifold X with b_{1}(X) = 0 and b_{2}^{+}(X) ≥ 1 is effective if X satisfies Conjecture 3.1 in [14]. This conjecture asserts that for a moduli space of ideal Seiberg-Witten monopoles, M_{𝔰}×Sym^{ℓ}(X), appearing in level ℓ ≥ 0 of M̄_{𝔱}, the intersection number

# (V̄(z) ∩ W̄^{η} ∩ L̄_{𝔱,𝔰}) (1.4)

is a multiple of the Seiberg-Witten invariant SW_{X}(𝔰). Here, V̄(z) and W̄ are the closures in M̄_{𝔱}/S^{1} of geometric representatives of Donaldson-type cohomology classes on the top stratum of M_{𝔱}/S^{1} and L̄_{𝔱,𝔰} is the link of (M_{𝔰} × Sym^{ℓ}(X)) ∩ M̄_{𝔱}/S^{1} in M̄_{𝔱}/S^{1}, and η ≥ 0 is an integer for which deg(z) + 2(η + 1) = dim M_{𝔱}. In particular, the intersection number (1.4) is zero when the Seiberg-Witten invariant SW_{X}(𝔰) is zero.

When b_{1}(X) > 0, one replaces the Seiberg-Witten invariants, SW_{X}(𝔰), mentioned in Definition 1.3 with values of the Seiberg-Witten function SW_{X,𝔰} — see [18, Equation (1.7) and §4.1] for definitions. The motivation for Conjecture 3.1 of [14] and a more detailed explanation appears in [14, §3.1] together with an explanation of its role in the proofs of the main results of that paper; see also [15]. It is almost certainly true that this conjecture holds for all four-manifolds, based on our work in [19, 20, 21], and it is a simple consequence of our Conjecture 1.6 for the general form of the pairings of Donaldson-type cohomology classes with links of ideal Seiberg-Witten moduli spaces in M̄_{𝔱}.

We verify Conjecture 3.1 in [14] by direct calculation in the present article (see Theorem 6.1 and Proposition 6.5) for Donaldson invariants defined by M_{κ}^{w} ↪ M_{𝔱} and Seiberg-Witten moduli spaces M_{𝔰} embedded in the first level, M_{𝔱_{1}} × X, while in [17, 18] we verified the conjecture for Seiberg-Witten moduli spaces M_{𝔰} contained in the top level, M_{𝔱}, of the Uhlenbeck compactification M̄_{𝔱} (see Theorem 4.13 and Proposition 4.22 in [18]). However, we strongly expect the conjecture to hold for Seiberg-Witten moduli spaces M_{𝔰} contained in any level of the compactification M̄_{𝔱}.

Equation (1.2) is a special case of a more general formula for Donaldson invariants which we now describe; the hypotheses still include an important restriction which guarantees that Seiberg-Witten moduli spaces with non-trivial invariants do not lie in the second or lower levels of the compactified SO(3)-monopole moduli space. For Λ ∈ H^{2}(X; ℤ), define

i(Λ) = Λ^{2} + c(X) + χ + σ. (1.5)

If S(X) ⊂ Spin^{c}(X) denotes the support of the function SW_{X} : Spin^{c}(X) → ℤ, let

(1.6) r(Λ, c_{1}(𝔰)) = −(c_{1}(𝔰) − Λ)^{2} − \frac{3}{4}(χ + σ) and r(Λ) = min_{𝔰∈S(X)}r(Λ, c_{1}(𝔰)).

See Remark 3.36 in [18] for a discussion of the significance of r(Λ, c_{1}(𝔰)) and r(Λ), while the significance of i(Λ) is explained in [18, §1.2 & §4.6]; see also [14]. For the statement of Theorem 1.4 below, we refer the reader to §2.5 for a definition of the Donaldson invariants, D_{X}^{w}(h^{δ−2m}x^{m}). Recall that P_{d}^{a,b}(ζ), the Jacobi polynomial [27, §8.960], is defined by

P_{d}^{a,b}(ζ) = \frac{1}{2^{d}} ∑_{v=0}^{d} \binom{d + a}{v}\binom{d + b}{d − v}(ζ − 1)^{d−v}(ζ + 1)^{v}, ζ ∈ ℂ, (1.7)

just as in [18, Equation (4.27)]; if d = 0, then P_{d}^{a,b}(0) = 1.

Theorem 1.4. Let X be a four-manifold with b_{1}(X) = 0 and odd b_{2}^{+}(X) ≥ 1. Assume X is effective. Suppose Λ, w ∈ H^{2}(X; ℤ) are classes such that w − Λ ≡ w_{2}(X) (mod 2) and, if b_{2}^{+}(X) = 1, the class w (mod 2) admits no torsion integral lifts. Suppose h ∈ H_{2}(X; ℝ), x ∈ H_{0}(X; ℤ) is the positive generator, and δ is a non-negative integer. If δ < i(Λ) and δ = r(Λ) + 4, then

D_{X}^{w}(h^{δ−2m}x^{m}) = 2^{1− \frac{1}{4} i(Λ)− \frac{3}{4} δ}(−1)^{m+ \frac{1}{2} (σ−w^{2})} ∑_{\begin{matrix}𝔰∈Spin^{c}(X) \\ r(Λ,c_{1}(𝔰))=δ \end{matrix}}(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·(w−Λ))}

× (−2)^{d_{s}(𝔰)/2}P_{d_{s}(𝔰)/2}^{a−1,b}(0)SW_{X}(𝔰)⟨c_{1}(𝔰) − Λ, h⟩^{δ−2m}

+ 2^{1− \frac{1}{4} i(Λ)− \frac{3}{4} δ}(−1)^{m+ \frac{1}{2} (σ−w^{2})} ∑_{\begin{matrix}𝔰∈Spin^{c}(X) \\ r(Λ,c_{1}(𝔰))=δ−4 \end{matrix}}(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·(w−Λ))}

× (−2)^{d_{s}(𝔰)/2}P_{d_{s}(𝔰)/2}^{a,b}(0)SW_{X}(𝔰)

× (a_{0}⟨c_{1}(𝔰) − Λ, h⟩^{δ−2m} + b_{0}⟨c_{1}(𝔰) − Λ, h⟩^{δ−2m−1}⟨Λ, h⟩

+a_{1}⟨c_{1}(𝔰) − Λ, h⟩^{δ−2m−2}Q_{X}(h, h)) , (1.8)

where all terms on the right which would have a negative exponent are omitted and the coefficients a_{0}(c_{1}(𝔰), Λ, δ, m), a_{1}(δ, m), and b_{0}(δ, m) are given by

\begin{aligned}a_{0} & = 3(c_{1}(𝔰) − Λ)^{2} + c_{1}^{2}(X) + 2(c_{1}(𝔰) − Λ) · Λ + 4δ − 12m, \\ b_{0} & = 2(δ − 2m)\frac{P_{d_{s}(𝔰)/2}^{a−1,b+1}(0)}{P_{d_{s}(𝔰)/2}^{a,b}(0)}, \\ a_{1} & = 4\binom{δ − 2m}{2}, \end{aligned} (1.9)

with

a = −\frac{1}{2}d_{s}(𝔰) + \frac{1}{4}(i(Λ) − δ) and b = −\frac{1}{2}d_{s}(𝔰) − \frac{1}{4}(χ + σ),

and P_{d_{s}(𝔰)/2}^{a,b}(0) given by definition (1.7). If d_{s}(𝔰) = 0, then P_{d_{s}(𝔰)/2}^{a,b}(0) = 1. If b_{2}^{+}(X) = 1, then all invariants in equation (1.8) are evaluated with respect to the chambers determined by the same period point in the positive cone of H^{2}(X; ℝ).

Theorem 1.2 in [18] provides a formula analogous to (1.8) under similar hypotheses for D_{X}^{w}(h^{δ−2m}x^{m}), but for δ = r(Λ), and a vanishing result for δ < r(Λ).

Specializing Theorem 1.4 to case where X has SW-simple type and Λ is in B^{⊥} and recalling that in this situation one has [18, §4.6]

r(Λ) = −Λ^{2} + c(X) − (χ + σ), (1.10)

yields

Theorem 1.5. Continue the hypotheses of Theorem 1.4. We further assume that X has b_{2}^{+}(X) > 1, Seiberg-Witten simple type and that Λ ∈ B^{⊥}. Then

D_{X}^{w}(h^{δ−2m}x^{m}) = 2^{− \frac{1}{2} (c(X)+δ)}(−1)^{m+ \frac{1}{2} (σ−w^{2})}

× ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+w·c_{1}(𝔰))}SW_{X}(𝔰)

× (a_{0}⟨c_{1}(𝔰) − Λ, h⟩^{δ−2m} + b_{0}⟨c_{1}(𝔰) − Λ, h⟩^{δ−2m−1}⟨Λ, h⟩

+a_{1}⟨c_{1}(𝔰) − Λ, h⟩^{δ−2m−2}Q_{X}(h, h)) , (1.11) where the coefficients a_{0}(Λ, δ, m), b_{0}(δ, m), and a_{1}(δ, m) are given by

\begin{aligned}a_{0} & = 4c_{1}^{2}(X) + Λ^{2} + 4δ − 12m, \\ b_{0} & = 2(δ − 2m), \\ a_{1} & = 4\binom{δ − 2m}{2}. \end{aligned} (1.12)

Theorem 1.4 in [18] provides a formula analogous to (1.11) under similar hypotheses for D_{X}^{w}(h^{δ−2m}x^{m}), but for δ = r(Λ), and a vanishing result for δ < r(Λ).

Our main result, Theorem 1.1, then follows from Theorem 1.5 and [18, Theorem 1.1].

1.2. Some applications to minimal surfaces of general type. If X is any closed, oriented, smooth four-manifold we may define (by analogy with their values when X is a complex surface)

c_{1}^{2}(X) = 2χ + 3σ, (1.13)

and

χ_{h}(X) = \frac{1}{4}(χ + σ). (1.14)

Then, the topological invariant c(X) acquires a more familiar interpretation:

c(X) = −\frac{1}{4}(7χ + 11σ) = χ_{h}(X) − c_{1}^{2}(X). (1.15)

Theorem 1.1 has some immediate applications to surfaces of general type: it can be used to compute previously unknown Donaldson invariants. If X is a simply-connected, minimal surface of general type, then, according to [3, Theorem VII.1.1(iv′)], these surfaces obey the Noether inequality:

χ_{h}(X) ≤ \begin{cases}\frac{1}{2}c_{1}^{2}(X) + 3, & if c_{1}^{2}(X) is even, \\ \frac{1}{2}c_{1}^{2}(X) + \frac{5}{2}, & if c_{1}^{2}(X) is odd. \end{cases}

Hence, Theorem 1.1 computes previously unknown Donaldson invariants D_{X}^{w}(x^{m}h^{δ−2m}) for minimal surfaces of general type for δ obeying δ ≤ c(X) and

δ ≤ \begin{cases}3 − \frac{1}{2}c_{1}^{2}(X), & if c_{1}^{2}(X) is even, \\ \frac{5}{2} − \frac{1}{2}c_{1}^{2}(X), & if c_{1}^{2}(X) is odd. \end{cases}

This gives non-trivial results for Donaldson invariants when c_{1}^{2}(X) ≤ 6.

1.3. Discussion of the hypotheses of Theorems 1.1, 1.4, and 1.5. To prove Theorem 1.4 (and thus Theorems 1.5 and 1.1), we employ the compactified moduli space of SO(3) monopoles, M̄_{𝔱}/S^{1}, as a cobordism between a link of the moduli space of anti-self-dual connections, M_{κ}^{w}, and the links of moduli spaces of ideal Seiberg-Witten monopoles, M_{𝔰} × Sym^{ℓ}(X). Our application of the cobordism method in this article requires that

1. The codimension of M_{κ}^{w} in M_{𝔱}, given by twice the complex index of a Dirac operator, is positive (used in Proposition 3.29 in [18]), and

2. Only the top or first levels of the Uhlenbeck compactification M̄_{𝔱} can contain Seiberg-Witten moduli spaces M_{𝔰}×Sym^{ℓ}(X) (where ℓ = 0, 1) with non-trivial invariants (used in §2.6 to eliminate the more intractable terms in the sum (2.52)).

In the proof of Theorem 1.4, one has 2δ = deg(z) = dim M_{κ}^{w} and the hypotheses δ < i(Λ) and δ = r(Λ) + 4 ensure that Conditions (1) and (2) hold, respectively.

The hypotheses imply that the cobordism M̄_{𝔱}/S^{1} yields an equality between pairings with the link of M_{κ}^{w} and a sum of pairings with the links of M_{𝔰} × Sym^{ℓ}(X) where ℓ = 0, 1. The same remarks apply to the hypotheses in Theorem 1.5.

The assumption that b_{1}(X) = 0 could be relaxed by generalizing our calculation of the Segre classes for the virtual normal bundle N_{𝔱,𝔰} → M_{𝔰} to the case b_{1}(X) > 0. We computed those Segre classes in [17, Corollary 3.32], [18, Lemma 4.7] when b_{1}(X) > 0, under a technical assumption on H^{1}(X; ℝ), but we have not used this generalization here as it would greatly complicate our main formulae.

When b_{2}^{+}(X) = 1, we assume that w (mod 2) does not admit a torsion integral lift in order to avoid complications in defining the chamber in the positive cone of H^{2}(X; ℝ) with respect to which the Donaldson and Seiberg-Witten invariants are computed. See the comments at the end of §3.4.2 in [18] and before Lemma 4.1 in [18] for further discussion.

The proof of Theorem 1.1 requires one to choose classes Λ ∈ B^{⊥} with optimally prescribed even square in order to obtain the indicated vanishing results for the Donaldson and Seiberg-Witten series, as well as compute the first non-vanishing terms. The hypothesis that X is abundant guarantees that one can find such classes, though such choices are also possible for some non-abundant four-manifolds [14]. The constraints on the pair w, Λ were discussed in §1.1.

1.4. Role of the present article in the SO(3)-monopole program. Since our series of articles on the SO(3)-monopole approach to the proof of Witten’s conjecture is quite long, it is perhaps worth mentioning how the present article fits into this program. We gave a broad (but now slightly dated) review of the program in [15], while we provided a more modern outline in the introductions to [17] and [18].

We proved the basic transversality and compactness properties of the moduli space of SO(3) monopoles in [16, 13]. The local gluing results for SO(3) monopoles — focusing on analytical steps involved in construction of the links of lower-level Seiberg-Witten moduli spaces — are the subject of [19, 20], while the global gluing results — focusing on topological steps involved in construction of these links — are considered in [21]. At the time of writing, work on [20] and [21] is still in progress, though we believe we have surmounted most of the difficult technicalities.

Turning to computational aspects of our work on Witten’s conjecture, we show in [17, 18] that, accepting some mild hypotheses, the Donaldson series is equal to 2^{2−c(X)}e^{\frac{1}{2} Q_{X}} times the Seiberg-Witten series, at least through terms h^{δ} of degree δ ≤ c(X) (compare Theorem 1.1 here). In our articles [17, 18] we did not consider contributions from Seiberg-Witten moduli spaces other than those in the top level, so we could not compare higher-degree terms in those articles. In order to compare terms of arbitrarily high degree we need to compute contributions of the form (1.4) from Seiberg-Witten moduli spaces in arbitrarily low levels: partial computations in this vein are given in [21]. Though we only compute contributions from the first level in the present article, thus extending the calculations of [17, 18], whenever possible we present the calculations in sufficient generality that they apply to arbitrary levels: see §5 and §6 for details. Finally, the article [14] is an application of the main conclusions of [18] and [21].

1.5. Comparison with [34]. We use our gluing theorems [19, 20] to describe the topology of an open neighborhood in M̄_{𝔱} of the Seiberg-Witten ‘stratum’ (M_{𝔰} × Sym^{ℓ}(X)) ∩ M̄_{𝔱}, restricting our attention in the present article to the case ℓ = 1. We then compute pairings with (see Theorem 6.1) the circle-quotient of the boundary of this open neighborhood, the link L̄_{𝔱,𝔰}, by generalizing techniques developed in [54] and [34] to compute wall-crossing contributions to Donaldson invariants of four-manifolds with b_{2}^{+} = 1 (see Lemma 5.24) to compute contributions to Donaldson invariants Seiberg-Witten strata in M̄_{𝔱}. There are two novel features here:

1. The moduli space of ‘reducibles’, M_{𝔰}, can be positive-dimensional.

2. There is an obstruction to gluing arising from the cokernel of a Dirac operator on a

complex-rank eight Clifford module over S^{4}. In [34, 54] the strata of ‘reducibles’ have the form ([A] × Sym^{ℓ}(X)) ∩ M̄_{κ}^{w}(g_{I}), where [A] is a single point represented by a reducible anti-self-dual connection and M_{κ}^{w}(g_{I}) is the parametrized (by a path of metrics g_{I} with I = (−1, 1)) moduli space of anti-self-dual connections on an SO(3) bundle over X with second Stiefel-Whitney class w (mod 2) and Pontrjagin number −κ/4. However, as we have already computed [17, 18] the Segre classes of a ‘virtual’ or ‘stabilized’ normal bundle of the stratum M_{𝔰} ↪ M_{𝔱_{ℓ}}, the problem that M_{𝔰} is positive-dimensional can be addressed via an elementary formula, Proposition 5.20, relating the Thom class of an equivariant normal bundle with the Segre classes. (The authors are not aware if this formula is already known — the derivation is straightforward, but it clarifies the relation between the equivariant localization computations of Seiberg-Witten wall-crossing formula in [9] and the Segre class computations of the same formula in [36] and [41].)

The second novel feature is the presence of an obstruction to gluing, described in §3.6.2, arising from the cokernel of the twisted Dirac operator on S^{4}: this produces a term in the formula (6.2) with coefficient b_{0}, which has no counterpart in the corresponding formulae of [34] and [54].

1.6. Outline of the computation of Seiberg-Witten link pairings. The central problem in the application of the cobordism method to a proof of Witten’s conjecture is to compute the intersection number (1.4). It seems worthwhile to outline the basic steps involved in this calculation for the case ℓ = 1. These steps comprise the proof of Theorem 6.1, the computational core of the current article. In our outline we try to draw a distinction between the technical points which are at the heart of the difficulty underlying the cobordism approach to a proof of Witten’s conjecture and those which are more tractable.

1.6.1. Defining the virtual link. The domain of the gluing map (see Theorem 3.8) is the space

M̄_{𝔱,𝔰}^{vir}/S^{1} = N_{𝔱_{1},𝔰}(ε) ×_{S^{1}} Ḡl_{𝔱_{1}}(δ), (1.16)

where N_{𝔱_{1},𝔰}(ε) → M_{𝔰} is a complex disk bundle (of radius ε) homeomorphic to a neighborhood of M_{𝔰} in a virtual moduli space containing a neighborhood of M_{𝔰} in the moduli space M_{𝔱_{1}} and Ḡl_{𝔱_{1}}(δ) is a space of gluing parameters, containing the moduli space of instantons on S^{4} and the bundle frames necessary to splice instantons onto SO(3) monopoles represented by points in N_{𝔱_{1},𝔰}(ε). We define a virtual link, L̄_{𝔱,𝔰}^{vir}, as the boundary of the domain (1.16).

1.6.2. Passage from an intersection product to a pairing of dual cohomology classes with the fundamental class of the virtual link. In §5.1, we prove that the intersection number (1.4) is equal to a pairing of a product of cohomology classes with a homology class [L̄_{𝔱,𝔰}^{vir}] which is, effectively, the fundamental class of the virtual link. We first show that the intersection number is equal to the pairing of a relative cohomology class defined by the geometric representatives and the relative Euler class of the obstruction bundle and obstruction section with the relative fundamental class of a codimension-zero subspace of the top stratum of the virtual link (see (5.23)). The proofs given in §5.1 are specific to the topology of the case ℓ = 1, but the results should hold for all ℓ > 0.

1.6.3. Integration over the fiber. We then consider a subspace, M_{𝔰} × ∂Ḡl_{𝔱_{1}}(δ)/S^{1}, of the virtual link, L̄_{𝔱,𝔰}^{vir}. We use division by the Poincaré dual of the subspace M_{𝔰} × ∂Ḡl_{𝔱_{1}}(δ)/S^{1} to relate pairings with [L̄_{𝔱,𝔰}^{vir}] to pairings with the fundamental class of this subspace (see equation (5.71)). An analogous subspace appears in the virtual link for when ℓ > 1 and the Poincaré dual division argument will translate to the general case.

1.6.4. Kunneth-type formula and Seiberg-Witten pairings. Next we write pairings with [M_{𝔰}× ∂Ḡl_{𝔱_{1}}(δ)/S^{1}] as a sum of products of pairings with [M_{𝔰}], yielding multiples of the Seiberg-Witten invariant, SW_{X}(𝔰), and pairings with [∂Ḡl_{𝔱_{1}}(δ)/S^{1}]; see equations (6.12), (6.13), and the identities that follow.

1.6.5. Instanton link pairings. The pairings with [∂Ḡl_{𝔱_{1}}(δ)/S^{1}] are almost identical with those found in [30]. It is the calculation of these pairings (and indeed the definition of the space Ḡl_{𝔱_{ℓ}}(δ) for arbitrary levels ℓ > 0) which causes by far the most difficulty. In this article we use the computations of [34], recorded here in equation (5.76), to compute these pairings; see equation (6.18).

1.7. Extension of results from level-one case to higher levels. Although Theorem 1.1 falls well short of a complete proof of Witten’s conjecture, it nonetheless provides further confirmation of the promise of the SO(3)-monopole approach. Our proof of Theorem 1.1 addresses many of the technical issues needed for our later work on a proof of Conjecture 1.6 and hence Witten’s conjecture. As explained in more detail in §1.7.2, our proof of Theorem 1.1 (in particular, Theorem 6.1) illustrates that a crucial step is to evaluate a certain ‘instanton link’ pairing (see (1.18)), of the kind that arise in the Göttsche-Kotschick-Morgan wall-crossing formula for Donaldson invariants [25, 30].

1.7.1. Level two. The most immediate extension would involve level-two Seiberg-Witten moduli spaces. The first main change would be to replace Lemma 5.24, which records formulae from [34] for pairings with the boundary of the level-one gluing-data bundle ∂Ḡl_{𝔱_{1}}(δ)/S^{1}, with the analogous results from [34] for the boundary of the level-two gluingdata ∂Ḡl_{𝔱_{2}}(δ)/S^{1}. The second main change would be to compute the Euler class of the instanton obstruction bundle for the level-two case, generalizing the current Lemma 4.12 which addresses the level-one case. For the Euler class calculation, the description of the cokernel of the twisted Dirac operator [12, Lemma 3.3.28] might provide a useful starting point. (Note that the description of this cokernel bundle in [2] does not suffice as we must work with an equivariant extension over the gluing-data bundle.) We would then need to replace in the proof of Theorem 6.1 the resulting formulae for pairings with [∂Ḡl_{𝔱_{1}}(δ)/S^{1}] by formulae for level-two pairings and replace the Euler class of the instanton obstruction bundle with its level-two counterpart. The end result would be versions of Theorems 1.4 and 1.5 valid for δ ≤ r(Λ) + 8 and an extension of Theorem 1.1 to a mod h^{c(X)+4} equivalence in equation (1.2) (using δ = c(X)+2 when Λ ∈ B^{⊥} obeys Λ^{2} = 6−(χ+σ), noting that δ must obey the inequality δ < i(Λ), with i(Λ) as given in definition (1.5), and obey δ ≡ c(X) + Λ^{2} (mod 4) to give non-zero Donaldson invariants.) From [14, Theorem 1.3] we know that the Donaldson invariants D_{X}^{w}(z) vanish, for suitable w (for example, w characteristic), when deg(z) < c(X) − 2 (at least for abundant four-manifolds of Seiberg-Witten simple type), so this calculation should give a direct verification that Kronheimer-Mrowka simple type implies Seiberg-Witten simple type for a non-trivial case with deg(z) ≤ c(X) + 2:

D_{X}^{w}(h^{c(X)−2}x^{2}) = 4D_{X}^{w}(h^{c(X)−2}).

Hence, this computation could be very useful for future attempts to understand the relationship between Kronheimer-Mrowka and Seiberg-Witten simple type.

1.7.2. Higher levels and comparison with the Kotschick-Morgan conjecture. When b_{2}^{+}(X) = 1, the work of Kotschick and Morgan (see [30, Theorem 3.0.1] or [25, Theorem 2.3]) expresses the difference in Donaldson invariants of X, defined by two different chambers C_{±} in the same connected component of the positive cone of H^{2}(X; ℝ), as an alternating sum

D_{X,C_{+}}^{w}(x^{m}h^{δ−2m}) − D_{X,C_{−}}^{w}(x^{m}h^{δ−2m})

= ∑_{ξ∈S(C_{+},C_{−},w,δ)}(−1)^{ε(w,ξ,δ)}δ_{X,ξ}^{w}(x^{m}h^{δ−2m}),

where the precise definition of the index set S(C_{+}, C_{−}, w, δ) ⊂ H^{2}(X; ℤ) and sign ε(w, ξ, δ) are not relevant to this discussion. The classes ξ ∈ S(C_{+}, C_{−}, w, δ) correspond to points [A_{ξ}] ∈ M_{κ−ℓ}^{w}(g_{0}) represented by reducible connections. The terms δ_{X,ξ}^{w}(x^{m}h^{δ−2m}) in the preceding sum are given by [30, Definition 5.1.1]

δ_{X,ξ}^{w}(x^{m}h^{δ−2m}) = ⟨μ̄(x^{m}h^{δ−2m}), [∂ (B_{ℂ}^{n}(ε) ×_{S^{1}} Ḡl_{ξ,ℓ}(δ))]⟩ , (1.17)

where B_{ℂ}^{n}(ε) ⊂ ℂ^{n} is the ball of radius ε centered at the origin. The space B_{ℂ}^{n}(ε) ×_{S^{1}} Ḡl_{ξ,ℓ}(δ) is homeomorphic to a neighborhood of the reducible ideal anti-self-dual connections [A_{ξ}] × Sym^{ℓ}(X) in a compactified moduli space M̄_{κ}^{w}(g_{I}) of anti-self-dual ideal connections parametrized by a path of metrics g_{I}. A neighborhood of the reducible connection [A_{ξ}] in the parametrized moduli space M_{κ−ℓ}^{w}(g_{I}) is homeomorphic to B_{ℂ}^{n}(ε)/S^{1}. The space of global gluing data, Ḡl_{ξ,ℓ}(δ), is (roughly) the union of the spaces of stratum-wise gluing data, Ḡl_{ξ,ℓ}(Σ, δ), as Σ ranges over the smooth strata of Sym^{ℓ}(X), patched together via (non-canonical) transition maps obeying a cocycle condition. Naturally, the case ℓ = 1 is simplest, as no transition map is needed, while if ℓ = 2 the transition map does not need to satisfy a cocycle condition: the real problem of constructing Ḡl_{ξ,ℓ}(δ) arises when ℓ ≥ 3. An approach to constructing the space of global gluing data Ḡl_{ξ,ℓ}(δ) is indicated in [30, Theorem 4.4.2], although the construction of the space of global gluing data and the global gluing map — in a form suitable for our purposes — is a difficult analytical and topological problem.

The terms δ_{X,ξ}^{w}(x^{m}h^{δ−2m}) are computed directly in [34] for the cases ℓ = 1 and ℓ = 2; we exploit the computation when ℓ = 1 in the present article, where the result is recorded as Lemma 5.24. It is natural then to ask if one can use the computations of δ_{X,ξ}^{w}(x^{m}h^{δ−2m}) in [25] to compute the intersection numbers (1.4) with links in the moduli space of SO(3) monopoles.

There are some differences worth noting between the link of [A_{ξ}] × Sym^{ℓ}(X) in M̄_{κ}^{w}(g_{I}),

L̄_{κ,ξ}^{w} := ∂(B_{ℂ}^{n}(ε) ×_{S^{1}} Ḡl_{ξ,ℓ}(δ)),

and the links L̄_{𝔱,𝔰}^{vir} of M_{𝔰}×Sym^{ℓ}(X) in M̄_{𝔱,𝔰}^{vir}/S^{1} or the links L̄_{𝔱,𝔰} of M_{𝔰}×Sym^{ℓ}(X) in M̄_{𝔱}/S^{1}. A neighborhood of the point [A_{ξ}] in the background moduli space M_{κ−ℓ}^{w}(g_{0}) is homeomorphic to B_{ℂ}^{n}(ε)/S^{1}, while a neighborhood of the Seiberg-Witten moduli space (which can have positive dimension) in the background virtual moduli space, M_{𝔱_{1},𝔰}^{vir}/S^{1}, is homeomorphic to the S^{1}-quotient of the complex disk bundle, N_{𝔱_{1},𝔰}(ε)/S^{1} → M_{𝔰}. However, the Poincaré-dual division argument mentioned in §1.6 reduces both the pairing (1.17) and the intersection number (1.4) to a sum of terms of the form

⟨ν_{𝔱_{ℓ}}^{n} ⌣ π_{X}^{∗}α, [∂Ḡl_{𝔱_{ℓ}}/S^{1}]⟩ , (1.18) where ν_{𝔱_{ℓ}} is the first Chern class of the circle bundle ∂Ḡl_{𝔱_{ℓ}} → ∂Ḡl_{𝔱_{ℓ}}/S^{1}, the map π_{X} is a projection ∂Ḡl_{𝔱_{ℓ}}/S^{1} → Sym^{ℓ}(X), and α is a cohomology class on Sym^{ℓ}(X). (One has ξ = Λ − c_{1}(𝔰) in the proof of Theorem 6.1; see equation (6.12).) Thus, when comparing the Kotschick-Morgan conjecture and conjectures about the intersection number (1.4), we will discuss pairings of the form (1.18).

We note that the terms δ_{X,ξ}^{w}(x^{m}h^{δ−2m}) in [25] are computed under the assumption that the four-manifold X has b_{2}^{+}(X) = 1, so the computational methods of [25] do not immediately apply to the pairings (1.18), where we allow b_{2}^{+}(X) ≥ 1.

While pairings with links of reducible SO(3) monopoles involve some cohomology classes differing from those appearing in (1.17) (specifically, a cohomology class (2.44) associated with the circle action on M_{𝔱}^{∗,0}), these pairings can still be expressed in terms of pairings of the form (1.18). Thus, no new difficulties are posed by those additional cohomology classes.

A final difference between computation of the pairings in [30] and the computation of pairings with Seiberg-Witten links is due to the presence of obstructions to gluing SO(3) monopoles. The link of a family of M_{𝔰} × Sym^{ℓ}(X) in M̄_{𝔱}/S^{1} is given not by the space L̄_{𝔱,𝔰}^{vir}, the SO(3)-monopole analogue of L̄_{κ,ξ}^{w}, but rather by the zero-locus of a section of an obstruction bundle over L̄_{𝔱,𝔰}^{vir}. However, in Lemmas 4.11 and 4.12 we prove that for ℓ = 1, the Euler class of this obstruction bundle can be expressed in terms of ν_{𝔱_{ℓ}} and π_{X}^{∗}Λ. We expect a similar result will hold for ℓ > 1. In addition, we expect the arguments of §5.1 representing the zero-locus of the obstruction section as dual to extensions of the Euler class of the obstruction bundle to hold for ℓ > 1. Thus, in spite of the obstruction to gluing, the pairings with the link of reducible SO(3) monopoles can still be understood in terms of the pairings (1.18).

The crux of the proof of Witten’s conjecture, via SO(3) monopoles, is then to define the link components ∂Ḡl_{𝔱_{ℓ}}(δ)/S^{1} and compute the pairings (1.18): one can see how they arise in the present article in the course of the proof of Theorem 6.1. Short of proving Witten’s conjecture outright, one can aim instead to prove that (by analogy with the conjecture of Kotschick and Morgan, as phrased by Göttsche [25, Remark 4.3]):

Conjecture 1.6. Let X be a closed, oriented, smooth four-manifold with b_{1}(X) = 0 and odd b_{2}^{+}(X) > 1. Suppose w, Λ ∈ H^{2}(X; ℤ) obey w − Λ ≡ w_{2}(X) (mod 2). Let δ, m be nonnegative integers for which m ≤ [δ/2] and δ ≡ −w^{2} − \frac{3}{4}(χ + σ) (mod 4). Then let 𝔱 be the spin^{u} structure over (X, g) with c_{1}(𝔱) = Λ, p_{1}(𝔱) determined by δ = −p_{1}(𝔱) − \frac{3}{4}(χ + σ), and w_{2}(𝔱) ≡ w (mod 2). (See §2.1 here or §2.1 and §2.2 in [17].) Suppose 𝔰 is a spin^{c} structure over (X, g) with dim M_{𝔰} = 0 and ℓ ≡ \frac{1}{4}(δ − r(Λ, c_{1}(𝔰))) ≥ 0. Let L̄_{𝔱,𝔰} denote the link of M_{𝔰} × Sym^{ℓ}(X) in M̄_{𝔱}/S^{1} and denote η = \frac{1}{4}(p_{1}(𝔱) + Λ^{2} − σ) − 1. Let x ∈ H_{0}(X; ℤ) be the positive generator. Then for any h ∈ H_{2}(X; ℝ), and z = h^{δ−2m}x^{m}, one has

\begin{matrix}# (V̄(z) ∩ W̄^{η} ∩ L̄_{𝔱,𝔰}) \\ = ±2^{a(χ,σ,δ,m,ℓ)}SW_{X}(𝔰) ∑_{j=0}^{q} p_{δ,m,,ℓ,j} (⟨c_{1}(𝔰), h⟩, ⟨Λ, h⟩) Q_{X}^{j}(h, h), \end{matrix} (1.19)

where q = min(ℓ, [δ/2] − m), a(·) is a linear polynomial, and p_{δ,m,ℓ,j} is a degree-(δ − 2m − 2j) homogeneous polynomial in two variables with coefficients which are degree-(ℓ − j) polynomials in 2χ ± 3σ, (c_{1}(𝔰) − Λ)^{2}, Λ^{2}, (c_{1}(𝔰) − Λ) · c_{1}(𝔰), and also depend on δ, m, ℓ.

The assumption that dim M_{𝔰} = 0 can be relaxed; the coefficients on the right-hand side of equation (1.19) will then additionally depend on d_{s}(𝔰) = dim M_{𝔰}. Conjecture 1.6 is motivated by our explicit calculations of the pairings (1.19) when ℓ = 0 in [18] and ℓ = 1 here, as well as partial calculations when ℓ ≥ 2. Our development of a proof of Conjecture 1.6 is work in progress [21]. Conjecture 1.6 also implies the ‘multiplicity conjecture’, namely Conjecture 2.5, on which our proof of the Mariño-Moore-Peradze conjecture [14] relies. Conjecture 1.6 is of interest to us because of

Conjecture 1.7. Witten’s formula (1.3) is implied by Conjecture 1.6.

Göttsche’s calculation of the wall-crossing formula for Donaldson invariants [25], assuming the Kotschick-Morgan conjecture [30] and Fintushel and Stern’s proof of the general blow-up formula for Donaldson invariants lends weight to our expectation that Conjecture 1.7 holds. Even though there are many unknown universal coefficients in our conjectured formula (1.19) for the pairings for arbitrary ℓ(𝔱, 𝔰) ≥ 0, that formula still yields a qualitative version of Witten’s conjecture — that Donaldson invariants are determined by Seiberg-Witten invariants — and implies that a formula (1.3) of the type predicted by Witten should exist.

1.8. Guide to the article. We now outline the contents of the remainder of this article and indicate the principal steps in the proofs of our principal results.

In §2 we recall the definition of the SO(3) monopole moduli space and its basic properties, describe the strata given by the moduli spaces of anti-self-dual SO(3) connections and Seiberg-Witten monopoles, and review the definitions of the Donaldson and Seiberg-Witten invariants. We briefly describe the links of these strata (deferring a detailed account of the Seiberg-Witten links until §3) and the resulting SO(3)-monopole cobordism formula relating the invariants.

In §3 we define the domain of the gluing map [19, 20], the topological model space which parametrizes a neighborhood of the stratum (M_{𝔰}×Sym^{ℓ}(X))∩M̄_{𝔱} when ℓ = 1; the detailed definition occupies §3.1 through §3.5. We identify the obstruction bundle in §3.6 and discuss the gluing theorem in §3.7. We define the virtual link in §3.8 as the boundary of the domain of the gluing map; the actual link is given by the image of the zero-locus of a section of the obstruction bundle under the gluing map. In §3.9, we define an orientation for the virtual link and compare this orientation to orientations previously defined in [18].

In §4 we identify and extend the universal cohomology classes. Specifically, in §4.1 we compute the pullback of the cohomology classes on M_{𝔱} (both Donaldson-type and those specific to the SO(3)-monopole moduli space) to the domain of the gluing map. In §4.2 we compute the Euler classes of the instanton and Seiberg-Witten components of the obstruction bundle.

The proof that the intersection number in (1.4) can be expressed cohomologically appears in §5.1. The topological computations described in §1.6 for the cohomology of the virtual link appear in §5.2.

We give the computations leading to Theorem 1.1 in §6. The main technical result of the article is Theorem 6.1 in which we compute the intersection number appearing on the left-hand-side of (1.19) for a level-one link. Theorem 6.1 follows from the results of §5 and some algebraic computations performed in §6.1. Because the Donaldson and Seiberg-Witten invariants are defined with the aid of the blow-up formula (to avoid technical difficulties associated with flat (reducible) SO(3) or U(1) connections), in §6.2 we prove a blow-up formula for level-one links. The proofs of Theorems 1.4, 1.5, and 1.1 are then completed in §6.3.

Acknowledgments. We are grateful for the generous support of the Institute for Advanced Study (Princeton), the Max Planck Institut für Mathematik (Bonn), and the Department of Mathematics at Columbia University, and Trinity College Dublin (half-year visit by Leness) where parts of this article were completed during visits to these institutions.

2. Preliminaries

We begin in §2.1 by recalling the definition of the moduli space of SO(3) monopoles and its basic properties [16, 13]. In §2.2 we describe the stratum of zero-section monopoles, or anti-self-dual connections. In §2.3 we discuss the strata of reducible, or Seiberg-Witten monopoles, together with their ‘virtual’ neighborhoods and normal bundles. In §2.4 we define the cohomology classes which will be paired with the links of the anti-self-dual and Seiberg-Witten moduli spaces. In §2.5 we review the definition of the Donaldson series. Lastly, in §2.6 we describe the basic relation between the pairings with links of the antiself-dual and Seiberg-Witten moduli spaces provided by the SO(3)-monopole cobordism.

2.1. The moduli space of SO(3) monopoles. Throughout this article, (X, g) will denote a closed, connected, oriented, smooth, Riemannian four-manifold.

2.1.1. Clifford modules. Let V be a Hermitian vector bundle over (X, g) and let ρ : T^{∗}X → End_{ℂ}(V) be a real-linear map satisfying

ρ(α)^{2} = −g(α, α)id_{V} and ρ(α)^{†} = −ρ(α), α ∈ C^{∞}(T^{∗}X). (2.1)

The map ρ uniquely extends to an algebra homomorphism, ρ : Λ^{•}(T^{∗}X) ⊗_{ℝ} ℂ → End_{ℂ}(V), and gives V the structure of a Hermitian Clifford module for the complex Clifford algebra ℂℓ(T^{∗}X). There is a splitting V = V^{+} ⊕ V^{−}, where V^{±} are the ∓1 eigenspaces of ρ(vol). A unitary connection A on V is spin if

[∇_{A}, ρ(α)] = ρ(∇α) on C^{∞}(V), (2.2)

for any α ∈ C^{∞}(T^{∗}X), where ∇ is the Levi-Civita connection.

A Hermitian Clifford module 𝔰 = (ρ, W) is a spin^{c} structure when W has complex rank four; it defines a class

c_{1}(𝔰) = c_{1}(W^{+}), (2.3)

and every class in H^{2}(X; ℤ) lifting the second Stiefel-Whitney class, w_{2}(X) ∈ H^{2}(X; ℤ/2ℤ), arises this way.

We call a Hermitian Clifford module 𝔱 = (ρ, V) a spin^{u} structure when V has complex rank eight. Recall that 𝔤_{V} ⊂ 𝔰𝔲(V) is the SO(3) subbundle given by the span of the sections of the bundle 𝔰𝔲(V) which commute with the action of ℂℓ(T^{∗}X) on V. We obtain a splitting

𝔰𝔲(V^{+}) ≅ ρ(Λ^{+}) ⊕ iρ(Λ^{+}) ⊗_{ℝ} 𝔤_{V} ⊕ 𝔤_{V} , (2.4)

and similarly for 𝔰𝔲(V^{−}). The fibers V_{x}^{+} define complex lines whose tensor-product square is det(V_{x}^{+}) and thus a complex line bundle over X,

det^{\frac{1}{2}} (V^{+}). (2.5)

A spin^{u} structure 𝔱 thus defines classes,

c_{1}(𝔱) = \frac{1}{2}c_{1}(V^{+}), p_{1}(𝔱) = p_{1}(𝔤_{V}), and w_{2}(𝔱) = w_{2}(𝔤_{V}). (2.6)

Given W, one has an isomorphism V ≅ W ⊗_{ℂ} E of Hermitian Clifford modules, where E is a rank-two Hermitian vector bundle [17, Lemma 2.3]; then

𝔤_{V} = 𝔰𝔲(E) and det^{\frac{1}{2}} (V^{+}) = det(W^{+}) ⊗_{ℂ} det(E).

2.1.2. SO(3) monopoles. We fix a smooth unitary connection A_{Λ} on the line bundle det^{\frac{1}{2}} (V^{+}), let k ≥ 2 be an integer, and let A_{𝔱} be the affine space of L_{k}^{2} spin connections on V which induce the connection 2A_{Λ} on det(V^{+}). If A is a spin connection on V then it defines an SO(3) connection Â on the subbundle 𝔤_{V} ⊂ 𝔰𝔲(V) [17, Lemma 2.5]; conversely, every SO(3) connection on 𝔤_{V} lifts to a unique spin connection on V inducing the connection 2A_{Λ} on det(V^{+}) [17, Lemma 2.11].

Let G_{𝔱} denote the group of L_{k+1}^{2} unitary automorphisms of V which commute with ℂℓ(T^{∗}X) and which have Clifford-determinant one (see [17, Definition 2.6]). Define

C̃_{𝔱}(X) = A_{𝔱}(X) × L_{k}^{2}(X, V^{+}) and C_{𝔱} = C̃_{𝔱}/G_{𝔱}. (2.7)

The action of G_{𝔱} on V induces an adjoint action on End_{ℂ}(V), acting as the identity on ρ(Λ_{ℂ}^{•}) ⊂ End_{ℂ}(V) and inducing an adjoint action on 𝔤_{V} ⊂ End_{ℂ}(V) (see [17, Lemma 2.7]). The space C̃_{𝔱} and hence C_{𝔱} carry circle actions induced by scalar multiplication on V:

S^{1} × V → V, (e^{iθ}, Φ) ↦ e^{iθ}Φ. (2.8)

Because this action commutes with that of G_{𝔱}, the action (2.8) also defines an action on C_{𝔱}. Note that −1 ∈ S^{1} acts trivially on C_{𝔱}. Let C_{𝔱}^{0} ⊂ C_{𝔱} be the subspace represented by pairs whose spinor components are not identically zero, let C_{𝔱}^{∗} ⊂ C_{𝔱} be the subspace represented by pairs where the induced SO(3) connections on 𝔤_{V} are irreducible, and let C_{𝔱}^{∗,0} = C_{𝔱}^{∗} ∩ C_{𝔱}^{0}.

We call a pair (A, Φ) in C̃_{𝔱} a SO(3) monopole if

\begin{aligned}ad^{−1}(F + ; Â) − τρ^{−1}(Φ ⊗ Φ^{∗})_{00} & = 0, \\ D_{A}Φ + ρ(ϑ) & = 0. \end{aligned} (2.9)

Here, D_{A} = ρ ∘ ∇_{A} : C^{∞}(X, V^{+}) → C^{∞}(X, V^{−}) is the Dirac operator; the isomorphism ad : 𝔤_{V} → 𝔰𝔬(𝔤_{V}) identifies the self-dual curvature F + ; Â, a section of Λ^{+} ⊗ 𝔰𝔬(𝔤_{V}), with ad^{−1}(F + ; Â), a section of Λ^{+} ⊗ 𝔤_{V}; the section τ of GL(Λ^{+}) is a perturbation close to the identity; the perturbation ϑ is a complex one-form close to zero; Φ^{∗} ∈ Hom(V^{+}, ℂ) is the pointwise Hermitian dual ⟨·, Φ⟩ of Φ; and (Φ ⊗ Φ^{∗})_{00} is the component of the section Φ ⊗ Φ^{∗} of i𝔲(V^{+}) lying in ρ(Λ^{+}) ⊗ 𝔤_{V} with respect to the splitting 𝔲(V^{+}) = i\underline{ℝ} ⊕ 𝔰𝔲(V^{+}) and decomposition (2.4) of 𝔰𝔲(V^{+}).

Equation (2.9) is invariant under the action of G_{𝔱}. We let M_{𝔱} ⊂ C_{𝔱} be the subspace represented by pairs satisfying equation (2.9) and write

M_{𝔱}^{∗} = M_{𝔱} ∩ C_{𝔱}^{∗}, M_{𝔱}^{0} = M_{𝔱} ∩ C_{𝔱}^{0}, and M_{𝔱}^{∗,0} = M_{𝔱} ∩ C_{𝔱}^{∗,0}. (2.10)

Since equation (2.9) is invariant under the circle action induced by scalar multiplication on V, the subspaces (2.10) of C_{𝔱} are also invariant under this action.

Theorem 2.1. [13, Theorem 1.1], [52] Let X be a closed, oriented, smooth four-manifold and let V be a complex rank-eight, Hermitian vector bundle over X. Then for parameters (ρ, g, τ, ϑ), which are generic in the sense of [13], and 𝔱 = (ρ, V), the space M_{𝔱}^{∗,0} is a smooth manifold of the expected dimension,

dim M_{𝔱}^{∗,0} = d_{a}(𝔱) + 2n_{a}(𝔱), where d_{a}(𝔱) = −2p_{1}(𝔱) − \frac{3}{2}(χ + σ),
n_{a}(𝔱) = \frac{1}{4}(p_{1}(𝔱) + c_{1}(𝔱)^{2} − σ).

(2.11)

For the remainder of the article, we assume that the perturbation parameters in (2.9) are chosen as indicated in Theorem 2.1. For each non-negative integer ℓ, let 𝔱_{ℓ} = (ρ, V_{ℓ}), where

c_{1}(V_{ℓ}) = c_{1}(V), p_{1}(𝔤_{V_{ℓ}}) = p_{1}(𝔤_{V}) + 4ℓ, and w_{2}(𝔤_{V_{ℓ}}) = w_{2}(𝔤_{V}).

We let M̄_{𝔱} denote the closure of M_{𝔱} in the space of ideal monopoles,

IM_{𝔱} = ⨆_{ℓ=0}^{∞}(M_{𝔱_{ℓ}} × Sym^{ℓ}(X)), (2.12)

with respect to an Uhlenbeck topology [16, Definition 4.19] and call the intersection of M̄_{𝔱} with M_{𝔱_{ℓ}} × Sym^{ℓ}(X) its ℓ-th level.

Theorem 2.2. [16, Theorem 1.1] Let X be a Riemannian four-manifold with spin^{u} structure 𝔱. Then there is a positive integer N, depending at most on the curvature of the chosen unitary connection on det(V^{+}) together with p_{1}(𝔱), such that the Uhlenbeck closure M̄_{𝔱} of M_{𝔱} in ⊔_{ℓ=0}^{N}(M_{𝔱_{ℓ}} × Sym^{ℓ}(X)) is a second countable, compact, Hausdorff space. The space M̄_{𝔱} carries a continuous circle action, which restricts to the circle action defined on M_{𝔱_{ℓ}} on each level.

2.2. Stratum of anti-self-dual or zero-section solutions. From equation (2.9), we see that the stratum of M_{𝔱} represented by pairs with zero spinor is identified with

{A ∈ A_{𝔱} : F + ; Â = 0}/G_{𝔱} ≅ M_{κ}^{w}(X, g),

the moduli space of g-anti-self-dual connections on the SO(3) bundle 𝔤_{V}, where κ = −\frac{1}{4}p_{1}(𝔱) and w ≡ w_{2}(𝔱) (mod 2). For a generic Riemannian metric g, the space M_{κ}^{w}(X, g) is a smooth manifold of the expected dimension −2p_{1}(𝔱) − \frac{3}{2}(χ + σ) = d_{a}(𝔱).

As explained in [17, §3.4.1], it is desirable to choose w (mod 2) so as to exclude points in M̄_{𝔱} with associated flat SO(3) connections, so we have a disjoint union,

M̄_{𝔱} ≅ M̄_{𝔱}^{∗,0} ⊔ M̄_{κ}^{w} ⊔ M̄_{𝔱}^{red}, (2.13)

where M̄_{𝔱}^{∗} ⊂ M̄_{𝔱} is the subspace represented by triples whose associated SO(3) connections are irreducible, M̄_{𝔱}^{0} ⊂ M̄_{𝔱} is the subspace represented by triples whose spinors are not identically zero, M̄_{𝔱}^{∗,0} = M̄_{𝔱}^{∗}∩M̄_{𝔱}^{0}, while M̄_{𝔱}^{red} ⊂ M̄_{𝔱} is the subspace M̄_{𝔱}−M̄_{𝔱}^{∗} represented by triples whose associated SO(3) connections are reducible. We recall the

Definition 2.3. [18, Definition 3.20] A class v ∈ H^{2}(X; ℤ/2) is good if no integral lift of v is torsion.

If w (mod 2) is good, then the union (2.13) is disjoint, as desired. In practice, rather than constraining w (mod 2) itself, we replace X by the blow-up X#\overline{ℂℙ}^{2} and w by w + PD[e] (where e ∈ H_{2}(X; ℤ) is the exceptional class), noting that w + PD[e] (mod 2) is always good, and define gauge-theoretic invariants of X in terms of moduli spaces on X#\overline{ℂℙ}^{2}. When w (mod 2) is good, we define [17, Definition 3.7] the link of M̄_{κ}^{w} in M̄_{𝔱}/S^{1} by

L_{𝔱,κ}^{w} = {[A, Φ, x] ∈ M̄_{𝔱}/S^{1} : ∥Φ∥_{L^{2}}^{2} = ε}, (2.14)

where ε is a small positive constant; for generic ε, the link L_{𝔱,κ}^{w} is a smoothly-stratified, codimension-one subspace of M̄_{𝔱}/S^{1}.

2.3. Strata of Seiberg-Witten or reducible solutions. We call a pair (A, Φ) ∈ C̃_{𝔱} reducible if the connection A on V respects a splitting,

V = W ⊕ W ⊗ L = W ⊗ (\underline{ℂ} ⊕ L), (2.15)

for some spin^{c} structure 𝔰 = (ρ, W) and complex line bundle L, in which case c_{1}(L) = c_{1}(𝔱) − c_{1}(𝔰). A spin connection A on V is reducible with respect to the splitting (2.15) if and only if Â is reducible with respect to the splitting 𝔤_{V} ≅ \underline{ℝ} ⊕ L, [17, Lemma 2.9]. If A is reducible, we can write A = B ⊕ B ⊗ A_{L}, where B is a spin connection on W and A_{L} is a unitary connection on L; then Â = d_{ℝ} ⊕ A_{L} and A_{L} = A_{Λ} ⊗ (B^{det})^{∗}, where B^{det} is the connection on det(W^{+}) induced by B on W.

2.3.1. Seiberg-Witten monopoles. Given a spin^{c} structure 𝔰 = (ρ, W) on X, let A_{𝔰} denote the affine space of L_{k}^{2} spin connections on W. Let G_{𝔰} denote the group of L_{k+1}^{2} unitary automorphisms of W, commuting with ℂℓ(T^{∗}X), which we identify with L_{k+1}^{2}(X, S^{1}). We then define

C̃_{𝔰} = A_{𝔰} × L_{k}^{2}(W^{+}) and C_{𝔰} = C̃_{𝔰}/G_{𝔰}, (2.16)

where G_{𝔰} acts on C̃_{𝔰} by

(s, (B, Ψ)) ↦ s(B, Ψ) = (B − (s^{−1}ds)id_{W} , sΨ). (2.17)

We call a pair (B, Ψ) ∈ C̃_{𝔰} a Seiberg-Witten monopole if

\begin{aligned}Tr(F_{B}^{+}) − τρ^{−1}(Ψ ⊗ Ψ^{∗})_{0} − F^{+}(A_{Λ}) & = 0, \\ D_{B}Ψ + ρ(ϑ)Ψ & = 0, \end{aligned} (2.18)

where Tr : 𝔲(W^{+}) → iℝ is defined by the trace on 2 × 2 complex matrices, (Ψ ⊗ Ψ^{∗})_{0} is the component of the section Ψ ⊗ Ψ^{∗} of i𝔲(W^{+}) contained in i𝔰𝔲(W^{+}), D_{B} : C^{∞}(W^{+}) → C^{∞}(W^{−}) is the Dirac operator, and A_{Λ} is a unitary connection on a line bundle with first Chern class Λ ∈ H^{2}(X; ℤ). The perturbations are chosen so that solutions to equation (2.18) are identified with reducible solutions to (2.9) when c_{1}(𝔱) = Λ. Let M̃_{𝔰} ⊂ C̃_{𝔰} be the subspace cut out by equation (2.18) and denote the moduli space of Seiberg-Witten monopoles by M_{𝔰} = M̃_{𝔰}/G_{𝔰}.

2.3.2. Seiberg-Witten invariants. We let C_{𝔰}^{0} ⊂ C_{𝔰} be the open subspace represented by pairs whose spinor components which are not identically zero and define a complex line bundle over C_{𝔰}^{0} × X by

𝕃_{𝔰} = C̃_{𝔰}^{0} ×_{G_{𝔰}} \underline{ℂ}, (2.19)

where \underline{ℂ} = X × ℂ and s ∈ G_{𝔰} acts on (B, Ψ) ∈ C̃_{𝔰} and (x, ζ) ∈ \underline{ℂ} by

((B, Ψ), (x, ζ)) ↦ (s(B, Ψ), (x, s(x)^{−1}ζ)). (2.20)

If x ∈ H_{0}(X; ℤ) denotes the positive generator, we set

μ_{𝔰} = c_{1}(𝕃_{𝔰})/x ∈ H^{2}(C_{𝔰}^{0}; ℤ). (2.21)

Equivalently, μ_{𝔰} is the first Chern class of the S^{1} base-point fibration over C_{𝔰}^{0}. If b_{1}(X) = 0, then c_{1}(𝕃_{𝔰}) = μ_{𝔰} × 1 by [17, Lemma 2.14].

For b_{2}^{+}(X) > 0 and generic Riemannian metrics on X, the space M_{𝔰} contains zero-section pairs if and only if c_{1}(𝔰) − Λ is a torsion class by [39, Proposition 6.3.1]. If M_{𝔰} contains no zero-section pairs then, for generic perturbations, it is a compact, oriented, smooth manifold of dimension

d_{s}(𝔰) = dim M_{𝔰} = \frac{1}{4}(c_{1}(𝔰)^{2} − 2χ − 3σ). (2.22)

Let X̃ = X#\overline{ℂℙ}^{2} denote the blow-up of X with exceptional class e ∈ H_{2}(X̃; ℤ) and denote its Poincaré dual by PD[e] ∈ H^{2}(X̃; ℤ). Let 𝔰^{±} = (ρ̃, W̃) denote the spin^{c} structure on X̃ with c_{1}(𝔰^{±}) = c_{1}(𝔰) ± PD[e] obtained by splicing the spin^{c} structure 𝔰 = (ρ, W) on X with the spin^{c} structure on \overline{ℂℙ}^{2} with first Chern class ± PD[e]. (See [18, §4.5] for an explanation of the relation between spin^{c} structures on X and X̃.) Now

c_{1}(𝔰) ± PD[e] − Λ ∈ H^{2}(X̃; ℤ)

is not a torsion class and so — for b_{2}^{+}(X) > 0, generic Riemannian metrics on X and related metrics on the connected sum X̃ — the moduli spaces M_{𝔰^{±}}(X̃) contain no zerosection pairs. Thus, for our choice of generic perturbations, the moduli spaces M_{𝔰^{±}}(X̃) are compact, oriented, smooth manifolds, both of dimension dim M_{𝔰}(X).

For b_{1}(X) = 0 and odd b_{2}^{+}(X) > 1, we define the Seiberg-Witten invariant by [18, §4.1]

SW_{X}(𝔰) = ⟨μ_{𝔰^{+}}^{d}, [M_{𝔰^{+}}(X̃)]⟩ = ⟨μ_{𝔰^{−}}^{d}, [M_{𝔰^{−}}(X̃)]⟩, (2.23)

where 2d = d_{s}(𝔰) = d_{s}(𝔰^{±}). When b_{2}^{+}(X) = 1 the pairing on the right-hand side of definition (2.23) depends on the chamber in the positive cone of H^{2}(X̃; ℝ) determined by the period point of the Riemannian metric on X̃. The definition of the Seiberg-Witten invariant for this case is also given in [18, §4.1]: we assume that the class w_{2}(X) − Λ (mod 2) is good to avoid technical difficulties involved in chamber specification. Since w ≡ w_{2}(X) − Λ (mod 2), this coincides with the constraint we use to define the Donaldson invariants in §2.5 when b_{2}^{+}(X) = 1. We refer to [18, Lemma 4.1 & Remark 4.2] for a comparison of the chamber structures required for the definition of Donaldson and Seiberg-Witten invariants when b_{2}^{+}(X) = 1.

2.3.3. Reducible SO(3) monopoles. If 𝔱 = (ρ, V) and 𝔰 = (ρ, W) with V = W ⊕ W ⊗ L, then there is an embedding

ι : C̃_{𝔰} ↪ C̃_{𝔱}, (B, Ψ) ↦ (B ⊕ B ⊗ A_{Λ} ⊗ B^{det,∗}, Ψ ⊕ 0), (2.24)

which is gauge-equivariant with respect to the homomorphism

ϱ : G_{𝔰} ↪ G_{𝔱}, s ↦ id_{W} ⊗ (\begin{matrix}s & 0 \\ 0 & s^{−1} \end{matrix}) . (2.25)

According to [17, Lemma 3.13], the map (2.24) defines a topological embedding ι : M_{𝔰}^{0} ↪ M_{𝔱}, where M_{𝔰}^{0} = M_{𝔰} ∩ C_{𝔰}^{0} and an embedding of M_{𝔰} if w_{2}(𝔱) ≠ 0 or b_{1}(X) = 0; its image in M_{𝔱} is represented by pairs which are reducible with respect to the splitting V = W ⊕W ⊗L. Henceforth, we shall not distinguish between M_{𝔰} and its image in M_{𝔱} under this embedding.

2.3.4. Circle actions. When V = W ⊕ W ⊗ L and 𝔱 = (ρ, V), the space C̃_{𝔱} inherits a circle action defined by

S^{1} × V → V, (e^{iθ}, Ψ ⊕ Ψ′) ↦ Ψ ⊕ e^{iθ}Ψ′, (2.26)

where Ψ ∈ C^{∞}(W) and Ψ′ ∈ C^{∞}(W ⊗ L). With respect to the splitting V = W ⊕ W ⊗ L, the actions (2.26) and (2.8) are related by

(\begin{matrix}1 & 0 \\ 0 & e^{i2θ} \end{matrix}) = e^{iθ}u, where u = (\begin{matrix}e^{−iθ} & 0 \\ 0 & e^{iθ} \end{matrix}) ∈ G_{𝔱}, (2.27)

and so, when we pass to the induced circle actions on the quotient C_{𝔱} = C̃_{𝔱}/G_{𝔱}, the actions (2.26) and (2.8) on C_{𝔱} differ only in their multiplicity. Recall [17, Lemma 3.11] that the image in C̃_{𝔱} of the map (2.24) contains all pairs which are fixed by the circle action (2.26).

2.3.5. The virtual normal bundle of the Seiberg-Witten moduli space. Suppose 𝔱 = (ρ, V) and 𝔰 = (ρ, W), with V = W ⊕ W ⊗ L, so we have a topological embedding M_{𝔰} ↪ M_{𝔱}; we assume M_{𝔰} contains no zero-section monopoles. Recall from [17, §3.5] that there exist finite-rank, complex vector bundles,

π_{Ξ} : Ξ_{𝔱,𝔰} → M_{𝔰} and π_{N} : N_{𝔱,𝔰} → M_{𝔰}, (2.28) with Ξ_{𝔱,𝔰} ≅ M_{𝔰} × ℂ^{r_{Ξ}}, called the obstruction bundles and virtual normal bundles of M_{𝔰} ↪ M_{𝔱}, respectively. For a small enough positive radius ε, there are a topological embedding [17, Theorem 3.21] of an open tubular neighborhood,

γ_{𝔰} : N_{𝔱,𝔰}(ε) ↪ C_{𝔱}, (2.29)

and a smooth section χ_{𝔰} of the pulled-back complex vector bundle,

π_{N}^{∗}Ξ_{𝔱,𝔰} → N_{𝔱,𝔰}(ε), (2.30)

such that the tubular map yields a homeomorphism

γ_{𝔰} : χ_{𝔰}^{−1}(0) ∩ N_{𝔱,𝔰}(ε) ≅ M_{𝔱} ∩ γ_{𝔰}(N_{𝔱,𝔰}(ε)), (2.31)

restricting to a diffeomorphism on the complement of M_{𝔰} and identifying M_{𝔰} with its image ι(M_{𝔰}) ⊂ M_{𝔱}. The image γ_{𝔰}(N_{𝔱,𝔰}(ε)) is an open subset of a virtual moduli space, M_{𝔱,𝔰}^{vir} ⊂ C_{𝔱}.

Our terminology is loosely motivated by that of [26] and [46], where the goal (translated to our setting) would be to construct a virtual fundamental class for M_{𝔱}, given by the cap product of the fundamental class of an ambient space containing M_{𝔱} with the Euler class of a vector bundle over this ambient space, where M_{𝔱} is the zero locus of a (possibly nontransversally vanishing) section. Here, M_{𝔱,𝔰}^{vir} plays the role of the ambient space and (the pushforward of) Ξ_{𝔱,𝔰} the vector bundle with zero section yielding (an open neighborhood in) M_{𝔱}. Then, N_{𝔱,𝔰} is the normal bundle of M_{𝔰} ↪ M_{𝔱,𝔰}^{vir}, while [N_{𝔱,𝔰}] − [Ξ_{𝔱,𝔰}] would more properly be called the ‘virtual normal bundle’ of M_{𝔰} ↪ M_{𝔱}, in the language of K-theory.

Recall that minus the index of the SO(3)-monopole elliptic deformation complex [17, Equations (2.47) & (3.35)] at a reducible solution can be written as

dim M_{𝔱} = 2n_{s}(𝔱, 𝔰) + d_{s}(𝔰), (2.32)

where d_{s}(𝔰) is the expected dimension of the Seiberg-Witten moduli space M_{𝔰} (see equation (2.22)), while n_{s}(𝔱, 𝔰) = n′_{s}(𝔱, 𝔰) + n′′_{s}(𝔱, 𝔰) is minus the complex index of the normal deformation operator [17, Equations (3.71) & (3.72)]), with

\begin{aligned}n′_{s}(𝔱, 𝔰) & = −(c_{1}(𝔱) − c_{1}(𝔰))^{2} − \frac{1}{2}(χ + σ), \\ n′′_{s}(𝔱, 𝔰) & = \frac{1}{8}(c_{1}(𝔰) − 2c_{1}(𝔱))^{2} − σ). \end{aligned} (2.33)

If r_{Ξ} is the complex rank of Ξ_{𝔱,𝔰} → M_{𝔰}, and r_{N}(𝔱, 𝔰) is the complex rank of N_{𝔱,𝔰} → M_{𝔰}, then

r_{N}(𝔱, 𝔰) = n_{s}(𝔱, 𝔰) + r_{Ξ}, (2.34)

as we can see from the dimension relation (2.32) and the topological model (2.31).

The map (2.29) is circle equivariant when the circle acts trivially on M_{𝔰}, by scalar multiplication on the fibers of N_{𝔱,𝔰}(ε), and by the action (2.26) on C_{𝔱}. The bundle (2.30) and section χ_{𝔰} are circle equivariant if the circle acts on N_{𝔱,𝔰} and the fibers of γ_{𝔰}^{∗}Ξ_{𝔱,𝔰} by scalar multiplication.

Let Ñ_{𝔱,𝔰} → M̃_{𝔰} be the pullback of N_{𝔱,𝔰} by the projection M̃_{𝔰} → M_{𝔰} = M̃_{𝔰}/G_{𝔰}, so Ñ_{𝔱,𝔰} is a G_{𝔰}-equivariant bundle, where G_{𝔰} acts on the base M̃_{𝔰} by the usual gauge group action (2.17) and the induced action on the total space,

Ñ_{𝔱,𝔰} ⊂ M̃_{𝔰} × L_{k}^{2}(Λ^{1} ⊗_{ℝ} L) ⊕ L^{2}(W^{+} ⊗ L) ⊂ M̃_{𝔰} × L_{k}^{2}(Λ^{1} ⊗_{ℝ} 𝔤_{V}) ⊕ L^{2}(V^{+}), (2.35)

via the embedding (2.25) of G_{𝔰} into G_{𝔱} and the splittings 𝔤_{V} ≅ \underline{ℝ} ⊕ L [17, Lemma 3.10] and V = W ⊕ W ⊗ L. Thus, s ∈ G_{𝔰} acts by scalar multiplication by s^{−2} on sections of Λ^{1} ⊗_{ℝ} L and by s^{−1} on sections of W^{+} ⊗ L [17, §3.5.4].

For a small enough positive ε, there is a smooth embedding [17, §3.5.4] of the open tubular neighborhood Ñ_{𝔱,𝔰}(ε),

γ̃_{𝔰} : Ñ_{𝔱,𝔰}(ε) → C̃_{𝔱}, (2.36)

which is gauge equivariant with respect to the preceding action of G_{𝔰}, and covers the topological embedding (2.29). The map (2.36) is circle equivariant, where the circle acts trivially on M̃_{𝔰}, by scalar multiplication on the fibers of Ñ_{𝔱,𝔰}(ε), and by the action (2.26) on C_{𝔱}. We note that the map (2.36) is also circle equivariant with respect to the action (2.8) on C_{𝔱}, if the circle acts on Ñ_{𝔱,𝔰} by

(e^{iθ}, (B, Ψ, β, ψ)) ↦ ϱ(e^{iθ})(B, Ψ, e^{2iθ}β, e^{2iθ}ψ) = (B, e^{iθ}Ψ, β, e^{iθ}ψ), (2.37)

where (B, Ψ) ∈ M̃_{𝔰}, (β, ψ) ∈ L_{k}^{2}(Λ^{1} ⊗_{ℝ} L) ⊕ L^{2}(W^{+} ⊗ L), so (B, Ψ, β, ψ) ∈ Ñ_{𝔱,𝔰}, and ϱ : G_{𝔰} → G_{𝔱} is the homomorphism (2.25). This equivariance follows from the relation (2.27) between the actions (2.26) and (2.8).

We recall the following calculation of the Segre classes of the virtual normal bundle N_{𝔱,𝔰}:

Lemma 2.4. [18, Lemma 4.11] Assume M_{𝔰} ⊂ M_{𝔱} contains no zero-section pairs and let μ_{𝔰} ∈ H^{2}(M_{𝔰}; ℤ) be the cohomology class (2.21). If b_{1}(X) = 0, then the Segre classes of the complex vector bundle N_{𝔱,𝔰} → M_{𝔰} are given by

s_{p}(N_{𝔱,𝔰}) = μ_{𝔰}^{p} ∑_{j=0}^{p} 2^{j}\binom{−n′_{s}}{j}\binom{−n′′_{s}}{p − j}, p = 0, 1, 2, … (2.38)

2.4. Cohomology classes on the moduli space of SO(3) monopoles. The identity (1.2) arises as an equality of pairings of suitable cohomology classes with a link in M̄_{𝔱}^{∗,0}/S^{1} of the anti-self-dual moduli subspace and the links of the Seiberg-Witten moduli subspaces in M̄_{𝔱}/S^{1}. We recall the definitions of these cohomology classes and their dual geometric representatives given in [18, §3].

The first kind of cohomology class is defined on M_{𝔱}^{∗}/S^{1}, via the associated SO(3) bundle,

𝔽_{𝔱} = C̃_{𝔱}^{∗}/S^{1} ×_{G_{𝔱}} 𝔤_{V} → C_{𝔱}^{∗}/S^{1} × X. (2.39)

The group G_{𝔱} acts diagonally in (2.39), with G_{𝔱} acting on the left on 𝔤_{V}. We define [18, §3.1]

μ_{p} : H_{•}(X; ℝ) → H^{4−•}(C_{𝔱}^{∗}/S^{1}; ℝ), β ↦ −\frac{1}{4}p_{1}(𝔽_{𝔱})/β. (2.40)

On restriction to M_{κ}^{w} ↪ M_{𝔱}, the cohomology classes μ_{p}(β) coincide with those used in the definition of Donaldson invariants [18, Lemma 3.1]. We assume b_{1}(X) = 0 and thus let

𝔸(X) = Sym (H_{even}(X; ℝ)) (2.41)

be the graded algebra, with z = β_{1}β_{2} ⋯ β_{r} having total degree deg(z) = ∑_{p}(4 − i_{p}), when β_{p} ∈ H_{i_{p}}(X; ℝ). A point x ∈ X gives a distinguished generator still called x in 𝔸(X) of degree four. Then μ_{p} extends in the usual way to a homomorphism of graded real algebras,

μ_{p} : 𝔸(X) → Sym(H^{even}(C_{𝔱}^{∗}/S^{1}; ℝ)),

which preserves degrees. Next, we define a complex line bundle over C_{𝔱}^{∗,0}/S^{1},

𝕃_{𝔱} = C_{𝔱}^{∗,0} ×_{(S^{1},×−2)} ℂ, (2.42)

where the circle action is given, for [A, Φ] ∈ C_{𝔱}^{∗,0} and ζ ∈ ℂ, by

([A, Φ], ζ) ↦ ([A, e^{iθ}Φ], e^{2iθ}ζ). (2.43)

Then we define the second kind of cohomology class on M_{𝔱}^{∗,0}/S^{1} by

μ_{c} = c_{1}(𝕃_{𝔱}) ∈ H^{2}(C_{𝔱}^{∗,0}/S^{1}; ℝ). (2.44)

For monomials z ∈ 𝔸(X), we constructed [18, §3.2] geometric representatives V(z) dual to μ_{p}(z) and W dual to μ_{c}, defined on M_{𝔱}^{∗}/S^{1} and M_{𝔱}^{∗,0}/S^{1}, respectively; their closures in M̄_{𝔱}/S^{1} are denoted by V̄(z) and W̄ [18, Definition 3.14]. When

deg(z) + 2η = dim(M_{𝔱}^{∗,0}/S^{1}) − 1,

and deg(z) ≥ dim M_{κ}^{w} it follows that [18, §3.3] the intersection

V̄(z) ∩ W̄^{η} ∩ M̄_{𝔱}^{∗,0}/S^{1}, (2.45)

is an oriented one-manifold (not necessarily connected) whose closure in M̄_{𝔱}/S^{1} can only intersect (M̄_{𝔱} − M_{𝔱})/S^{1} at points in M_{𝔱}^{red} ≅ ∪(M_{𝔰} × Sym^{ℓ}(X)), where the union is over ℓ ≥ 0 and 𝔰 ∈ Spin^{c}(X) [18, Corollary 3.18].

2.5. Donaldson invariants. We first recall the definition [31, §2] of the Donaldson series when b_{1}(X) = 0 and b_{2}^{+}(X) > 1 is odd, so that, in this case, χ + σ ≡ 0 (mod 4). See also §3.4.2 in [18], especially for a definition of the Donaldson invariants when b_{2}^{+}(X) = 1. For any choice of w ∈ H^{2}(X; ℤ), the Donaldson invariant is a linear function

D_{X}^{w} : 𝔸(X) → ℝ.

Let X̃ = X#\overline{ℂℙ}^{2} be the blow-up of X and let e ∈ H_{2}(X̃; ℤ) be the exceptional class, with Poincaré dual PD[e] ∈ H^{2}(X̃; ℤ). If z ∈ 𝔸(X) is a monomial, we define D_{X}^{w}(z) = 0 unless

deg(z) ≡ −2w^{2} − \frac{3}{2}(χ + σ) (mod 8). (2.46)

If deg(z) obeys equation (2.46), we let κ ∈ \frac{1}{4}ℤ be defined by

deg(z) = 8κ − \frac{3}{2} (χ + σ) .

There exists an SO(3) bundle over X̃ with first Pontrjagin number −4κ − 1 and second Stiefel-Whitney class w + PD[e] (mod 2). One then defines the Donaldson invariant on monomials by

D_{X}^{w}(z) = # (V̄(ze) ∩ M̄_{κ+1/4}^{w+PD[e]}(X̃)) , (2.47)

and extends to a real linear function on 𝔸(X). Note that w + PD[e] (mod 2) is good in the sense of Definition 2.3. If w′ ≡ w (mod 2), then [11]

D_{X}^{w′} = (−1)^{\frac{1}{4} (w′−w)^{2}}D_{X}^{w}. (2.48)

The Donaldson series is a formal power series,

D_{X}^{w}(h) = D_{X}^{w}((1 + \frac{1}{2}x)e^{h}), h ∈ H_{2}(X; ℝ). (2.49)

By equation (2.46), the series D_{X}^{w} is even if

−w^{2} − \frac{3}{4}(χ + σ) ≡ 0 (mod 2),

and odd otherwise. A four-manifold has KM-simple type if for some w and all z ∈ 𝔸(X),

D_{X}^{w}(x^{2}z) = 4D_{X}^{w}(z).

According to [31, Theorem 1.7], when X has KM-simple type the series D_{X}^{w}(h) is an analytic function of h and there are finitely many characteristic cohomology classes K_{1}, …, K_{m} (the KM-basic classes) and constants a_{1}, …, a_{m} (independent of w) so that

D_{X}^{w}(h) = e^{\frac{1}{2} h·h} ∑_{i=1}^{r}(−1)^{\frac{1}{2} (w^{2}+K_{i}·w)}a_{i}e^{⟨K_{i},h⟩}.

Witten’s conjecture (1.3) then relates the Donaldson and Seiberg-Witten series for fourmanifolds of simple type.

When b_{2}^{+}(X) = 1 the pairing on the right-hand side of definition (2.47) depends on the chamber in the positive cone of H^{2}(X̃; ℝ) determined by the period point of the Riemannian metric on X̃, just as in the case of Seiberg-Witten invariants described in §2.3.2. We refer to §3.4.2 in [18] for a detailed discussion of this case and, as in [18], we assume that the class w (mod 2) is good in order to avoid technical difficulties involved in chamber specification.

2.6. Links and the cobordism. Since the endpoints of the components of the onemanifold (2.45) either lie in M_{κ}^{w} or M_{𝔰} × Sym^{ℓ}(X), for some 𝔰 and ℓ(𝔱, 𝔰) ≥ 0 for which 𝔱_{ℓ} = 𝔰 ⊕ 𝔰 ⊗ L, we have when w_{2}(𝔱) is good in the sense of Definition 2.3 that

# (V̄(z) ∩ W̄^{n_{a}−1} ∩ L̄_{𝔱,κ}^{w}) = − ∑_{𝔰∈Spin^{c}(X)}# (V̄(z) ∩ W̄^{n_{a}−1} ∩ L̄_{𝔱,𝔰}) , (2.50)

where L̄_{𝔱,κ}^{w} is the link of M̄_{κ}^{w} in M̄_{𝔱}/S^{1} (see [17, Definition 3.7]) and where L̄_{𝔱,𝔰} is empty if ℓ(𝔱, 𝔰) < 0 and is the boundary of an open neighborhood of the Seiberg-Witten stratum M_{𝔰}×Sym^{ℓ}(X) in M̄_{𝔱}/S^{1} if ℓ = ℓ(𝔱, 𝔰) ≥ 0. By construction, the intersection of L̄_{𝔱,𝔰} with the top stratum of M̄_{𝔱}/S^{1} is a smooth manifold and the intersection of L̄_{𝔱,𝔰} with the geometric representatives is in this top stratum. The precise definition of L̄_{𝔱,𝔰} is given in [17, Definition 3.22] for ℓ = 0, in (3.79) for ℓ = 1, and will be given in [21] for ℓ > 1.

When deg(z) = dim M_{κ}^{w}, the intersection of the one-manifold (2.45) with the link L̄_{𝔱,κ}^{w} is given by [18, Lemma 3.30]

2^{1−n_{a}}# (V̄(z) ∩ W̄^{n_{a}−1} ∩ L̄_{𝔱,κ}^{w}) = # (V̄(z) ∩ M̄_{κ}^{w}) . (2.51)

Applying this identity to the blow-up, X#\overline{ℂℙ}^{2}, when n_{a}(𝔱) > 0, we recover the Donaldson invariant D_{X}^{w}(z) on the right-hand side of (2.51) via definition (2.47).

Our proof of Theorem 1.1 relies on the following conjecture whose motivation is discussed in [14, p. 179]:

Conjecture 2.5. [14, Conjecture 3.1] When b_{1}(X) = 0 and η ≥ 0 is an integer for which deg(z) + 2(η + 1) = dim M_{𝔱}, the intersection number

# (V̄(z) ∩ W̄^{η} ∩ L̄_{𝔱,𝔰})

is a multiple of SW_{X}(𝔰) and thus vanishes if SW_{X}(𝔰) = 0.

We need only assume this conjecture holds for ℓ(𝔱, 𝔰) > 1, as we shall prove it here when ℓ(𝔱, 𝔰) = 1; the relevant statement for b_{1}(X) ≥ 0 is given as Conjecture 3.34 in [18], which we proved when ℓ(𝔱, 𝔰) = 0 [18, Remark 4.15]. The identities (2.51) and (2.50) yield

2^{n_{a}−1}# (V̄(z) ∩ M̄_{κ}^{w}) = − ∑_{𝔰∈Spin^{c}(X)}# (V̄(z) ∩ W̄^{n_{a}−1} ∩ L̄_{𝔱,𝔰}) , (2.52)

where, assuming Conjecture 2.5, the intersection numbers on the right-hand side are nonzero only when SW_{X}(𝔰) ≠ 0 (and ℓ(𝔱, 𝔰) ≥ 0). The hypotheses of Theorem 1.1 ensure that we only have ℓ(𝔱, 𝔰) = 0, 1 in (2.52) when SW_{X}(𝔰) ≠ 0 and, since we addressed the case ℓ = 0 in [18], the remainder of the present article concerns the case ℓ = 1.

3. Gluing SO(3) monopoles

Our goal in this section is to construct a topological model for an open neighborhood in M̄_{𝔱}, and thus a link, of the Seiberg-Witten stratum

(M_{𝔰} × Sym^{ℓ}(X)) ∩ M̄_{𝔱} ⊂ M̄_{𝔱},

where M_{𝔰} ⊂ M_{𝔱_{ℓ}} and p_{1}(𝔱_{ℓ}) = p_{1}(𝔱) + 4ℓ with ℓ ≥ 0; in this article, we shall only carry out this construction when ℓ = 1, whereas in [21] we consider the general case ℓ ≥ 1. For the remainder of this article, however, it will be more convenient to denote the pair (𝔱_{ℓ}, 𝔱) by (𝔱, 𝔱′) in order to minimize the number of subscripts and simplify notation. Hence, for the remainder of the article, we shall denote the Seiberg-Witten stratum of interest by

(M_{𝔰} × Sym^{ℓ}(X)) ∩ M̄_{𝔱′} ⊂ M̄_{𝔱′}, (3.1)

where M_{𝔰} ⊂ M_{𝔱} and p_{1}(𝔱) = p_{1}(𝔱′) + 4ℓ with 𝔱 = (ρ, V), 𝔱′ = (ρ′, V′), and ℓ ≥ 0.

The topological models for open neighborhoods in M̄_{𝔱′} of the strata (3.1) are constructed by gluing SO(3) monopoles. In §3.4 we describe the cut-and-paste process of splicing instantons from the four-sphere (corresponding to SO(3) monopoles with zero-spinor) onto a family of SO(3) monopoles over X parametrized by a tubular neighborhood N_{𝔱,𝔰}(ε) of M_{𝔰}. The supporting technical details required to define this splicing process are discussed in §3.1, where we review the required Clifford module and spin group representation theory, in §3.2 where we describe the Clifford frame bundles and their structure groups, and in §3.3 where we describe the connections and spinors on the four-sphere to be spliced onto the four-manifold X. The space of gluing data and splicing map γ′ are defined in §3.5, while in §3.7 we recall a gluing result (Theorem 3.8) from [19, 20] which asserts that the splicing map can be perturbed to a gluing map, thus giving a topological model for an open neighborhood of the stratum (3.1) in M̄_{𝔱′} when ℓ = 1; the extension to the general case ℓ ≥ 1 is discussed in [21]. The obstruction bundle appearing in this model is described in §3.6. In particular, the model allows us to define in §3.8 a link L̄_{𝔱′,𝔰} of the stratum M_{𝔰} × X in M̄_{𝔱′}. The relationships among the orientations of the link and the associated moduli spaces are described in §3.9.

3.1. Clifford modules and representation theory. We recall the representation theory we shall need for the complex Clifford algebra ℂℓ(ℝ^{4}). (In Lawson-Michelsohn [32], this algebra is denoted ℂℓ_{4} = Cℓ_{4} ⊗_{ℝ}ℂ, with Cℓ_{4} = Cℓ_{4,0}.) Recall from [32, Theorem I.4.3] that ℂℓ(ℝ^{4}) ≅ M_{4}(ℂ), where M_{d}(ℂ) denotes the algebra of complex d × d matrices. Then the natural representation of M_{4}(ℂ) on Δ = ℂ^{4} is the only irreducible complex representation of M_{4}(ℂ) [32, Theorems I.5.6–7], up to equivalence in the sense of [32, Definition I.5.1]. In particular, Δ is the only irreducible complex representation of ℂℓ(ℝ^{4}) [32, Theorems I.5.8].

If 𝕍 is a complex vector space of dimension 4d and ℂℓ(ℝ^{4})-module, then 𝕍 is isomorphic to Δ ⊕ ⋯ ⊕ Δ (d times), or Δ ⊗ ℂ^{d}, as a ℂℓ(ℝ^{4})-module by [32, Proposition I.5.4]. From the remarks surrounding [17, Equation (2.28)] for the case d = 2, the map

M_{d}(ℂ) → End_{ℂℓ(ℝ^{4})}(Δ ⊗ ℂ^{d}), M ↦ id_{Δ} ⊗ M, (3.2)

is an isomorphism of complex algebras.

Recall that a representation ρ : ℂℓ(ℝ^{4}) → End_{ℂ}(𝕍) is unitary, in the sense implied by [32, Theorem I.5.16], when 𝕍 is equipped with a Hermitian metric such that

⟨ρ(α)Φ, ρ(α)Ψ⟩ = ⟨Φ, Ψ⟩,

for all α ∈ ℝ^{4} with |α| = 1 and Φ, Ψ ∈ 𝕍.

Lemma 3.1. Let 𝕍 be a complex vector space of dimension 4d and ℂℓ(ℝ^{4})-module. If F_{1}, F_{2} : Δ ⊗ ℂ^{d} → 𝕍 are isomorphisms of ℂℓ(ℝ^{4})-modules, then

F_{1} = F_{2} ∘ (id_{Δ} ⊗ M),

where M ∈ GL(d, ℂ); if F_{1}, F_{2} respect Hermitian metrics, then M ∈ U(d).

Proof. Given ℂℓ(ℝ^{4})-module isomorphisms F_{1}, F_{2}, then F_{2}^{−1}F_{1} is a ℂℓ(ℝ^{4})-automorphism of the ℂℓ(ℝ^{4})-module Δ ⊗ ℂ^{d} and so F_{2}^{−1}F_{1} = id_{Δ} ⊗ M ∈ GL(d, ℂ) by equation (3.2). If F_{1}, F_{2} respect Hermitian metrics, then M^{†}M = id_{ℂ^{d}} and so M ∈ U(d).

If 𝕍 is a complex rank-eight vector space and complex ℂℓ(ℝ^{4})-module, then 𝕍 = 𝕍^{+}⊕𝕍^{−} and we have an associated complex determinant line det(𝕍^{+}) = Λ^{4}(𝕍^{+}). Given a complex rank-four vector space and ℂℓ(ℝ^{4})-module 𝕎, there is an isomorphism F : 𝕎 → Δ of ℂℓ(ℝ^{4})-modules (determined up to multiplication by some e^{iθ} ∈ S^{1}). Setting 𝔼 = Hom_{ℂℓ(ℝ^{4})}(𝕎, 𝕍) (now determined up to multiplication by e^{−iθ} ∈ S^{1}), we obtain an isomorphism of ℂℓ(ℝ^{4})-modules,

𝕎 ⊗_{ℂ} 𝔼 → 𝕍,

induced by the map Φ ⊗ H ↦ H(Φ), where Φ ∈ 𝕎 and H ∈ E. We then obtain a square root of det(𝕍^{+}),

(3.3) det^{\frac{1}{2}} (𝕍^{+}) = det(𝕎^{+}) ⊗_{ℂ} det(𝔼) = Λ^{2}(𝕎^{+}) ⊗_{ℂ} Λ^{2}(𝔼),

which is independent of the choice of 𝕎.

3.2. Structure groups and associated bundles. Given a Hermitian Clifford module (ρ, V) over (X, g), where V has complex rank 4d, the Hermitian subbundles V^{±} ⊂ V are defined as the ∓1 eigenspaces of ρ(vol) on V. The determinant bundles det(V^{±}) are canonically isomorphic via an isomorphism det(ρ(α)) : det(V^{+}) → det(V^{−}), α ∈ C^{∞}(U, T^{∗}X) with |α| = 1, which is independent of the choice of local one-forms [48, §5.2].

We let Fr(T^{∗}X) → X be the principal SO(4) bundle with fiber Isom_{ℝ}^{+}(ℝ^{4}, T^{∗}X|_{x}) over x ∈ X given by the space of orientation-preserving isometries, with SO(4) action induced by the standard action on ℝ^{4}, and analogously define Fr(T X).

Next, we define the frame bundles for the Clifford modules. Given Δ ⊗_{ℂ} ℂ^{d} with its ℂℓ(ℝ^{4})-module structure induced by the Clifford map ρ_{0} : ℝ^{4} → End_{ℂ}(Δ ⊗_{ℂ} ℂ^{d}), and V_{x} with its ℂℓ(T^{∗}X|_{x})-module structure induced by the Clifford map ρ : T^{∗}X|_{x} → End_{ℂ}(V_{x}), for each x ∈ X we let Isom_{ℂℓ}(Δ ⊗_{ℂ} ℂ^{d}, V_{x}) be the space of unitary isomorphisms F̃ : Δ ⊗ ℂ^{d} → V_{x} which are Clifford-module isomorphisms with respect to a Clifford-algebra isomorphism F : ℂℓ(ℝ^{4}) → ℂℓ(T^{∗}X|_{x}) determined by a frame F : ℝ^{4} → T^{∗}X|_{x}. Thus each F̃ ∈ Isom_{ℂℓ}(Δ ⊗_{ℂ} ℂ^{d}, V_{x}) covers a frame F ∈ Isom_{ℝ}^{+}(ℝ^{4}, T^{∗}X|_{x}) and corresponding Clifford-algebra isomorphism yielding a commutative diagram, for all α ∈ ℂℓ(ℝ^{4}),

Δ ⊗_{ℂ} ℂ^{d} ⟶^{F̃} V_{x}
ρ_{0}(α)↓  ↓^{ρ(F α)}
Δ ⊗_{ℂ} ℂ^{d} ⟶^{F̃} V_{x}

Hence, we define the Clifford-module frame bundle for V,

π : Fr_{ℂℓ(T^{∗}X)}(V) → Fr(T^{∗}X), (3.4) to have fibers Isom_{ℂℓ}(Δ ⊗_{ℂ} ℂ^{d}, V_{x}), where F̃ ∈ Isom_{ℂℓ}(Δ ⊗_{ℂ} ℂ^{d}, V_{x}) and π(F̃) = F obey

F̃ρ_{0}(α)F̃^{−1} = ρ(F α), α ∈ ℂℓ(ℝ^{4}). (3.5)

If F̃_{1}, F̃_{2} ∈ Fr_{ℂℓ(T^{∗}X)}(V_{x}) both lie in the fiber of the projection (3.4) over F ∈ Fr(T^{∗}X|_{x}), then F̃_{2}^{−1}F̃_{1} is a Hermitian ℂℓ(ℝ^{4})-module automorphism of Δ ⊗_{ℂ} ℂ^{d} and hence F̃_{2}^{−1}F̃_{1} = id_{Δ} ⊗ M, for some M ∈ U(d) by Lemma 3.1. Therefore, we have

Lemma 3.2. The space Fr_{ℂℓ(T^{∗}X)}(V) is a principal U(d) bundle over Fr(T^{∗}X).

We now consider the case d = 1 and let (ρ, W) be a spin^{c} structure over X.

Lemma 3.3. The space Fr_{ℂℓ(T^{∗}X)}(W) is a principal Spin^{c}(4) bundle over X.

Proof. If F̃_{1}, F̃_{2} ∈ Fr_{ℂℓ(T^{∗}X)}(W_{x}) lie in the fibers over F_{1}, F_{2} ∈ Fr(T^{∗}X|_{x}) of the projection (3.4), respectively, then F_{2}^{−1}F_{1} = R ∈ SO(4) is an isometric automorphism of ℝ^{4} while F̃_{2}^{−1}F̃_{1} = R̃ ∈ U(4) is a Hermitian automorphism of Δ, yielding a commutative diagram

Δ ⟶^{R̃} Δ
ρ_{0}(α)↓  ↓^{ρ_{0}(Rα)}
Δ ⟶^{R̃} Δ

Thus, just as in (3.5), these automorphisms obey

R̃ρ_{0}(α)R̃^{−1} = ρ_{0}(Rα), α ∈ ℂℓ(ℝ^{4}). (3.6)

Because R has determinant one, it preserves the volume form. Thus identity (3.6) implies that R̃ preserves the splitting Δ = Δ^{+} ⊕ Δ^{−} and hence, we may write

R̃ = (R̃_{+}, R̃_{−}) ∈ U(2) × U(2) ⊂ U(4).

Identity (3.6) then implies that R̃_{−}ρ_{0}(α)R̃_{+}^{−1} = ρ_{0}(Rα) on Δ^{+}, for α ∈ ℝ^{4}, and since the isomorphism det(ρ_{0}(α)) : Λ^{2}(Δ^{+}) ≅ Λ^{2}(Δ^{−}) of complex lines is independent of the choice of unit-norm α ∈ ℝ^{4}, we must have

det(R̃_{+}) = det(R̃_{−}).

Recall that Spin(4) = SU(2) × SU(2). Thus,

R̃ ∈ {(M, N) ∈ U(2) × U(2) : det(M) = det(N)}
= (SU(2) × SU(2) × S^{1})/{±1}
= (Spin(4) × S^{1})/{±1}
= Spin^{c}(4).

(3.7)

Therefore, Fr_{ℂℓ(T^{∗}X)}(W) → X is a principal Spin^{c}(4) bundle, as claimed.

For A ∈ Spin^{c}(4), Ad(A) denotes an element of SO(End(Δ)). However, we note that Ad(A) preserves ρ(ℝ^{4}) ⊂ End(Δ) and thus defines a linear transformation of ℝ^{4} which is an element of SO(4). This defines a homomorphism

Ad^{c} : Spin^{c}(4) → SO(4).

By [32, Corollary I.5.19] the homomorphism Ad^{c} is equal to the projection Spin^{c}(4) → SO(4) implicit in the description of Spin^{c}(4) in (3.7), recalling that Spin(4) = (SU(2) × SU(2))/{±1}. The S^{1} factor in the presentation (3.7) of Spin^{c}(4) is the kernel of the homomorphism Ad^{c}. Since Fr_{ℂℓ(T^{∗}X)}(W)/S^{1} ≅ Fr(T^{∗}X) by Lemma 3.2, we have

(3.8) Fr_{ℂℓ(T^{∗}X)}(W) ×_{(Spin^{c}(4),Ad^{c})} ℝ^{4} ≅ T^{∗}X.

From the description of Spin^{c}(4) in (3.7), we see that there is a homomorphism,

det^{c} : Spin^{c}(4) → S^{1}, (M, N) ↦ det(M) = det(N),

and thus,

(3.9) Fr_{ℂℓ(T^{∗}X)}(W) ×_{(Spin^{c}(4),det^{c})} ℂ ≅ det(W^{+}).

This completes our discussion of the Clifford frame bundle for W and its associated vector bundles.

We now determine the structure group of the bundle Fr_{ℂℓ(T^{∗}X)}(V) over X, where (ρ, V) is a complex-rank eight Hermitian Clifford module over X. Suppose F̃_{1}, F̃_{2} ∈ Fr_{ℂℓ(T^{∗}X)}(V_{x}) lie in the fibers over F_{1}, F_{2} ∈ Fr(T^{∗}X|_{x}) of the projection (3.4), so that F_{2}^{−1}F_{1} = R = Ad^{c}(R̃) ∈ SO(4), for some R̃ ∈ Spin^{c}(4). Therefore, according to (3.5), we have

F̃_{2}ρ_{0}(α)F̃_{2}^{−1} = ρ(F_{2}α), α ∈ ℂℓ(ℝ^{4}),

and hence, replacing α by Rα and applying identity (3.6), we see that

F̃_{2}R̃ρ_{0}(α)(F̃_{2}R̃)^{−1} = F̃_{2}ρ_{0}(Rα)F̃_{2}^{−1}
= ρ(F_{2}Rα)
= ρ(F_{1}α) α ∈ ℂℓ(ℝ^{4}),

so that F̃_{2}R̃ ∈ Fr_{ℂℓ(T^{∗}X)}(V_{x}) also lies in the fiber of the projection (3.4) over F_{1} ∈ Fr(T^{∗}X|_{x}). Thus, (F̃_{2}R̃)^{−1}F̃_{1} = R̃^{−1}F̃_{2}^{−1}F̃_{1} is a Hermitian ℂℓ(ℝ^{4})-module automorphism of Δ ⊗_{ℂ} ℂ^{2} and hence, by Lemma 3.1, we have

(R̃^{−1} ⊗ id_{ℂ^{2}}) ∘ F̃_{2}^{−1}F̃_{1} = id_{Δ} ⊗ M,

for some M ∈ U(2). Therefore,

F̃_{1} = F̃_{2} ∘ (R̃ ⊗ M).

Hence, any two elements of the fiber of Fr_{ℂℓ(T^{∗}X)}(V) over x ∈ X differ by an element of

Spin^{u}(4) = (Spin^{c}(4) × U(2))/S^{1}
= Spin(4) ×_{{±1}} U(2),
= (SU(2) × SU(2)) ×_{{±1}} (SU(2) ×_{{±1}} S^{1}),

(3.10)

noting that U(2) = (SU(2) × S^{1})/{±1}. Therefore, we have

Lemma 3.4. The space Fr_{ℂℓ(T^{∗}X)}(V) is a principal Spin^{u}(4) bundle over X.

The composition of projection onto the factor S^{1} of Spin^{u}(4) in the presentation (3.10) and squaring defines a homomorphism

det^{u} : Spin^{u}(4) → S^{1}.

Equivalently, writing elements of Spin^{u}(4) as R̃ ⊗ M, with R̃ ∈ Spin^{c}(4) and M ∈ U(2) using the first isomorphism in (3.10), this homomorphism is given by

(3.11) det^{u} : Spin^{u}(4) → S^{1}, R̃ ⊗ M ↦ det^{c}(R̃) det(M).

Definition (3.11), the identification (3.9) of the line bundle associated to det^{c}, and the definition (3.3) of the line bundle det^{\frac{1}{2}} (V^{+}) imply that

(3.12) Fr_{ℂℓ(T^{∗}X)}(V) ×_{(Spin^{u}(4),det^{u})} ℂ ≅ det^{\frac{1}{2}} (V^{+}).

The homomorphisms

Ad_{SO(4)}^{u} : Spin^{u}(4) → SO(4), R̃ ⊗ M ↦ Ad^{c}(R̃),
Ad_{SO(3)}^{u} : Spin^{u}(4) → SO(3), R̃ ⊗ M ↦ Ad(M),

(3.13)

induce bundle isomorphisms

Fr_{ℂℓ(T^{∗}X)}(V) ×_{(Spin^{u}(4),Ad_{SO(4)}^{u})} ℝ^{4} ≅ T^{∗}X,
Fr_{ℂℓ(T^{∗}X)}(V) ×_{(Spin^{u}(4),Ad_{SO(3)}^{u})} 𝔰𝔲(2) ≅ 𝔤_{V} . (3.14)

The first isomorphism in (3.14) follows from the argument giving the isomorphism (3.8). To prove the second isomorphism in (3.14), observe that one has a bundle isomorphism

(3.15) Fr_{ℂℓ(T^{∗}X)}(V) ×_{(Spin^{u}(4),Ad)} 𝔰𝔲(Δ ⊗_{ℂ} ℂ^{2}) ≅ 𝔰𝔲(V),

defined via the adjoint representation,

Ad : Spin^{u}(4) → SO(𝔰𝔲(Δ ⊗_{ℂ} ℂ^{2})), U ↦ U( ·)U^{†}.

Recall from [17, Equation (2.18)] that the SO(3) subbundle 𝔤_{V} ⊂ 𝔰𝔲(V) is characterized as the span of the sections which commute with the sections ρ(ω) of End_{ℂ}(V) for all ω ∈ ℂℓ(T^{∗}X). In the decomposition,

𝔰𝔲(Δ ⊗_{ℂ} ℂ^{2}) ≅ 𝔰𝔲(Δ) ⊕ 𝔰𝔲(Δ) ⊗ 𝔰𝔲(2) ⊕ 𝔰𝔲(2), (3.16)

one has a similar characterization of the subspace

𝔰𝔲(2) ≅ id_{Δ} ⊗ 𝔰𝔲(2) ⊂ 𝔰𝔲(Δ ⊗_{ℂ} ℂ^{2}). (3.17)

Because any frame F̃ ∈ Fr_{Cℓ(T^{∗}X)}(V_{x}) defines an isomorphism of Hermitian Clifford modules, Δ ⊗_{ℂ} ℂ^{2} ≅ V_{x}, the induced isomorphism 𝔰𝔲(Δ ⊗_{ℂ} ℂ^{2}) ≅ 𝔰𝔲(V_{x}) preserves the splitting (3.16) and restricts to an isomorphism 𝔰𝔲(2) ≅ 𝔤_{V} |_{x}. Hence, the isomorphism (3.15) gives an isomorphism of subbundles,

(3.18) Fr_{ℂℓ(T^{∗}X)}(V) ×_{(Spin^{u}(4),Ad)} 𝔰𝔲(2) ≅ 𝔤_{V} .

For R̃ ∈ Spin^{c}(4) and M ∈ U(2), one can write

Ad(R̃ ⊗ M) = Ad^{c}(R̃) ⊕ (Ad^{c}(R̃) ⊗ Ad(M)) ⊕ Ad(M)

with respect to the decomposition (3.16), the restrictions of Ad(R̃⊗M) and Ad_{SO(3)}^{u}(R̃⊗M) to the subspace 𝔰𝔲(2) in (3.17) agree, so the isomorphism (3.18) yields the second bundle isomorphism in (3.14).

3.3. Anti-self-dual and spin connections over the four-sphere. Assume S^{4} ⊂ ℝ^{5} has the standard embedding, with round metric of radius one. Up to equivalence, there is a unique spin^{c} structure (ρ, W) over S^{4} and unitary connection on W which is spin with respect to the Levi-Civita connection on T^{∗}S^{4} and Clifford map ρ : T^{∗}S^{4} → End_{ℂ}(W). The sphere S^{4} has north pole n = (0, 0, 0, 0, 1), south pole s = (0, 0, 0, 0, −1), and local parametrizations

\begin{aligned}φ_{n} : ℝ^{4} & → S^{4} \ \{s\}, x ↦ φ_{n}(x), \\ φ_{s} : ℝ^{4} & → S^{4} \ \{n\}, y ↦ φ_{s}(y), \end{aligned} (3.19)

defined by inverse stereographic projection from the south and north poles, respectively.

Let E → S^{4} be a Hermitian, rank-two bundle with c_{2}(E) = k ≥ 1 and set V = W ⊗ E, yielding a complex rank-eight, Hermitian Clifford module (ρ, V). We may fix isomorphisms of complex line bundles det(W^{±}) ≅ S^{4} × ℂ and det(E) ≅ S^{4} × ℂ.

Let A_{k}(S^{4}) be the space of spin connections on V, which induce the product connection on det(V^{+}) ≅ S^{4} × ℂ, let G_{k} be the group of determinant-one, unitary automorphisms of E, identify G_{k} with the group of spin^{u} automorphisms of V via u ↦ id_{W} ⊗ u, and set

B_{k}^{s}(S^{4}) = A_{k}(S^{4}) ×_{G_{k}} Fr(𝔤_{V}|_{s}) (3.20)

where Fr(𝔤_{V}) denotes the frame bundle for the SO(3) bundle 𝔤_{V}. This is a principal SO(3) bundle over the quotient space B_{k}(S^{4}) of unframed spin connections on V.

A point [A] ∈ B_{k}(S^{4}) has an associated ‘center’ and ‘scale’ defined by [51, pp. 343–344]

\begin{aligned}z[A] & = \frac{1}{8π^{2}k} ∫_{ℝ^{4}} x|φ_{n}^{∗}F (Â)|^{2} d^{4}x, \\ λ^{2}[A] & = \frac{1}{8π^{2}k} ∫_{ℝ^{4}} |x − z[A]|^{2}|φ_{n}^{∗}F (Â)|^{2} d^{4}x. \end{aligned} (3.21)

Let M_{k}^{s}(S^{4}) ⊂ B_{k}^{s}(S^{4}) be the moduli space of pairs consisting of frames for the fiber 𝔤_{V}|_{s} and ‘instanton’ connections on V — the spin connections A on V corresponding to antiself-dual SO(3) connections Â on 𝔤_{V} = 𝔰𝔲(E). Let M_{k}^{♮}(S^{4}) ⊂ M_{k}^{s}(S^{4}) denote the subspace of ‘mass-centered’ instantons, so z[A] = 0 for [A] ∈ M_{k}^{♮}(S^{4}) and similarly define M_{k}^{s,♮}(S^{4}). Let

M_{k}^{♮}(S^{4}, δ) ⊂ M_{k}^{♮}(S^{4}) (3.22)

denote the subspace of points [A] ∈ M_{k}^{♮}(S^{4}) with scales λ[A] in (0, δ], where δ is a small positive constant, and similarly define M_{k}^{s,♮}(S^{4}, δ). Hence, points in M_{κ}^{s,♮}(S^{4}, δ) have connections with curvature density with center of mass at and energy concentrated near the north pole.

Now suppose k = 1. Because there is, up to gauge transformation and rescaling, a unique mass-centered anti-self-dual connection on 𝔤_{V}, there is diffeomorphism [19, §3.1], [51, §3],

M_{1}^{s,♮}(S^{4}, δ) ≅ (0, δ] × SO(3). (3.23)

The Uhlenbeck compactification M̄_{1}^{s,♮}(S^{4}, δ) of M_{1}^{s,♮}(S^{4}, δ) is obtained by adjoining the ideal point {n}×[Θ] ∈ S^{4} ×M_{0}(S^{4}), where Θ is the tensor product of the Levi-Civita connection on W with the product connection on S^{4} × ℂ^{2}, and forgetting the frame for 𝔤_{V}|_{s}. Thus, we have a homeomorphism

M̄_{1}^{s,♮}(S^{4}, δ) ≅ c(SO(3)), (3.24)

where c(SO(3)) is the cone on SO(3).

An element R ∈ SO(4) acts by rotation on ℝ^{4} and hence on S^{4} ⊂ ℝ^{5} via the fixed stereographic projection from the south pole,

S^{4} \ {s} ≅ ℝ^{4}, (3.25)

where the north pole is identified with the origin in ℝ^{4}. The group SO(4) in turn acts by pullback via (3.25) on sections of and spin connections on V → S^{4}.

3.4. Splicing Clifford modules, connections, and spinors. In this section we define the process of splicing instantons from the four-sphere onto SO(3) monopoles over X, together with the required gluing data choices.

3.4.1. Splicing Clifford modules. Let B(x_{0}, r_{0}) be a geodesic ball in X with center x_{0} ∈ X and radius r_{0}. Choose a frame F ∈ Fr(T^{∗}X|_{x_{0}}), that is, an isomorphism of oriented real inner-product spaces,

F : ℝ^{4} ≅ T^{∗}X|_{x_{0}}. (3.26)

This determines a dual isomorphism ℝ^{4} ≅ T X|_{x_{0}} and an isomorphism of Clifford algebras,

F : ℂℓ(ℝ^{4}) ≅ ℂℓ(T^{∗}X|_{x_{0}}). (3.27)

Extend the frame (3.26) by parallel translation along radial geodesics for the Levi-Civita connection on (X, g) to a local frame for T^{∗}X (and dual frame for T X) over B(x_{0}, r_{0}) and geodesic, normal coordinates on this ball, yielding a geodesic, normal coordinate chart

φ_{x_{0}}^{−1} : B(x_{0}, r_{0}) ⊂ X → ℝ^{4}, (3.28)

and trivializations

T^{∗}X|_{B(x_{0},r_{0})} ≅ B(x_{0}, r_{0}) × ℝ^{4} and ℂℓ(T^{∗}X)|_{B(x_{0},r_{0})} ≅ B(x_{0}, r_{0}) × ℂℓ(ℝ^{4}), (3.29)

induced by parallel translation with respect to the Levi-Civita connection along radial geodesics of the frame F ∈ Fr(T^{∗}X|_{x_{0}}).

Let (ρ, V) be a complex-rank eight, Hermitian-Clifford module over (X, g), equipped with unitary connection A which is spin with respect to the Levi-Civita connection on T^{∗}X and induces the fixed connection 2A_{Λ} on the fixed determinant line bundle det(V^{+}), where Λ = det^{\frac{1}{2}} (V^{+}). Choose a frame F̃ ∈ Fr(V_{x_{0}}) in the fiber of the bundle (3.4) over F, so F̃ is an isomorphism of Hermitian-Clifford modules with respect to the isomorphism (3.27),

F̃ : Δ ⊗_{ℂ} ℂ^{2} ≅ V_{x_{0}}. (3.30)

It will be useful to note the

Lemma 3.5. Let (U, ρ) be a ℂℓ(T^{∗}Y)-module, of arbitrary complex rank, over an oriented, Riemannian manifold (Y, g). Then parallel translation with respect to a spin connection on U gives ℂℓ(T^{∗}Y)-module isomorphisms of the fibers of U → Y, with respect to the isomorphisms of the fibers of T^{∗}Y → Y given by parallel translation with respect to the Levi-Civita connection on T^{∗}Y.

Proof. Suppose γ is a smooth curve in Y with initial point y_{0}. Let α ∈ Ω^{1}(Y, ℂ), let Φ ∈ C^{∞}(Y, U), and let ∇ be a spin connection on U. Because ∇ is a Clifford module derivation of C^{∞}(Y, U), we have (see equation (2.2))

∇γ̇(ρ(α)Φ) = ρ(∇γ̇α)Φ + ρ(α)∇γ̇Φ,

where ∇ also denotes the Levi-Civita connection on T^{∗}X. Thus, if α and Φ are parallel along γ with respect to ∇, then ρ(α)Φ must be parallel along γ with respect to ∇. Let P_{y} denote parallel translation, with respect to the Levi-Civita connection on T^{∗}Y or spin connection on U, along γ from y_{0} to a point y ∈ Y. Therefore, α_{y} = P_{y}α_{0} and Φ_{y} = P_{y}Φ_{0}. The sections of U over γ ⊂ Y given by

ρ_{y}(P_{y}α_{0})P_{y}Φ_{0} and P_{y}(ρ_{0}(α_{0})Φ_{0}), y ∈ γ,

agree when y = y_{0} and are parallel along the curve γ from y_{0}. Hence, parallel translation along curves in Y with respect to spin connections commutes with Clifford multiplication and thus yields Clifford-module isomorphisms of the fibers of U → Y.

Therefore, given a spin connection A on V and a choice of frame F̃ : Δ ⊗_{ℂ} ℂ^{2} ≅ V_{x_{0}}, parallel translation with respect to A along radial geodesics in X from x_{0} yields a Cliffordmodule isomorphism,

V |_{B(x_{0},r_{0})} ≅ B(x_{0}, r_{0}) × Δ ⊗_{ℂ} ℂ^{2}, (3.31)

with respect to the Clifford algebra isomorphism (3.29). The trivialization (3.31) also induces a trivialization

𝔤_{V} |_{B(x_{0},r_{0})} ≅ B(x_{0}, r_{0}) × 𝔰𝔲(2). (3.32)

This completes our discussion of the Clifford modules and bundles over X.

Over S^{4}, we have the standard spin^{c} structure (ρ, W) with Clifford multiplication ρ : T^{∗}S^{4} → End_{ℂ}(W) and unitary connection which is spin with respect to the Levi-Civita connection on T^{∗}S^{4} for the standard round metric of radius one. We may fix, once and for all, a unit-norm ℂℓ(T^{∗}S|_{n})-frame Δ ≅ W|_{n} for W|_{n} and a trivialization defined by parallel radial translation which is a Clifford-module isomorphism (see Lemma 3.5),

W|_{S^{4}\{s}} ≅ ℝ^{4} × Δ, (3.33)

with respect to the Clifford algebra isomorphism

ℂℓ(T^{∗}S^{4})|_{S^{4}\{s}} ≅ S^{4} \ {s} × ℂℓ(ℝ^{4}). (3.34)

A spin connection A on V = W ⊗_{ℂ} E → S^{4} and a choice of frame 𝔰𝔲(2) ≅ 𝔤_{V|_{s}}, that is, an isomorphism of oriented real inner product spaces, define a trivialization of SO(3) bundles,

𝔤_{V}|_{S^{4}\{n}} ≅ ℝ^{4} × 𝔰𝔲(2), (3.35)

via parallel translation with respect to the SO(3) connection Â on 𝔤_{V}. Up to an ambiguity corresponding to the action of {±1}, the frame 𝔰𝔲(2) ≅ 𝔰𝔲(E)|_{s} = 𝔤_{V|_{s}} lifts to a frame ℂ^{2} ≅ E_{s}, an isomorphism of complex inner product spaces. This ambiguity will vanish when we take the quotient by the gauge group. Thus, up to this ambiguity, we obtain an isomorphism lifting (3.35),

E|_{S^{4}\{n}} ≅ ℝ^{4} × ℂ^{2}, (3.36)

coinciding with the trivialization defined by the isomorphism ℂ^{2} ≅ E_{s} and radial parallel translation with respect to the unique lift of Â to an SU(2) connection on E.

The isomorphisms (3.33) and (3.36) combine to give a Clifford-module isomorphism

V|_{S^{4}\{n,s}} ≅ ℝ^{4} \ {0} × Δ ⊗_{ℂ} ℂ^{2}, (3.37)

with respect to the Clifford algebra isomorphism (3.34).

We then define a spin^{u} structure 𝔱′ = (ρ, V′) over (X, g) by setting

V′ = (V |_{X\{x_{0}}})#(V|_{S^{4}\{s}}), (3.38)

where the bundles V and V are identified via the trivializations (3.31) and (3.37) and the embedding

φ_{n} ∘ φ_{x_{0}}^{−1} : B(x_{0}, r_{0}) → S^{4} \ {s}, (3.39)

identifying a ball in X centered at x_{0} with a ball in S^{4} centered at the north pole. Because these trivializations are Clifford module isomorphisms, the bundle V′ has a ℂℓ(T^{∗}X)-module structure. The Clifford module (3.38) induces an associated SO(3)-bundle 𝔤_{V′},

𝔤_{V′} = (𝔤_{V} |_{X\{x_{0}}})#(𝔤_{V}|_{S^{4}\{s}}), (3.40) where the bundles 𝔤_{V} and 𝔤_{V} are identified over small annuli centered at x ∈ X and n ∈ S^{4} by the trivializations (3.32) and (3.35) and the embedding (3.39). The characteristic classes of 𝔱 = (ρ, V) and 𝔱′ = (ρ, V′) are related by

p_{1}(𝔤_{V′}) = p_{1}(𝔤_{V}) − 4, c_{1}(V′) = c_{1}(V), and w_{2}(𝔤_{V′}) = w_{2}(𝔤_{V}), (3.41)

as we shall later exploit, often without comment.

We give a more detailed description of where the bundles V′ and 𝔤_{V′} are spliced below. Define a smooth cutoff function χ_{x_{0},ε} : X → [0, 1] by setting

χ_{x_{0},ε}(x) := χ(dist(x, x_{0})/ε), (3.42)

where χ : ℝ → [0, 1] is a smooth function such that χ(t) = 1 for t ≥ 1 and χ(t) = 0 for t ≤ 1/2. Thus, we have

χ_{x_{0},ε}(x) = \begin{cases}1 & for x ∈ X − B(x_{0}, ε), \\ 0 & for x ∈ B(x_{0}, ε/2). \end{cases}

The orientation-preserving embedding (3.39) therefore identifies the annulus φ_{n}(Ω(0, \frac{1}{2} \sqrt{λ}, 2\sqrt{λ})) in S^{4}, where

Ω (0, \frac{1}{2} \sqrt{λ}, 2\sqrt{λ}) := \{x ∈ ℝ^{4} : \frac{1}{2} \sqrt{λ} < |x| < 2\sqrt{λ}\} ⊂ ℝ^{4},

with the annulus in X,

Ω (x_{0}, \frac{1}{2} \sqrt{λ}, 2\sqrt{λ}) := \{x ∈ X : \frac{1}{2} \sqrt{λ} < dist_{g}(x, x_{0}) < 2\sqrt{λ}\} ⊂ X.

Hence, the spliced bundle (3.38) is defined explicitly by setting

V′ = \begin{cases}V & over X − B(x_{0}, \frac{1}{2} \sqrt{λ}), \\ V & over B(x_{0}, 2\sqrt{λ}). \end{cases}

The bundles V and V are identified over the annulus Ω(x_{0}, \frac{1}{2} \sqrt{λ}, 2\sqrt{λ}) in X via the orientation-preserving diffeomorphism (3.39) identifying the annulus Ω(x_{0}, \frac{1}{2} \sqrt{λ}, 2\sqrt{λ}) with the corresponding annulus in S^{4} and the bundle map defined by the trivializations (3.31) and (3.37).

Similarly, the spliced SO(3) bundle (3.40) is defined explicitly by setting

𝔤_{V′} = \begin{cases}𝔤_{V} & over X − B(x_{0}, \frac{1}{2} \sqrt{λ}), \\ 𝔤_{V} & over B(x_{0}, 2\sqrt{λ}). \end{cases}

The bundles 𝔤_{V} and 𝔤_{V} are identified over the annulus Ω(x_{0}, \frac{1}{2} \sqrt{λ}, 2\sqrt{λ}\}) in X via the diffeomorphism (3.39) and the SO(3) bundle map defined by the trivializations (3.32) and (3.35).

3.4.2. Splicing connections and cutting off background spinors. To construct a spliced spin connection on the Clifford module V′, we choose the following data:

• An SO(3) connection Â on 𝔤_{V} → X,

• An SO(4) frame F ∈ Fr(T^{∗}X|_{x_{0}}), and induced chart B(x_{0}, r_{0}) ⊂ X → ℝ^{4} as in (3.28),

• An SO(3) frame for Fr(𝔤_{V} |_{x_{0}}), and corresponding trivialization 𝔤_{V} |_{B(x_{0},r_{0})} ≅ B(x_{0}, r_{0})× 𝔰𝔲(2) as in (3.32),

• An SO(3) connection Â on 𝔤_{V} → S^{4}, and associated scale parameter λ ∈ (0, δ],

• An SO(3) frame for 𝔤_{V}|_{s}, and corresponding trivialization 𝔤_{V}|_{S^{4}\{n}} ≅ S^{4} \{n}×𝔰𝔲(2) as in (3.35), together with the fixed chart S^{4} \ {s} ≅ ℝ^{4} as in (3.19).

This yields a spliced connection Â′ on the spliced SO(3) bundle 𝔤_{V′} over X, as in (3.40).

We give a more detailed description of where the SO(3) connections are spliced below. We first define a cut-off SO(3) connection on the bundle 𝔤_{V} over X by setting

(3.43) χ_{x_{0},4\sqrt{λ}}Â := \begin{cases}Â & over X − B(x_{0}, 4\sqrt{λ}), \\ Γ + χ_{x_{0},4\sqrt{λ}}σ_{0}^{∗}Â & over Ω(x_{0}, 2\sqrt{λ}, 4\sqrt{λ}), \\ Γ & over B(x_{0}, 2\sqrt{λ_{0}}), \end{cases}

where Γ denotes the product connection on B(x_{0}, r_{0}) × SO(3) and σ_{0} is the section of 𝔤_{V} over B(x_{0}, r_{0}) defined by the trivialization (3.32).

We define a cut-off SO(3) connection on the bundle 𝔤_{V} over S^{4}, with mass center at the north pole, by setting

(3.44) (1 − χ_{n,\sqrt{λ}/2})Â := \begin{cases}Â & over φ_{n}(B(0, \frac{1}{4} \sqrt{λ})), \\ Γ + (1 − χ_{n,\sqrt{λ}/2})τ_{0}^{∗}Â & over φ_{n}(Ω(0, \frac{1}{4} \sqrt{λ}, \frac{1}{2} \sqrt{λ})), \\ Γ & over S^{4} − φ_{n}(B(0, \frac{1}{2} \sqrt{λ})), \end{cases}

where Γ denotes the product connection on (S^{4} − {n}) × SO(3) and τ_{0} is the section of 𝔤_{V} over S^{4} − {n} defined by the trivialization (3.35).

Hence, we define a spliced connection Â′ on 𝔤_{V′} by setting

(3.45)  Â′ := \begin{cases}Â & over X − B(x_{0}, 4\sqrt{λ}), \\ Γ + χ_{x_{0},4\sqrt{λ}}Â + (1 − χ_{x_{0},\sqrt{λ}/2})Â & over Ω(x_{0}, \frac{1}{4} \sqrt{λ}, 4\sqrt{λ}), \\ Â & over B(x_{0}, \frac{1}{4} \sqrt{λ}), \end{cases}

where the cut-off connections are defined as above; the bundle and annulus identifications are understood.

According to Lemma 2.11 in [17], we then obtain a unitary connection A′ on V′ defined by the SO(3) connection Â′ on 𝔤_{V′} and by the

• Frame F̃ ∈ Fr_{ℂℓ(T^{∗}X)}(V_{x_{0}}) covering F ∈ Fr(T^{∗}X|_{x_{0}}), and corresponding Cliffordmodule isomorphism V |_{B(x_{0},r_{0})} ≅ B(x_{0}, r_{0}) × Δ ⊗_{ℂ} ℂ^{2} as in (3.31),

• Fixed U(1) connection 2A_{Λ} on det(V′^{+}), and the

• Requirement that A′ be spin with respect to the Levi-Civita connection on T^{∗}X for the metric g and Clifford map ρ′ : T^{∗}X → End_{ℂ}(V′).

Finally, given a section Φ of V → X, we obtain a section Φ′ of V′ → X by cutting off Φ in a ball around x_{0} ∈ X, with radius determined by λ.

More explicitly, we define the spinor Φ′ on the bundle V′ over X by setting

(3.46) Φ′ ≡ χ_{x_{0},8λ^{1/3}}Φ := \begin{cases}Φ & over X − B(x_{0}, 8λ^{1/3}), \\ χ_{x_{0},8λ^{1/3}}Φ & over Ω(x_{0}, 4λ^{1/3}, 8λ^{1/3}), \\ 0 & over B(x_{0}, 4λ^{1/3}). \end{cases}

The reason for the different choice of annulus radii is explained in [19].

3.4.3. Splicing in spinors from the four-sphere. To construct a spliced section of the Clifford module V′, given a spinor over the four-sphere, we choose the following data:

• A spin connection A on V → X,

• An SO(4) frame F ∈ Fr(T^{∗}X|_{x_{0}}) and induced chart B(x_{0}, r_{0}) ⊂ X → ℝ^{4} as in (3.28),

• A frame F̃ ∈ Fr_{ℂℓ(T^{∗}X)}(V_{x_{0}}) covering F ∈ Fr(T^{∗}X|_{x_{0}}), and corresponding Cliffordmodule isomorphism V |_{B(x_{0},r_{0})} ≅ B(x_{0}, r_{0}) × Δ ⊗_{ℂ} ℂ^{2} as in (3.31), • A spin connection A on V → S^{4}, and associated scale parameter λ ∈ (0, δ],

• An oriented, orthonormal frame for E_{s}, and corresponding trivialization E|_{S^{4}\{n}} ≅ S^{4} \ {n} × ℂ^{2}, together with the fixed chart S^{4} \ {s} ≅ ℝ^{4} as in (3.19), fixed unitnorm ℂℓ(T^{∗}S^{4}|_{n})-frame for W|_{n}, and fixed Clifford-module isomorphism W|_{S^{4}\{s}} ≅ S^{4}\{s}×Δ. The bundle isomorphisms combine to give a Clifford-module isomorphism V|_{S^{4}\{n,s}} ≅ S^{4} \ {n, s} × Δ ⊗ ℂ^{2}, which we employ in conjunction with the chart S^{4} \ {s} ≅ ℝ^{4}.

Spinors Ψ on V → S^{4} are then spliced from ℝ^{4} ≅ S^{4} \ {s} onto the ball B(x_{0}, r_{0}) ⊂ X, via the preceding data, with the north pole n ∈ S^{4} being identified with x_{0} ∈ X.

More explicitly, we define the spinor Ψ′ on the bundle V′ over X by setting

Ψ′ := (1 − χ_{n,\sqrt{λ}/2})Ψ

:= \begin{cases}Ψ & over φ_{n}(B(0, \frac{1}{4} \sqrt{λ})), \\ (1 − χ_{n,\sqrt{λ}/2})Ψ & over φ_{n}(Ω(0, \frac{1}{4} \sqrt{λ}, \frac{1}{2} \sqrt{λ})), \\ 0 & over X − B(x_{0}, \frac{1}{2} \sqrt{λ}), \end{cases}

(3.47)

where the identification of the bundles V over the annulus Ω(x_{0}, \frac{1}{4} \sqrt{λ}, \frac{1}{2} \sqrt{λ}) in X and the bundle V over the annulus φ_{n}(Ω(0, \frac{1}{4} \sqrt{λ}, \frac{1}{2} \sqrt{λ})) in S^{4}, together with these annuli, are described following the definition (3.38) of V′.

3.5. Space of gluing data and the definition of the splicing map. The patching constructions described in §3.4 yield a pre-splicing map

γ̃′ : Ñ_{𝔱,𝔰}(ε) × Fr(𝔤_{V}) ×_{X} Fr(T^{∗}X) × Fr(𝔤_{V}|_{s}) × M̃_{1}^{♮}(S^{4}, δ) → C̃_{𝔱′}, (3.48)

where

M̃_{1}^{♮}(S^{4}, δ) ⊂ A_{1}(S^{4})

is the preimage of the subspace M_{1}^{♮}(S^{4}, δ) under the projection A_{1}(S^{4}) → B_{1}(S^{4}). The map (3.48) is equivariant with respect to the

1. Diagonal action of G_{𝔰} on Ñ_{𝔱,𝔰}(ε) × Fr(𝔤_{V}), where G_{𝔰} acts on Ñ_{𝔱,𝔰} as described following equation (2.35) and on Fr(𝔤_{V}) by the inclusion (2.25) of G_{𝔰} in G_{𝔱},

2. Diagonal action of G_{1} on Fr(𝔤_{V}|_{s}) × M̃_{1}^{♮}(S^{4}, δ),

3. Diagonal action of SO(4) on Fr(T^{∗}X) × M̃_{1}^{♮}(S^{4}, δ) (where the action of SO(4) is as described following (3.25)),

4. Diagonal action of SO(3) on Fr(𝔤_{V}) × Fr(𝔤_{V}|_{s}). Therefore, if we define the space

(3.49)  G̃l_{𝔱}(δ) = (Fr(𝔤_{V}) ×_{X} Fr(T^{∗}X) × Fr(𝔤_{V}|_{s}) × M̃_{1}^{♮}(S^{4}, δ)) /(SO(3) × SO(4)),

where SO(3) × SO(4) act as described prior to (3.49), the pre-splicing map (3.48) descends to a map

γ̃′ : Ñ_{𝔱,𝔰}(ε) × G̃l_{𝔱}(δ) → C̃_{𝔱′}^{∗}. (3.50)

The map (3.50) is gauge equivariant and thus, if we define

Gl_{𝔱}(δ) = G̃l_{𝔱}(δ)/G_{1}
= (Fr(𝔤_{V}) ×_{X} Fr(T^{∗}X) × M_{1}^{s,♮}(S^{4}, δ)) /(SO(3) × SO(4)),

(3.51)

(recalling that M_{1}^{s,♮}(S^{4}, δ) = Fr(𝔤_{V}|_{s}) ×_{G_{1}} M̃_{1}^{♮}(S^{4}, δ)), the pre-splicing map descends to the gauge group quotients and gives the splicing map

γ′ : M_{𝔱′,𝔰}^{vir} → C_{𝔱′}, (3.52)

where we have defined

M_{𝔱′,𝔰}^{vir} = Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Gl_{𝔱}(δ). (3.53)

We refer to the space (3.53) as a virtual moduli space; it is defined by the stabilizing bundle used to construct N_{𝔱,𝔰}. It can be shown [19, 20] that γ′ is a smooth embedding, provided ε and δ are sufficiently small.

By analogy with the definition of the Uhlenbeck compactification in §2.1.2, we define

(3.54)  Ḡl_{𝔱}(δ) = (Fr(𝔤_{V}) ×_{X} Fr(T^{∗}X) × M̄_{1}^{s,♮}(S^{4}, δ)) /(SO(3) × SO(4)).

Points in Ḡl_{𝔱}(δ)−Gl_{𝔱}(δ) then correspond to the cone point in (3.24). The action of SO(4)× SO(3) in (3.54) is trivial on these cone points, so Ḡl_{𝔱}(δ) − Gl_{𝔱}(δ) = X. The cone completion of Gl_{𝔱}(δ) then gives an ‘Uhlenbeck compactification’

M̄_{𝔱′,𝔰}^{vir} = Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Ḡl_{𝔱}(δ) (3.55)

of the space M_{𝔱′,𝔰}^{vir}, where G_{𝔰} acts trivially on Ḡl_{𝔱}(δ) − Gl_{𝔱}(δ).

The splitting 𝔤_{V} ≅ \underline{ℝ} ⊕ L, induced by the splitting V = W ⊕ W ⊗ L, and the circle action given in (2.26) induce a circle action on Fr(𝔤_{V}) and thus on Ḡl_{𝔱}(δ) by scalar multiplication on L. If we write this action as (e^{iθ}, g) ↦ e^{iθ}g, where e^{iθ} ∈ S^{1} and g ∈ Ḡl_{𝔱}(δ), then the description of the action of G_{𝔰} on 𝔤_{V} following equation (2.35) implies that

(3.56) ϱ(e^{iθ})g = e^{−2iθ}g, for e^{iθ} ∈ S^{1} and g ∈ Ḡl_{𝔱}(δ).

If we set

C̄_{𝔱′} = ⨆_{ℓ=0}^{∞}(C_{𝔱′_{ℓ}} × Sym^{ℓ}(X)),

and extend the splicing map γ′ to a map

γ′ : M̄_{𝔱′,𝔰}^{vir} → C̄_{𝔱′}, (3.57)

by setting γ′ equal to γ_{𝔰} × id_{X} : N_{𝔱,𝔰}(ε) × X → C_{𝔱} × X on the cone points, where γ_{𝔰} is defined in (2.29), then this extension is continuous with respect to the Uhlenbeck topology. The extended splicing map (3.57) is S^{1}-equivariant if the circle acts by the action (2.8) on C̄_{𝔱′} and diagonally with weight two on each factor in the product Ñ_{𝔱,𝔰} × Ḡl_{𝔱}(δ):

Lemma 3.6. The following two circle actions on Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Ḡl_{𝔱}(δ) are equivalent:

1. The action (2.37) on Ñ_{𝔱,𝔰}(ε) and the trivial action on Ḡl_{𝔱}(δ), and

2. The diagonal action with weight two on the fibers of Ñ_{𝔱,𝔰}(ε) and with weight two on

Ḡl_{𝔱}(δ). Furthermore, the extended splicing and gluing maps

γ, γ′ : Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Ḡl_{𝔱}(δ) → C̄_{𝔱′}

are circle-equivariant if the circle acts on C̄_{𝔱′} by the action (2.8) and on the domain by either of the above equivalent actions.

Proof. The embedding Ñ_{𝔱,𝔰}(ε) → C̃_{𝔱} is S^{1}-equivariant if the circle acts on C̃_{𝔱} by the action (2.8) and on Ñ_{𝔱,𝔰}(ε) by the composition of multiplication with weight two on the fibers of Ñ_{𝔱,𝔰}(ε) and the circle action (2.37). The splicing map is S^{1}-equivariant if the circle acts by scalar multiplication on the sections both before and after cutting off the sections. Hence, the splicing map γ′ is S^{1}-equivariant if the circle acts on C_{𝔱′} by the action (2.8) and by the action (1) in the lemma statement on the domain, namely: for (B, Ψ, η) ∈ Ñ_{𝔱,𝔰} and g ∈ Gl_{𝔱}(δ), by

[(B, Ψ, η), g] ↦ [ϱ(e^{iθ})(B, Ψ, e^{i2θ}η), g].

Because G_{𝔰} acts diagonally in the definition of Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Ḡl_{𝔱}(δ), the action (1) on the domain is equal to the circle action given by scalar multiplication with weight two on the fibers of Ñ_{𝔱,𝔰}(ε) and by the action of the constant S^{1} in G_{𝔰} with weight negative one on the factor Ḡl_{𝔱}(δ). This last action is equal to the action (2) on the domain in the statement of the lemma by the equation (3.56) comparing the action of the constant S^{1} subgroup of G_{𝔰} on Ḡl_{𝔱}(δ) with the standard action:

[ϱ(e^{iθ})(B, Ψ, e^{i2θ}η), g] = [(B, Ψ, e^{i2θ}η), ϱ(e^{−iθ})g] = [(B, Ψ, e^{i2θ}η), e^{2iθ}g].

This completes the proof.

We note that there is a projection map,

Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}×S^{1}} Ḡl_{𝔱}(δ) → M_{𝔰} × X, (3.58)

defined via the projection maps Ḡl_{𝔱}(δ) → X (given by the fiber bundle structure in (3.54)) and N_{𝔱,𝔰} → M_{𝔰}.

3.6. The obstruction bundle. The SO(3)-monopole obstruction bundle is a finite-rank, S^{1}-equivariant subbundle,

\begin{matrix}Υ_{𝔱′,𝔰} & ⟶^{φ} & 𝔙_{𝔱′} \\ ↓ & & ↓ \\ Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Gl_{𝔱}(δ) & ⟶^{γ′} & C_{𝔱′}^{∗,0} \end{matrix} (3.59)

of the S^{1}-equivariant vector bundle

𝔙_{𝔱′} = C̃_{𝔱′}^{∗,0} ×_{G_{𝔱′}} L_{k−1}^{2}(Λ^{+} ⊗ 𝔤_{V′} ⊕ V′^{−}) → C_{𝔱′}^{∗,0}. (3.60)

The circle actions in (3.60) are given by the diagonal circle action, with the action (2.8) on C̃_{𝔱′}^{∗,0}, the trivial action on L_{k−1}^{2}(Λ^{+} ⊗ 𝔤_{V′}), and by scalar multiplication on L_{k−1}^{2}(V′). As described in [19], the fiber Υ_{[(B,Ψ,η),g]} over a point [(B, Ψ, η), g] = (γ′)^{−1}([A′, Φ′]) in the base of the obstruction bundle, Υ_{𝔱′,𝔰}, is given by

Υ_{[(B,Ψ,η),g]} ≅ Υ_{[B,Ψ,η]}^{s} ⊕ Υ_{[A]}^{i}, (3.61)

where Υ_{[B,Ψ,η]}^{s} is a fiber of the background obstruction bundle (see §3.6.1), and Υ_{[A]}^{i} is

a fiber of the instanton obstruction bundle (see §3.6.2), and [A] is a point in M_{k}^{s,♮}(S^{4}, δ) associated with [g] ∈ Gl_{𝔱}(δ).

3.6.1. Background or Seiberg-Witten component of the obstruction bundle. In [19], the Seiberg-Witten component of the SO(3)-monopole obstruction bundle was identified with the subbundle constructed in [17, §3.5.2], namely π_{N}^{∗}Ξ_{𝔱,𝔰} ≅ N_{𝔱,𝔰}(ε) × ℂ^{r_{Ξ}} where π_{N} : N_{𝔱,𝔰} → M_{𝔰} is the projection, and there is an embedding of S^{1}-equivariant vector bundles,

\begin{matrix}π_{N}^{∗}Ξ_{𝔱,𝔰} & ⟶ & 𝔙_{𝔱} \\ ↓ & & ↓ \\ N_{𝔱,𝔰}(ε) & ⟶ & C_{𝔱} \end{matrix} (3.62)

The bundle map π_{N}^{∗}Ξ_{𝔱,𝔰} → 𝔙_{𝔱} in (3.62) is S^{1}-equivariant if the circle acts diagonally on N_{𝔱,𝔰}(ε)×ℂ^{r_{Ξ}} by scalar multiplication with weight two on both the fibers of N_{𝔱,𝔰} and on ℂ^{r_{Ξ}}, while the circle acts on 𝔙_{𝔱} by the action described following (3.60).

The cut-and-paste construction in §3.5 defines an embedding of vector bundles,

\begin{matrix}Υ_{𝔱′,𝔰}^{s} & ⟶^{φ_{s}} & 𝔙_{𝔱′} \\ ↓ & & ↓ \\ Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Gl_{𝔱}(δ) & ⟶^{γ′} & C_{𝔱′} \end{matrix} (3.63)

where Υ_{𝔱′,𝔰}^{s} is the pullback by the projection Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Gl_{𝔱}(δ) → N_{𝔱,𝔰}(ε) of the obstruction bundle π_{N}^{∗}Ξ_{𝔱,𝔰} appearing in (2.30), so that

Υ_{𝔱′,𝔰}^{s} ≅ ℂ^{r_{Ξ}} × Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Gl_{𝔱}(δ).

Because this cut-and-paste construction is circle equivariant, the embedding (3.63) has the same circle-equivariance properties as the embedding (3.62), where the circle acts on 𝔙_{𝔱′} as previously described and diagonally on ℂ^{r_{Ξ}} × Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Gl_{𝔱}(δ), with weight two on ℂ^{r_{Ξ}} and on Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Gl_{𝔱}(δ) with the action given in Lemma 3.6.

To describe the intersection of the image of the extended gluing map, associated with the extended splicing map (3.57) (see §3.7) with M̄_{𝔱′}, we need to extend the background obstruction bundle over the space of extended gluing data (3.54). Thus, we define the extension by

Ῡ_{𝔱′,𝔰}^{s} ≅ ℂ^{r_{Ξ}} × Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Ḡl_{𝔱}(δ). (3.64)

The restriction of Ῡ_{𝔱′,𝔰}^{s} to the stratum N_{𝔱,𝔰}(ε) × X of the space of extended gluing data is given by the pullback of the bundle in the left-hand column of (3.62) to N_{𝔱,𝔰}(ε) × X and is embedded into 𝔙_{𝔱} via the diagram (3.62).

3.6.2. Instanton component of the obstruction bundle. In [19], the instanton components of the SO(3)-monopole gluing obstruction space were identified with cokernels of the Dirac operators,

D_{A} : C^{∞}(S^{4}, V^{+}) → C^{∞}(S^{4}, V^{−}),

where the induced connections Â on the SO(3) bundle 𝔤_{V} over S^{4} are anti-self-dual. Because the standard round metric on S^{4} has positive scalar curvature, one has Ker D_{A} = {0} and as Index_{ℂ} D_{A} = −1 (see, for example, [16, p. 314]), the cokernel bundles,

(3.65) Coker D̃_{V} → M̃_{1}^{s,♮}(S^{4}, δ) and Coker D_{V} → M_{1}^{s,♮}(S^{4}, δ),

are complex line bundles with fibers Coker D_{V}|_{A} = Coker D_{A}. The total space of the bundle Coker D_{V} is identified by the diffeomorphism

(3.66) Coker D_{V} ≅ (U(2) × (0, δ]) ×_{S^{1}} ℂ ≅ SU(2) ×_{{±1}} ℂ × (0, δ], where S^{1} acts diagonally on U(2) × ℂ. The bundle Coker D_{V} → M_{1}^{s,♮}(S^{4}, δ) is non-trivial, but torsion. There is an action of Spin^{u}(4) on Coker D_{V}, covering the action of SO(4) × SO(3) on M_{1}^{s,♮}(S^{4}, δ): this action is most easily understood using the trivializations (3.37) of V|_{S^{4}\{n,s}}. The section Ψ, in this trivialization, is a map

ℝ^{4} − {0} → Δ ⊗_{ℂ} ℂ^{2}.

Then R̃ ⊗ M ∈ Spin^{u}(4) acts by

Ψ ↦ (R̃ ⊗ M) ∘ Ψ ∘ Ad^{c}(R̃)^{−1}.

The procedure for splicing spinors over S^{4} onto X defines a map (see §3.4.3)

(3.67)  φ̃_{i} : Ñ_{𝔱,𝔰} × Fr_{ℂℓ(T^{∗}X)}(V) × Coker D̃_{V} → C̃_{𝔱′} × Γ(Λ^{+} ⊗ 𝔤_{V′} ⊕ V′^{−}).

The map (3.67) is invariant under the diagonal action of Spin^{u}(4) on Fr_{ℂℓ(T^{∗}X)}(V) × Coker D̃_{V}. Note that this Spin^{u}(4) action covers the diagonal SO(4) action on the component Fr(T^{∗}X) × M̃_{1}^{♮}(S^{4}, δ) of the base and the diagonal SO(3) action on Fr(𝔤_{V}) × Fr(𝔤_{V}|_{s}). In addition, the map (3.67) is equivariant with respect to the action of G_{𝔱′} on the image and

• The diagonal action of G_{𝔰} on Ñ_{𝔱,𝔰}×Fr_{ℂℓ(T^{∗}X)}(V), where G_{𝔰} acts on Fr_{ℂℓ(T^{∗}X)}(V) by the homomorphism G_{𝔰} ↪ G_{𝔱} in (2.25), covering the diagonal action of G_{𝔰} on Ñ_{𝔱,𝔰} × Fr(𝔤_{V}),

• The action of G_{1} on Coker D̃_{V}, covering its diagonal action on the component M̃_{1}^{s,♮}(S^{4}, δ)

of the base. Finally, the map (3.67) is S^{1}-equivariant with respect to the circle action 𝔙_{𝔱′} described after (3.60) and the diagonal circle action on Ñ_{𝔱,𝔰} × Coker D_{A}, where the circle acts on Coker D_{A} by scalar multiplication and on Ñ_{𝔱,𝔰} by the action (2.37). We define a vector bundle over Ñ_{𝔱,𝔰} ×_{G_{𝔰}} Gl_{𝔱}(δ) by setting

(3.68) Υ_{𝔱′,𝔰}^{i} = Ñ_{𝔱,𝔰} ×_{G_{𝔰}} Fr_{ℂℓ(T^{∗}X)}(V) ×_{Spin^{u}(4)} Coker D_{V},

where Spin^{u}(4) acts diagonally on Fr_{ℂℓ(T^{∗}X)}(V) × Coker D_{V} and trivially on Ñ_{𝔱,𝔰}, while G_{𝔰} acts diagonally on Ñ_{𝔱,𝔰} × Fr_{ℂℓ(T^{∗}X)}(V) as described above and trivially on Coker D_{V}. We then have the

Lemma 3.7. The map φ̃_{i} descends to an embedding of S^{1}-equivariant vector bundles,

\begin{matrix}Υ_{𝔱′,𝔰}^{i} & ⟶^{φ_{i}} & 𝔙_{𝔱′} \\ ↓ & & ↓ \\ Ñ_{𝔱,𝔰} ×_{G_{𝔰}} Gl_{𝔱}(δ) & ⟶^{γ′} & C_{𝔱′} \end{matrix} (3.69)

where the group actions are as described in the preceding paragraph.

3.7. Construction of the gluing map. The Uhlenbeck compactification (3.55) of the virtual moduli space model M_{𝔱′,𝔰}^{vir} has a smooth, circle invariant stratification (where the strata are manifolds with boundary),

\begin{aligned}M̄_{𝔱′,𝔰}^{vir} & = Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Gl_{𝔱}(δ) \\ & ⊔ (N_{𝔱,𝔰}(ε) − M_{𝔰}) × X \\ & ⊔ M_{𝔰} × X. \end{aligned} (3.70)

If r_{N} is the complex rank of N_{𝔱,𝔰} → M_{𝔰} and d_{s}(𝔰) = dim M_{𝔰}, then because the fiber of Gl_{𝔱}(δ) → X is four-dimensional, the dimensions of the strata in (3.70) are given by (see the rank and dimension relations (2.32) and (2.34))

dim (Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Gl_{𝔱}(δ)) = 2r_{N} + d_{s}(𝔰) + 8,
dim ((N_{𝔱,𝔰}(ε) − M_{𝔰}) × X) = 2r_{N} + d_{s}(𝔰) + 4,
dim (M_{𝔰} × X) = d_{s}(𝔰) + 4.

The restriction of the splicing map γ′ to the top stratum M_{𝔱′,𝔰}^{vir} takes values in C_{𝔱′}^{∗,0}; on the middle stratum, (N_{𝔱,𝔰}(ε) − M_{𝔰}) × X, the splicing map γ′ restricts to the map γ_{𝔱,𝔰} × id_{X}, which takes values in C_{𝔱}^{∗,0}× X; on the lowest stratum, M_{𝔰}× X, the splicing map γ′ restricts to the identity map on M_{𝔰} × X. The stratum M_{𝔰} × X is the fixed point set of the circle action described in Lemma 3.6.

The splicing map γ′ may be deformed S^{1}-equivariantly to a ‘gluing map’ γ with the properties described in the following

Theorem 3.8. [19, 20] For small enough positive ε and δ, there is a topological embedding,

γ : M̄_{𝔱′,𝔰}^{vir} → C̄_{𝔱′} = C_{𝔱′} ⊔ (C_{𝔱} − M_{𝔰}) × X ⊔ M_{𝔰} × X, (3.71)

restricting to a smooth embedding of the top stratum of (3.70) into C_{𝔱′}^{∗,0}, the smooth embedding γ_{𝔱,𝔰} × id_{X} of the middle stratum into C_{𝔱}^{∗,0} × X, and the identity map on the lowest stratum, M_{𝔰}×X ⊂ C_{𝔱}^{0}×X. There is a smooth, circle-equivariant section χ_{i} of the instanton obstruction bundle (3.68),

Υ_{𝔱′,𝔰}^{i} → M_{𝔱′,𝔰}^{vir},

and a continuous, circle-equivariant section χ_{s} of the background obstruction bundle (3.64),

Ῡ_{𝔱′,𝔰}^{s} → M̄_{𝔱′,𝔰}^{vir},

which is smooth when restricted to each stratum of M̄_{𝔱′,𝔰}^{vir} in (3.70) such that, if χ = χ_{s} ⊕χ_{i}, then

γ(M_{𝔱′,𝔰}^{vir} ∩ χ^{−1}(0)) = γ(M_{𝔱′,𝔰}^{vir}) ∩ M_{𝔱′},

and

γ (((N_{𝔱,𝔰}(ε) − M_{𝔰}) × X) ∩ χ_{s}^{−1}(0)) = γ((N_{𝔱,𝔰}(ε) − M_{𝔰}) × X) ∩ M̄_{𝔱′}.

The sections χ_{s} ⊕ χ_{i} and χ_{s} of the vector bundles,

Ῡ_{𝔱′,𝔰}^{s} ⊕ Υ_{𝔱′,𝔰}^{i} → M_{𝔱′,𝔰}^{vir} and Ῡ_{𝔱′,𝔰}^{s} → (N_{𝔱,𝔰} − M_{𝔰}) × X,

respectively, vanish transversely.

We shall formally extend the section χ_{i} over the lower strata of M̄_{𝔱′,𝔰}^{vir} by setting it equal to zero on the lower strata. This is done so that we may discuss χ^{−1}(0) as a subspace of M̄_{𝔱′,𝔰}^{vir}; we make no assumptions about the continuity or transversality of this formal extension of χ_{i} over the lower strata.

3.8. Link of a level-one Seiberg-Witten stratum. We define the Seiberg-Witten and instanton components of the virtual link of the stratum M_{𝔰} × X in M̄_{𝔱′,𝔰}^{vir}/S^{1} by

L̄_{𝔱′,𝔰}^{vir,s} = (∂Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Ḡl_{𝔱}(δ)) /S^{1},
L_{𝔱′,𝔰}^{vir,i} = (Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} ∂Ḡl_{𝔱}(δ)) /S^{1},

(3.72) with (see (3.51) and (3.54) for the definitions of Gl_{𝔱}(δ) and Ḡl_{𝔱}(δ), respectively),

(3.73) ∂Ḡl_{𝔱}(δ) = (Fr(𝔤_{V}) ×_{X} Fr(T^{∗}X) × λ^{−1}(δ) ∩ M_{1}^{s,♮}(S^{4})) /(SO(3) × SO(4)),

and where the circle action on M̄_{𝔱′,𝔰}^{vir} is induced by the circle action (2.37) on Ñ_{𝔱,𝔰} and the trivial action on Ḡl_{𝔱}(δ) (as described in the first action in Lemma 3.6). The map

λ : B_{1}(S^{4}) → (0, ∞)

appearing in the gluing-data boundary (3.73) is defined by the scale definition (3.21). We define the virtual link of the stratum M_{𝔰} × X in M̄_{𝔱′,𝔰}^{vir}/S^{1} by setting

(3.74)  L̄_{𝔱′,𝔰}^{vir} = L̄_{𝔱′,𝔰}^{vir,s} ∪ L_{𝔱′,𝔰}^{vir,i},

We let

(3.75) L_{𝔱′,𝔰}^{vir} = L̄_{𝔱′,𝔰}^{vir} ∩ M_{𝔱′,𝔰}^{vir}/S^{1} and L_{𝔱′,𝔰}^{vir,s} = L̄_{𝔱′,𝔰}^{vir,s} ∩ M_{𝔱′,𝔰}^{vir}/S^{1}

denote the intersection of these subspaces with the top stratum M_{𝔱′,𝔰}^{vir}/S^{1} of M̄_{𝔱′,𝔰}^{vir}/S^{1}. Note that the top level, L_{𝔱′,𝔰}^{vir}, of the virtual link is only a topological and not a smooth manifold because of the ‘edge’:

(3.76) L_{𝔱′,𝔰}^{vir,i} ∩ L_{𝔱′,𝔰}^{vir,s}.

Let L_{𝔱′,𝔰}^{low} be the intersection of L̄_{𝔱′,𝔰}^{vir} with the union of the lower levels of M̄_{𝔱′,𝔰}^{vir}/S^{1}:

L_{𝔱′,𝔰}^{low} = L̄_{𝔱′,𝔰}^{vir} − L_{𝔱′,𝔰}^{vir}
= L̄_{𝔱′,𝔰}^{vir} ∩ (M̄_{𝔱′,𝔰}^{vir} − M_{𝔱′,𝔰}^{vir}) /S^{1}
= ∂N_{𝔱,𝔰}(ε)/S^{1} × X,

(3.77)

We then have a stratification:

(3.78)  L̄_{𝔱′,𝔰}^{vir} = L_{𝔱′,𝔰}^{vir} ⊔ L_{𝔱′,𝔰}^{low}.

In this article, where ℓ(𝔱′, 𝔰) = 1, we see from (3.77) that L_{𝔱′,𝔰}^{low} is a closed, smooth manifold. We define the link of (M_{𝔰} × X) ∩ M̄_{𝔱′}/S^{1} in M̄_{𝔱′}/S^{1} and its top stratum by

(3.79)  L̄_{𝔱′,𝔰} = γ (χ^{−1}(0) ∩ L̄_{𝔱′,𝔰}^{vir}) and L_{𝔱′,𝔰} = L̄_{𝔱′,𝔰} ∩ M_{𝔱′}^{∗,0}/S^{1}.

For generic choices of the parameters ε and δ (which we shall henceforth assume) defining L_{𝔱′,𝔰}^{vir}, the subspaces

L_{𝔱′,𝔰} ∩ L_{𝔱′,𝔰}^{vir,i} and L_{𝔱′,𝔰} ∩ L_{𝔱′,𝔰}^{vir,s}

will be smooth submanifolds of L_{𝔱′,𝔰}^{vir,i} and L_{𝔱′,𝔰}^{vir,s} respectively, transverse to the common boundary (3.76). The following lemma is the first step in showing that the intersection numbers on the right-hand-side of (2.50) are well-defined.

Lemma 3.9. Assume w ∈ H^{2}(X; ℤ) is such that w (mod 2) is good in the sense of Definition 2.3. Given a Riemannian metric on X and a pair (𝔱′, 𝔰) with ℓ(𝔱′, 𝔰) = 1 and w_{2}(𝔱′) ≡ w (mod 2), there are positive constants ε_{0} and δ_{0} such that the following hold for all generic choices of ε ≤ ε_{0} and δ ≤ δ_{0} defining L̄_{𝔱′,𝔰}^{vir}.

• The link L̄_{𝔱′,𝔰} is disjoint from M̄_{κ}^{w} and M̄_{𝔱′}^{red} in the stratification (2.13) of M̄_{𝔱′}/S^{1}.

• For all z ∈ 𝔸(X), the intersection,

V̄(z) ∩ W̄^{η} ∩ L̄_{𝔱′,𝔰} (3.80)

is contained in the top stratum L_{𝔱′,𝔰} of L̄_{𝔱′,𝔰} ⊂ M̄_{𝔱′}/S^{1}, and is disjoint from the image of the edge (3.76) under the gluing map γ.

Proof. Because w (mod 2) is good, the union of strata M̄_{κ}^{w} is disjoint from the union of strata M̄_{𝔱′}^{red} (see remarks in §2.2). Hence, for sufficiently small parameters δ and ε in the definition of L̄_{𝔱′,𝔰}^{vir}, the link L̄_{𝔱′,𝔰} is disjoint from these strata.

The geometric representatives V̄(z)∩W̄^{η} do not intersect the lower levels (M̄_{𝔱′}−M_{𝔱′})/S^{1} of M̄_{𝔱′}/S^{1} except at points in M̄_{𝔱′}^{red} or at points in M̄_{κ}^{w} by [18, Corollary 3.18]. Therefore, the intersection (3.80) is contained in the top stratum, L_{𝔱′,𝔰}.

The geometric representatives are transverse to M_{𝔱′}^{∗,0}/S^{1}. Hence, the intersection

γ^{−1}(V̄(z) ∩ W̄^{η}) ∩ χ^{−1}(0),

is a smooth submanifold of M_{𝔱′,𝔰}^{vir,∗}/S^{1}. For generic values of ε and δ, the preceding intersection will be transverse to the edge and thus, by dimension-counting, disjoint from the edge.

Because the intersection (3.80) is contained in the locus of smooth points of L̄_{𝔱′,𝔰}, it will be possible to define an intersection number once we have discussed the orientation of the link.

We note that the construction of the link in this section applies to links L̄_{𝔱′,𝔰} of M_{𝔰} × Sym^{ℓ}(X) when ℓ(𝔱′, 𝔰) > 1. The main difference is that the additional dilation parameters needed to describe a neighborhood of the strata M_{𝔰} × Sym^{ℓ}(X) result in more complicated ‘edges’ in the boundaries of this neighborhood. However, for generic choices of dilation parameters, these edges will be smooth submanifolds of codimension-one or greater and the argument in Lemma 3.9 will show that the geometric representatives are disjoint from these edges.

3.9. Orientations. We now discuss the orientations of the moduli spaces and links appearing in the SO(3)-monopole cobordism — in particular links of level-one Seiberg-Witten moduli spaces — with respect to which the pairings are defined. Orientations for moduli spaces and links of top-level Seiberg-Witten moduli spaces were discussed in [18, §2].

An orientation for M_{𝔱′} determines one for L_{𝔱′,𝔰} through the convention introduced in [18, Equations (2.16), (2.16) & (2.25)] by considering L_{𝔱′,𝔰} as a boundary of (M_{𝔱′}−γ(M_{𝔱′,𝔰}^{vir}))/S^{1}. Specifically, at a point [A, Φ] ∈ L_{𝔱′,𝔰}, if

• r⃗ ∈ T M_{𝔱′}^{∗,0} is an outward-pointing radial vector with respect to the open neighborhood M_{𝔱′} ∩ γ(M_{𝔱′,𝔰}^{vir}) and complementary to the tangent space of L_{𝔱′,𝔰},

• v_{S^{1}} ∈ T M_{𝔱′}^{∗,0} is tangent to the orbit of [A, Φ] under the (free) circle action (where S^{1} ⊂ ℂ has its usual orientation), and

• λ_{M} ∈ det(T M_{𝔱′}^{∗,0}) is an orientation for T M_{𝔱′} at [A, Φ], then we define an orientation λ_{L} for T L_{𝔱′,𝔰} at [A, Φ] by

λ_{M} = −v_{S^{1}} ∧ r⃗ ∧ λ̃_{L}, (3.81)

where the lift λ̃_{L} ∈ Λ^{max −2}(T M_{𝔱′}^{∗,0}) at [A, Φ] of λ_{L} ∈ det(T L_{𝔱′,𝔰}) ⊂ Λ^{max −1}(T (M_{𝔱′}^{∗,0}/S^{1})), obeys π_{∗}λ̃_{L} = λ_{L}, if π : M_{𝔱′} → M_{𝔱′}/S^{1} is the quotient map.

Definition 3.10. If O is an orientation for M_{𝔱′}, we call the orientation for L_{𝔱′,𝔰} related to O by equation (3.81) the boundary orientation defined by O.

We note that the orientation convention (3.81) can be applied more generally to define an orientation for a quotient ∂M/S^{1}, given an orientation for a manifold with boundary M with a free circle action.

We now introduce an orientation for L_{𝔱′,𝔰}, based on a choice of orientation for the component L_{𝔱′,𝔰}^{vir,i} (see (3.72)), which is useful for cohomological computations. First, observe that the gluing map γ identifies a relatively open subspace of L_{𝔱′,𝔰} with the zero-locus of the obstruction section χ in L_{𝔱′,𝔰}^{vir,i}. Because this section vanishes transversely, the normal bundle of the zero locus in L_{𝔱′,𝔰}^{vir,i} is identified with the obstruction bundle, Υ_{𝔱′,𝔰}/S^{1}, which has a complex orientation. If this normal bundle is given the complex orientation, orientations of L_{𝔱′,𝔰} are thus determined by orientations for L_{𝔱′,𝔰}^{vir,i}. From its definition (3.72), we see that L_{𝔱′,𝔰}^{vir,i} can be viewed as a complex disk bundle:

(3.82) L_{𝔱′,𝔰}^{vir,i} = Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}×S^{1}} ∂Ḡl_{𝔱}(δ) → M̃_{𝔰} ×_{G_{𝔰}×S^{1}} ∂Ḡl_{𝔱}(δ).

The circle action in (3.82), described as the second circle action in Lemma 3.6, is trivial on M̃_{𝔰}. Because the circle acts trivially on M̃_{𝔰} and G_{𝔰} acts trivially on the quotient ∂Ḡl_{𝔱}(δ)/S^{1}, we can identify the base of the bundle (3.82) with the following product, a compact manifold:

BL_{𝔱′,𝔰}^{vir,i} : = M̃_{𝔰} ×_{G_{𝔰}} ∂Ḡl_{𝔱}(δ)/S^{1}
≅ M_{𝔰} × ∂Ḡl_{𝔱}(δ)/S^{1}.

(3.83)

We may further assume without loss of generality that the point [A, Φ] ∈ L_{𝔱′,𝔰} ⊂ M_{𝔱′}/S^{1} — at which we choose to compare orientations — is identified, via the gluing map γ : M_{𝔱′,𝔰}^{vir} → C_{𝔱′}^{∗,0}, with a point

γ^{−1}([A, Φ]) = [[B, Ψ, 0], [g]],

in the base BL_{𝔱′,𝔰}^{vir,i} of the bundle (3.82), where [B, Ψ, 0] ∈ M_{𝔰} ⊂ N_{𝔱,𝔰}(ε) and g ∈ ∂Ḡl_{𝔱}(δ). The commutative diagram (3.89) shows that the fiber of the disk bundle (3.82) over the point [[B, Ψ, 0], [g]] is identified with the fiber of N_{𝔱,𝔰}(ε) over [B, Ψ]. We therefore have an isomorphism of tangent spaces at this point:

(3.84) T_{γ^{−1}([A,Φ])}L_{𝔱′,𝔰}^{vir,i} ≅ T_{0}(N_{𝔱,𝔰}|_{[B,Ψ]}) ⊕ T_{[B,Ψ]}M_{𝔰} ⊕ T_{[g]}(∂Ḡl_{𝔱}(δ)/S^{1}).

We define the standard orientation for ∂Ḡl_{𝔱}(δ)/S^{1} by applying the convention (3.81) to an orientation for Gl_{𝔱}(2δ), considering ∂Ḡl_{𝔱}(δ)/S^{1} as a boundary of Gl_{𝔱}(2δ)/S^{1} − Gl_{𝔱}(δ)/S^{1}. From its definition, we see that the space Gl_{𝔱}(2δ) is a locally-trivial fiber-bundle over X with fiber (0, 2δ) × SO(3), so the orientation for X and an orientation for (0, 2δ) × SO(3) determine an orientation for Gl_{𝔱}(2δ). The standard orientation for ∂Ḡl_{𝔱}(δ)/S^{1} will then be given through the convention (3.81) by taking the orientation of Gl_{𝔱}(2δ) induced by the orientation on (0, 2δ) × SO(3) given by identifying its tangent spaces with the space of quaternions, ℍ, with the tangent spaces to SO(3) being identified with Im(ℍ). (This agrees with the convention given in [11, §3(c) & §3(d)].)

Definition 3.11. The standard orientation for L_{𝔱′,𝔰}^{vir,i} is determined, through the isomorphism (3.84), by the

• Standard orientation for ∂Ḡl_{𝔱}(δ)/S^{1},

• Complex orientation of the fibers of the bundle (3.82), and the

• Orientation of M_{𝔰} induced by a homology orientation Ω [39, §6.6].

Definition 3.12. The standard orientation for L_{𝔱′,𝔰} is determined by the

• Standard orientation for L_{𝔱′,𝔰}^{vir,i},

• Complex orientation of the normal bundle of χ^{−1}(0) ∩ L_{𝔱′,𝔰}^{vir,i} in L_{𝔱′,𝔰}^{vir,i}, and the

• Diffeomorphism γ : χ^{−1}(0) ∩ M_{𝔱′,𝔰}^{vir} → M_{𝔱′} ∩ γ(M_{𝔱′,𝔰}^{vir}).

We now relate the two orientations of L_{𝔱′,𝔰} which we defined above:

Lemma 3.13. Suppose that the spin^{u} structure 𝔱 admits a splitting 𝔱 = 𝔰 ⊕ 𝔰 ⊗ L. Then the boundary orientation for L_{𝔱′,𝔰} defined by the orientation O^{asd}(Ω, c_{1}(L)) for M_{𝔱′} (see [18, Definition 2.3]) agrees with the standard orientation for L_{𝔱′,𝔰}.

Proof. The standard orientation for L_{𝔱′,𝔰} ⊂ M_{𝔱′}/S^{1} is defined, through the diffeomorphism

γ : χ^{−1}(0) ∩ M_{𝔱′,𝔰}^{vir} → M_{𝔱′} ∩ γ(M_{𝔱′,𝔰}^{vir}),

by applying the convention (3.81) to the submanifold χ^{−1}(0) ∩ M_{𝔱′,𝔰}^{vir} of M_{𝔱′,𝔰}^{vir}, while the orientation O^{asd}(Ω, c_{1}(L)) is defined by applying it to M_{𝔱′}. Thus, to compare the two orientations of L_{𝔱′,𝔰}, it suffices to compare the orientations of χ^{−1}(0) ∩ M_{𝔱′,𝔰}^{vir} and M_{𝔱′} which induce these orientations of L_{𝔱′,𝔰}.

An orientation for M_{𝔱′} is given by an orientation for the index bundle of the deformation operator D defined in [17, Equation (2.62)], whose kernel gives the tangent spaces of M_{𝔱′}^{∗,0}. Suppose that the point (A, Φ) ∈ M̃_{𝔱′} at which we do the orientation comparisons is obtained by gluing a framed spin^{u} connection A ∈ M̃_{1}^{s,♮}(S^{4}, δ) onto the background pair (B, Ψ) ∈ M̃_{𝔰} at a point x ∈ X:

(A, Φ) = (B, Ψ)#(A, 0). (3.85)

Note that we abuse notation here and omit explicit mention of the frame for 𝔤_{V}|_{s}. Then the excision argument [12, §7.1] yields the isomorphism

(3.86) det D_{[A,Φ]} ≅ det D_{[B,Ψ]} ⊗ det D_{[A,0]}.

There are isomorphisms (see [18, Equations (2.5), (2.11), and (2.26)]),

det D_{[B,Ψ]} ≅ det(N_{𝔱,𝔰}|_{[B,Ψ]}) ⊗ det(T_{[B,Ψ]}M_{𝔰}) ⊗ det(Ξ_{𝔱,𝔰}|_{[B,Ψ]})^{∗},
det D_{[A,0]} ≅ det(T_{[A]}M_{1}^{s}(S^{4}, δ)) ⊗ det(Coker D_{A})^{∗}
≅ det(T_{[A]}M_{1}^{s,♮}(S^{4}, δ)) ⊗ det(T_{x}X) ⊗ det(Coker D_{A})^{∗},

(3.87)

where in the last isomorphism we have used the identification M_{1}^{s}(S^{4}, δ) ≅ M_{1}^{s,♮}(S^{4}, δ) × ℝ^{4} and the identification of a ball in ℝ^{4} with a ball around x in X via the gluing map. Lemma 2.6 in [18] implies that the orientation O^{asd}(Ω, c_{1}(L)) of det D_{[A,Φ]} is given through the isomorphisms (3.86) and (3.87) by the complex orientations of N_{𝔱,𝔰}|_{[B,Ψ]}, (Ξ_{𝔱,𝔰}|_{[B,Ψ]})^{∗}, and (Coker D_{A})^{∗}, the orientation of T_{[B,Ψ]}M_{𝔰} determined by the homology orientation Ω [39, §6.6], the standard orientation of T_{[A]}M_{1}^{s,♮}(S^{4}, δ) defined in [11, p. 413], and the orientation for T_{x}X given by that of X.

We now describe the orientation of χ^{−1}(0) ∩ M_{𝔱′,𝔰}^{vir} inducing the standard orientation of L_{𝔱′,𝔰}. First, the orientation of χ^{−1}(0) ∩ M_{𝔱′,𝔰}^{vir} is induced by one for M_{𝔱′,𝔰}^{vir} by identifying the normal bundle of χ^{−1}(0) ∩ M_{𝔱′,𝔰}^{vir} in M_{𝔱′,𝔰}^{vir} with the obstruction bundle Υ_{𝔱′,𝔰} and using the complex orientation of this obstruction bundle. With [A, Φ] as in (3.85), then by (3.61), (3.63), and (3.68) there is an isomorphism of obstruction bundle fibers,

(3.88) Υ_{𝔱′,𝔰}|_{γ^{−1}([A,Φ])} ≅ Coker D_{A} ⊕ Ξ_{𝔱,𝔰}|_{[B,Ψ]}.

Hence, the complex orientation of the obstruction bundle matches the complex orientation of the factors Coker D_{A} and Ξ_{𝔱,𝔰}|_{[B,Ψ]} in (3.87).

We now compare the tangent space T_{γ^{−1}([A,Φ])}M_{𝔱′,𝔰}^{vir} with the remaining factors on the right-hand-sides of the isomorphisms (3.87). The moduli space M_{𝔱′,𝔰}^{vir} can also be written as the disk bundle appearing on the left-hand-side of the following diagram:

Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Gl_{𝔱}(δ) ⟶ N_{𝔱,𝔰}(ε) × X
π_{N,Gl}↓  ^{π_{N,X}} ↓
M̃_{𝔰} ×_{G_{𝔰}} Gl_{𝔱}(δ) ⟶^{π_{Gl}} M_{𝔰} × X

(3.89)

We further suppose, without loss of generality, that [A, Φ] ∈ L_{𝔱′,𝔰} corresponds, via the gluing map γ, to a point γ^{−1}([A, Φ]) in the base of the bundle on the left-hand-side of the diagram (3.89), so

π_{Gl}(γ^{−1}([A, Φ])) = ([B, Ψ], x).

The fiber of the projection π_{Gl} in diagram (3.89) is M_{1}^{s,♮}(S^{4}, δ), while the same diagram identifies the fiber of the projection π_{N,Gl} with the fiber N_{𝔱,𝔰}(ε)|_{[B,Ψ]} of the projection π_{N,X}. Thus,

T_{γ^{−1}([A,Φ])}M_{𝔱′,𝔰}^{vir} ≅ N_{𝔱,𝔰}|_{[B,Ψ]} ⊕ T_{γ^{−1}([A,Φ])}(M̃_{𝔰} ×_{G_{𝔰}} Gl_{𝔱}(δ))
≅ N_{𝔱,𝔰}|_{[B,Ψ]} ⊕ T_{[B,Ψ]}M_{𝔰} ⊕ T_{[A]}M_{1}^{s,♮}(S^{4}, δ) ⊕ T_{x}X.

(3.90)

If we compare the isomorphisms (3.90) with (3.84), we see that the standard orientation of L_{𝔱′,𝔰} is induced, through the convention (3.81), by the orientation of χ^{−1}(0) ∩ M_{𝔱′,𝔰}^{vir} given by the complex orientation of the normal bundle of χ^{−1}(0) ∩ M_{𝔱′,𝔰}^{vir} and the orientation of M_{𝔱′,𝔰}^{vir} defined through the isomorphism (3.90) by the complex orientation of N_{𝔱,𝔰}|_{[B,Ψ]}, the orientation of T_{[B,Ψ]}M_{𝔰} determined by the homology orientation Ω, the standard orientation of T_{[A]}M_{1}^{s,♮}(S^{4}, δ), and the orientation of T_{x}X. This orientation matches the orientation O^{asd}(Ω, c_{1}(L)) as described in the paragraph following (3.87). This completes the proof.

We shall work with a fixed orientation O^{asd}(Ω, w) of M_{𝔱′} in the sum (2.52) and thus we include the following lemma on how the orientations change as the spin^{c} structure 𝔰 varies.

Lemma 3.14. If 𝔱 is a spin^{u} structure on X, let 𝔱′ be the spliced spin^{u} structure (3.38). Let Ω be a homology orientation and let w be an integral lift of w_{2}(𝔱′). If 𝔱 admits a splitting 𝔱 = 𝔰 ⊕ 𝔰 ⊗ L, then the standard orientation for L_{𝔱′,𝔰} and the boundary orientation for L_{𝔱′,𝔰} defined through the orientation O^{asd}(Ω, w) for M_{𝔱′} differ by a factor of

(−1)^{o_{𝔱′}(w,𝔰)}, where o_{𝔱′}(w, 𝔰) = \frac{1}{4}(w − c_{1}(L))^{2}. (3.91)

Proof. The result follows from Lemma 3.13 and the identity

O^{asd}(Ω, w) = (−1)\frac{1}{4}(w − c_{1}(L))^{2}O^{asd}(Ω, c_{1}(L))

given in [18, Lemma 2.6].

4. Pullback and extension of cohomology classes. Euler classes of obstruction bundles

In §4.1 we compute the pullbacks of the cohomology classes μ_{p}(β) and μ_{c} to M_{𝔱′,𝔰}^{vir}/S^{1} with respect to the gluing map γ : M_{𝔱′,𝔰}^{vir}/S^{1} → C_{𝔱′}^{∗,0}/S^{1}. When no confusion can arise, we denote the pullbacks γ^{∗}μ_{p}(β) and γ^{∗}μ_{c} by μ_{p}(β) and μ_{c}, respectively. We then observe that these pullbacks are the restrictions of cohomology classes μ̄_{p}(β) and μ̄_{c} on M̄_{𝔱′,𝔰}^{vir,∗}/S^{1}, where M̄_{𝔱′,𝔰}^{vir,∗} is defined in (4.1). In §4.2 we calculate Euler classes of the obstruction bundle (3.59) over M_{𝔱′,𝔰}^{vir}/S^{1}. We will see that the background component (3.63) of the obstruction bundle (3.59) (and hence its Euler class) extends over M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} while the Euler class of the instanton component (3.69) extends over M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} as a rational cohomology class. Although we only perform the calculations relevant to the stratum

(M_{𝔰} × Sym^{ℓ}(X)) ∩ M̄_{𝔱′} ⊂ M̄_{𝔱′}

when ℓ(𝔱′, 𝔰) = 1 in the present article, we shall usually indicate the nature of the changes required to address the general case ℓ ≥ 1.

4.1. Pullbacks of cohomology classes. By analogy with our definition (following (2.13)) of M̄_{𝔱}^{∗} as the subspace of M̄_{𝔱} represented by ideal SO(3) monopoles with reducible associated SO(3) connections, we define

M̄_{𝔱′,𝔰}^{vir,∗} = M̄_{𝔱′,𝔰}^{vir} − (M_{𝔰} × X), (4.1)

together with an inclusion map

ι : M_{𝔱′,𝔰}^{vir}/S^{1} → M̄_{𝔱′,𝔰}^{vir,∗}/S^{1}. (4.2)

The circle action on M̄_{𝔱′,𝔰}^{vir}, given in Lemma 3.6, is free on M̄_{𝔱′,𝔰}^{vir,∗} and trivial on M_{𝔰} × X.

Definition 4.1. Let ν be the first Chern class of the circle bundle

M̄_{𝔱′,𝔰}^{vir,∗} → M̄_{𝔱′,𝔰}^{vir,∗}/S^{1}, (4.3)

where the circle acts diagonally on M̄_{𝔱′,𝔰}^{vir} = Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Ḡl_{𝔱}(δ), acting on Ñ_{𝔱,𝔰}(ε) by scalar multiplication on the fibers and on Ḡl_{𝔱}(δ) by the action described before equation (3.56).

Recall that μ_{c} ∈ H^{2}(C_{𝔱′}^{∗,0}/S^{1}; ℤ) is the first Chern class (2.44) of the line bundle 𝕃_{𝔱′} → C_{𝔱′}^{∗,0}/S^{1} defined in (2.42); we now identify its pullback to M_{𝔱′,𝔰}^{vir}/S^{1}.

Lemma 4.2. Let μ_{c} be as in the preceding paragraph, let ν be as in Definition 4.1, and let ι be the inclusion (4.2). Then

γ^{∗}μ_{c} = −ι^{∗}ν.

Proof. From (2.44) and (2.42), the cohomology class μ_{c} is the first Chern class of

𝕃_{𝔱′} = C_{𝔱′}^{∗,0} ×_{(S^{1},×−2)} ℂ.

By Lemma 3.6, the gluing map γ : M_{𝔱′,𝔰}^{vir} → C_{𝔱′}^{∗,0} is circle-equivariant when the circle acts on C_{𝔱′}^{∗,0} by the action (2.8) and on M_{𝔱′,𝔰}^{vir} by the action in Definition 4.1, but with multiplicity two. Thus,

γ^{∗}𝕃_{𝔱′} = M_{𝔱′,𝔰}^{vir} ×_{(S^{1},−1)} ℂ,

and the conclusion follows.

Lemma 4.2 and its proof translate easily to the case ℓ ≥ 1.

Next, we identify the pullbacks to M_{𝔱′,𝔰}^{vir} of the cohomology classes μ_{p}(β) on C_{𝔱′}^{∗,0}. Let

(4.4) π_{𝔰} : M̄_{𝔱′,𝔰}^{vir,∗} → M_{𝔰} and π_{X} : M̄_{𝔱′,𝔰}^{vir,∗} → X denote the restrictions of the components of the projection M̄_{𝔱′,𝔰}^{vir} → M_{𝔰} × X given in (3.58) to M̄_{𝔱′,𝔰}^{vir,∗}. We define some additional projections:

π_{M} : M̄_{𝔱′,𝔰}^{vir,∗} × X → M̄_{𝔱′,𝔰}^{vir,∗},
π_{X,2} : M̄_{𝔱′,𝔰}^{vir,∗} × X → X,
π_{𝔰,1} = π_{𝔰} ∘ π_{M} : M̄_{𝔱′,𝔰}^{vir,∗} × X → M_{𝔰},

(4.5)

We shall use the same notation for the projections when the space M̄_{𝔱′,𝔰}^{vir,∗} above is replaced, for example, by its circle quotient, M̄_{𝔱′,𝔰}^{vir,∗}/S^{1}.

Recall from definition (2.39) that

𝔽_{𝔱′} = C̃_{𝔱′}^{∗}/S^{1} ×_{G_{𝔱′}} 𝔤_{V′}

is a universal SO(3) bundle over C_{𝔱′}^{∗}/S^{1} ×X. We now define an analogous SO(3) bundle over M_{𝔱′,𝔰}^{vir}/S^{1} × X. Using ‘cl’ for convenience here to indicate the ‘Uhlenbeck compactification’ implicit when Gl_{𝔱}(δ) is replaced by Ḡl_{𝔱}(δ), we denote

clM̃_{𝔱′,𝔰}^{vir,∗} = Ñ_{𝔱,𝔰}(ε) × Ḡl_{𝔱}(δ) − (M̃_{𝔰} × X) , (4.6)

so that M̄_{𝔱′,𝔰}^{vir,∗} = clM̃_{𝔱′,𝔰}^{vir,∗}/G_{𝔰}, where G_{𝔰} acts diagonally on Ñ_{𝔱,𝔰}(ε) × Ḡl_{𝔱}(δ). We set

(4.7)  𝔽̄_{𝔱′,𝔰}^{vir,∗} = clM̃_{𝔱′,𝔰}^{vir,∗} ×_{G_{𝔰}×S^{1}} (\underline{ℝ} ⊕ L) → M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} × X,

where G_{𝔰} acts diagonally on clM̃_{𝔱′,𝔰}^{vir,∗} × 𝔤_{V} — acting on 𝔤_{V} ≅ \underline{ℝ} ⊕ L by multiplication with weight negative two on L and on clM̃_{𝔱′,𝔰}^{vir,∗} by the action described above; the circle acts diagonally on clM̃_{𝔱′,𝔰}^{vir,∗} × 𝔤_{V} — by scalar multiplication on Ñ_{𝔱,𝔰}(ε), by the action on Ḡl_{𝔱}(δ) described before equation (3.56), and on 𝔤_{V} by scalar multiplication with weight one on L.

Lemma 4.3 below compares the restriction of the bundle 𝔽̄_{𝔱′,𝔰}^{vir,∗} with the restriction of the pulled-back SO(3) bundle (γ × id_{X})^{∗}𝔽_{𝔱′} to the complement of (π_{X} × id_{X})^{−1}(Δ) in M_{𝔱′,𝔰}^{vir}/S^{1} × X, where Δ ⊂ X × X is the diagonal. We restrict the pullback of the universal bundle to the complement of the subspace (π_{X} × id_{X})^{−1}(Δ) because the splicing process at a point x ∈ X only identifies the restrictions of the bundles 𝔤_{V} and 𝔤_{V′} to X \ {x}.

Lemma 4.3. Suppose the spin^{u} structure 𝔱 = (ρ, V) admits a splitting 𝔱 = 𝔰 ⊕ 𝔰 ⊗ L, so that 𝔤_{V} ≅ \underline{ℝ} ⊕ L, where \underline{ℝ} = X × ℝ. If O denotes the complement of (π_{X} × id_{X})^{−1}(Δ) in M_{𝔱′,𝔰}^{vir}/S^{1} × X, then there is an isomorphism of SO(3) bundles:

(γ × id_{X})^{∗}𝔽_{𝔱′}|_{O} ≅ 𝔽̄_{𝔱′,𝔰}^{vir,∗}|_{O} .

Proof. Let 𝔱′ = (ρ, V′) be the spin^{u} structure (3.38) obtained by splicing, over a neighborhood of x ∈ X, the spin^{u} structure 𝔱 = (ρ, V) over X with the spin^{u} structure (ρ, V) over S^{4}, where −\frac{1}{4}p_{1}(𝔤_{V}) = 1. We thus obtain an associated SO(3) bundle 𝔤_{V′} as in (3.40) and a bundle isomorphism,

ι_{V,V′} : 𝔤_{V} |_{X\{x}} → 𝔤_{V′}|_{X\{x}}.

Hence, if Õ ⊂ Ñ_{𝔱,𝔰}(ε) × G̃l_{𝔱}(δ) × X is the pre-image of O under the obvious projection, we have a bundle map

γ̃′ × ι_{V,V′} : (Ñ_{𝔱,𝔰}(ε) × G̃l_{𝔱}(δ) × 𝔤_{V} )|Õ → C̃_{𝔱′}^{∗} × 𝔤_{V′}, (4.8) where γ̃′ is the pre-splicing map (3.50). The map γ̃′ is gauge equivariant with respect to the action of G_{𝔰} × G_{1} on the domain and the action of G_{𝔱′} on the range. The group G_{1} of gauge transformations over S^{4} (see §3.3) acts trivially on the restriction of 𝔤_{V} to the complement of the splicing point x ∈ X. From definitions (3.51) and (4.6) we have

Ñ_{𝔱,𝔰}(ε) × G̃l_{𝔱}(δ)/G_{1} = ι^{∗}clM̃_{𝔱′,𝔰}^{vir,∗},

(where ι is the inclusion in (4.2)) and so the bundle map (4.8) descends to a bundle map on gauge-group quotients,

γ′ × ι_{V,V′} : (clM̃_{𝔱′,𝔰}^{vir,∗} ×_{G_{𝔰}} 𝔤_{V} )|_{O} → C̃_{𝔱′}^{∗} ×_{G_{𝔱′}} 𝔤_{V′}. (4.9)

By Lemma 3.6, the bundle map (4.9) is circle-equivariant with respect to the circle action on the domain induced by the action (2.37) on the factor Ñ_{𝔱,𝔰}(ε) in clM̃_{𝔱′,𝔰}^{vir,∗} together with the trivial action on Gl_{𝔱}(δ) and the circle action on C̃_{𝔱′}^{∗,0} induced by (2.8). The bundle map (4.9) thus descends to a bundle map on circle quotients,

γ′ × ι_{V,V′} : (clM̃_{𝔱′,𝔰}^{vir,∗}/S^{1} ×_{G_{𝔰}} 𝔤_{V} )|_{O} → 𝔽_{𝔱′}. (4.10)

The argument in the proof of Lemma 3.6 which shows that the two circle actions on M_{𝔱′,𝔰}^{vir} (described in its hypothesis) are equal then implies that the circle action on clM̃_{𝔱′,𝔰}^{vir,∗} × 𝔤_{V} induced by the circle action (2.37) on the factor Ñ_{𝔱,𝔰}(ε) in clM̃_{𝔱′,𝔰}^{vir,∗} × 𝔤_{V} and trivial actions on Gl_{𝔱}(δ) and on 𝔤_{V} is equal to the twice the circle action described following (4.7). The multiplicity of this action does not affect the quotient, so the bundle given as the domain of the map (4.10) is isomorphic to 𝔽̄_{𝔱′,𝔰}^{vir,∗} and hence the restrictions of (γ′ × id_{X})^{∗}𝔽_{𝔱′} and 𝔽̄_{𝔱′,𝔰}^{vir,∗} to O are isomorphic. Finally, because the gluing map γ and the splicing map γ′ are S^{1}-equivariantly homotopic, there is a bundle isomorphism (γ × id_{X})^{∗}𝔽_{𝔱′} ≅ (γ′ × id_{X})^{∗}𝔽_{𝔱′}. This completes the proof of the lemma.

Since μ_{p}(β) = −\frac{1}{4}p_{1}(𝔽_{𝔱′})/β, our next task is to compare the Pontrjagin classes of the bundles ι^{∗}𝔽̄_{𝔱′,𝔰}^{vir,∗} and (γ × id_{X})^{∗}𝔽_{𝔱′} over M_{𝔱′,𝔰}^{vir} × X:

Lemma 4.4. Continue the hypotheses of Lemma 4.3. Let PD[Δ] be the Poincaré dual of the diagonal Δ ⊂ X × X. Then, on M_{𝔱′,𝔰}^{vir} × X, we have

(4.11) (γ × id_{X})^{∗}p_{1}(𝔽_{𝔱′}) = (ι × id_{X})^{∗} (p_{1}(𝔽̄_{𝔱′,𝔰}^{vir,∗}) − 4(π_{X} × id_{X})^{∗} PD[Δ]) ,

where ι is the inclusion (4.2).

Proof. The restriction of the difference,

(γ × id_{X})^{∗}p_{1}(𝔽_{𝔱′}) − (ι × id_{X})^{∗}p_{1}(𝔽̄_{𝔱′,𝔰}^{vir,∗}), (4.12)

to the subspace O defined in Lemma 4.3 vanishes by that lemma. Therefore, by considering the exact sequence of the pair,

(M_{𝔱′,𝔰}^{vir}/S^{1} × X, O) , (4.13)

we see that the difference (4.12) lies in the image of the homomorphism

H^{4} (M_{𝔱′,𝔰}^{vir}/S^{1} × X, O; ℝ) → H^{4} (M_{𝔱′,𝔰}^{vir}/S^{1} × X; ℝ) (4.14)

appearing in the exact sequence of the pair (4.13). Because the map π_{X} × id_{X} is transverse to Δ, the image of the homomorphism (4.14) (see [5, p. 69] and [10, Proposition VIII.11.10]) is generated by (π_{X} × id_{X})^{∗} PD[Δ]. The difference (4.12) is therefore a multiple of (π_{X} × id_{X})^{∗} PD[Δ]. One can calculate this multiple by evaluating the difference (4.12) on a chain which intersects (π_{X} × id_{X})^{−1}(Δ) transversely at a single point. Such computations are carried out in the proof of [23, Theorem III.6.1] and in [12, §5.3], giving equation (4.11)

We see from (4.7) that the cohomology class on the right-hand side of (4.11) is defined on M̄_{𝔱′,𝔰}^{vir,∗}/S^{1}×X and thus extends the class (γ×id_{X})^{∗}p_{1}(𝔽_{𝔱′}), which is defined on M_{𝔱′,𝔰}^{vir}/S^{1}×X.

Definition 4.5. We define extensions of cohomology classes from M_{𝔱′,𝔰}^{vir}/S^{1} to M̄_{𝔱′,𝔰}^{vir,∗}/S^{1}:

• Let μ̄_{c} = −ν be the extension of μ_{c} given by Lemma 4.2.

• For β ∈ H_{•}(X; ℝ), let μ̄_{p}(β) be the extension of (the pullback by γ of) the cohomology class μ_{p}(β) = −\frac{1}{4}p_{1}(𝔽_{𝔱′})/β given by replacing (γ × id_{X})^{∗}p_{1}(𝔽_{𝔱′}) with the cohomology class on the right-hand side of (4.11).

In order to use equation (4.11) to compute (γ × id_{X})^{∗}p_{1}(𝔽_{𝔱′}), we must identify the Pontrjagin class p_{1}(𝔽̄_{𝔱′,𝔰}^{vir,∗}). The bundle 𝔽̄_{𝔱′,𝔰}^{vir,∗} admits a reduction 𝔽̄_{𝔱′,𝔰}^{vir,∗} ≅ \underline{ℝ} ⊕ 𝕃̄_{𝔱′,𝔰}^{vir,∗}, where \underline{ℝ} = M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} × X × ℝ and

(4.15)  𝕃̄_{𝔱′,𝔰}^{vir,∗} = clM̃_{𝔱′,𝔰}^{vir,∗} ×_{G_{𝔰}×S^{1}} L → M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} × X.

The actions of S^{1} and G_{𝔰} in the definition of the line bundle 𝕃̄_{𝔱′,𝔰}^{vir,∗} are described in Lemma 4.3. Since p_{1}(𝔽̄_{𝔱′,𝔰}^{vir,∗}) = c_{1}(𝕃̄_{𝔱′,𝔰}^{vir,∗})^{2}, it suffices to compute c_{1}(𝕃̄_{𝔱′,𝔰}^{vir,∗}) and for this purpose we shall use the following technical tool:

Lemma 4.6. [17, Lemma 3.27] If Q_{i}, for i = 1, 2, are circle bundles over a topological space M and L_{i} = Q_{i} ×_{S^{1}} ℂ are the associated complex line bundles, and k ∈ ℤ, then the following hold:

1. If V is a complex vector bundle over M, and e^{iθ} ∈ S^{1} acts on the fiber product Q_{1}×_{M} V by e^{iθ} · (q_{1}, v) = (e^{iθ}q_{1}, e^{ikθ}v), then

(Q_{1} ×_{M} V)/S^{1} ≅ L_{1}^{−k} ⊗ V.

2. If e^{iθ} ∈ S^{1} acts on the fiber product Q_{1} ×_{M} Q_{2} by e^{iθ} · (q_{1}, q_{2}) = (e^{iθ}q_{1}, e^{ikθ}q_{2}), then the first Chern class of the circle bundle (Q_{1} ×_{M} Q_{2})/S^{1} → M is

c_{1}((Q_{1} ×_{M} Q_{2})/S^{1}) = c_{1}(Q_{2}) − kc_{1}(Q_{1}),

where the circle action on (Q_{1} ×_{M} Q_{2})/S^{1} is induced by the circle action on Q_{2} of weight one.

Recall that μ_{𝔰} ∈ H^{2}(M_{𝔰}; ℤ) is the first Chern class of the base-point fibration over M_{𝔰} (see the remark following equation (2.21)); when there is no ambiguity we may write μ_{𝔰} for the pullback π_{𝔰}^{∗}μ_{𝔰} to the total space M̄_{𝔱′,𝔰}^{vir}/S^{1}. Recall from §2.3.1 that if b_{1}(X) = 0 then μ_{𝔰}×1 = c_{1}(𝕃_{𝔰}), where 𝕃_{𝔰} → C_{𝔰}^{0}×X is the line bundle (2.19). Moreover, c_{1}(L) = c_{1}(𝔱)−c_{1}(𝔰) in the definition (4.15) of the line bundle 𝕃̄_{𝔱′,𝔰}^{vir,∗}.

In the following we will use 𝕃_{𝔰} to denote the restriction of the bundle 𝕃_{𝔰} → C_{𝔰}^{0} × X to the subspace M_{𝔰} × X.

Lemma 4.7. Continue the hypotheses of Lemma 4.3. Let

𝕃_{ν} → M̄_{𝔱′,𝔰}^{vir,∗}/S^{1}

be the complex line bundle associated to the circle bundle (4.3). Then there is an isomorphism of complex line bundles over M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} × X:

𝕃̄_{𝔱′,𝔰}^{vir,∗} ≅ π_{M}^{∗}𝕃_{ν}^{−1} ⊗ (π_{𝔰,1} × id_{X})^{∗}𝕃_{𝔰}^{⊗2} ⊗ π_{X,2}^{∗}L. (4.16)

Proof. The argument yielding equation (3.68) in [17] implies that if G_{𝔰} acts diagonally on M̃_{𝔰} × L with weight negative two on L, then we have an isomorphism of line bundles over M_{𝔰} × X:

M̃_{𝔰} ×_{G_{𝔰}} L ≅ 𝕃_{𝔰}^{⊗2} ⊗ π_{X,𝔰}^{∗}L, (4.17)

where π_{X,𝔰} : M_{𝔰} × X → X is the projection. By the preceding isomorphism and the definition (4.5) of π_{𝔰} × id_{X}, we have an isomorphism of line bundles over M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} × X:

(π_{𝔰} × id_{X})^{∗}(𝕃_{𝔰}^{⊗2} ⊗ π_{X,𝔰}^{∗}L) ≅ (M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} × X) ×_{M_{𝔰}×X} (M̃_{𝔰} ×_{G_{𝔰}} L) .

Consequently, from this isomorphism we see that, as bundles over M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} × X,

(M̄_{𝔱′,𝔰}^{vir,∗} × X) ×_{M̄_{𝔱′,𝔰}^{vir,∗}/S^{1}×X} (π_{𝔰,1} × id_{X})^{∗}(𝕃_{𝔰}^{⊗2} ⊗ π_{X,𝔰}^{∗}L)
≅ (M̄_{𝔱′,𝔰}^{vir,∗} × X) ×_{M_{𝔰}×X} (M̃_{𝔰} ×_{G_{𝔰}} L) .

(4.18)

Now, consider the circle quotient of the bundle on the right-hand side of the preceding isomorphism,

(M̄_{𝔱′,𝔰}^{vir,∗} × X ×_{M_{𝔰}×X} (M̃_{𝔰} ×_{G_{𝔰}} L)) /S^{1} → M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} × X, (4.19)

where the circle acts diagonally on the factors M̄_{𝔱′,𝔰}^{vir,∗} and L. The definition of 𝕃_{ν} as the line bundle associated to the circle bundle (4.3), Lemma 4.6 and the isomorphism (4.18) then yield the bundle isomorphisms over M̄_{𝔱′,𝔰}^{vir,∗} × X:

\begin{matrix}((M̄_{𝔱′,𝔰}^{vir,∗} × X) ×_{M_{𝔰}×X} (M̃_{𝔰} ×_{G_{𝔰}} L)) /S^{1} \\ ≅ π_{M}^{∗}𝕃_{ν}^{−1} ⊗ (π_{𝔰,1} × id_{X})^{∗}(𝕃_{𝔰}^{⊗2} ⊗ π_{X,𝔰}^{∗}L) \\ ≅ π_{M}^{∗}𝕃_{ν}^{−1} ⊗ (π_{𝔰,1} × id_{X})^{∗}𝕃_{𝔰}^{⊗2} ⊗ π_{X,2}^{∗}L. \end{matrix} (4.20)

One can check that the map defined below is an isomorphism of line bundles over M̄_{𝔱′,𝔰}^{vir,∗}×X,

(4.21)  𝕃̄_{𝔱′,𝔰}^{vir,∗} = clM̃_{𝔱′,𝔰}^{vir,∗} ×_{G_{𝔰}×S^{1}} L ≅ ((M̄_{𝔱′,𝔰}^{vir,∗} × X) ×_{M_{𝔰}×X} (M̃_{𝔰} ×_{G_{𝔰}} L)) /S^{1},

given for (B, Ψ) ∈ M̃_{𝔰}, (B, Ψ, η) ∈ Ñ_{𝔱,𝔰}(ε), g ∈ Gl_{𝔱}(δ), and ζ ∈ L|_{x} by

[(B, Ψ, η), g, ζ] ↦ (([(B, Ψ, η), g], x), [B, Ψ, ζ]) .

Therefore, the isomorphisms (4.21) and (4.20) give the desired isomorphism (4.16).

Corollary 4.8. Continue the hypotheses of Lemma 4.7. Then, on M̄_{𝔱′,𝔰}^{vir,∗} × X, we have

c_{1}(𝕃̄_{𝔱′,𝔰}^{vir,∗}) = 2(π_{𝔰,1} × id_{X})^{∗}c_{1}(𝕃_{𝔰}) − π_{M}^{∗}ν + π_{X,2}^{∗}c_{1}(L). (4.22)

Lemma 4.4, equation (4.22), and the relation p_{1}(𝔽̄_{𝔱′,𝔰}^{vir,∗}) = c_{1}(𝕃̄_{𝔱′,𝔰}^{vir,∗})^{2} then yield:

Corollary 4.9. Continue the hypotheses of Lemma 4.3. Let PD[Δ] be the Poincaré dual of the diagonal Δ ⊂ X × X, let ν be the class in Definition 4.1, and let ι be the inclusion (4.2). Then, on M_{𝔱′,𝔰}^{vir}/S^{1}, we have

(γ × id_{X})^{∗}p_{1}(𝔽_{𝔱}) = (ι × id_{X})^{∗} (2(π_{𝔰,1} × id_{X})^{∗}c_{1}(𝕃_{𝔰}) − π_{M}^{∗}ν + π_{X,2}^{∗}c_{1}(L))^{2}

− 4(ι × id_{X})^{∗}(π_{X} × id_{X})^{∗} PD[Δ]. (4.23)

The equation (4.23) for (γ × id_{X})^{∗}p_{1}(𝔽_{𝔱′}) yields the following formulae for the pullback of μ_{p}(β) = −\frac{1}{4}p_{1}(𝔽_{𝔱′})/β to M_{𝔱′,𝔰}^{vir}/S^{1}:

Lemma 4.10. Continue the hypotheses of Lemma 4.3. Let ι be the inclusion (4.2), let x ∈ H_{0}(X; ℤ) be the positive generator, and let h ∈ H_{2}(X; ℝ). Let ν, π_{𝔰}^{∗}μ_{𝔰} ∈ H^{2}(M̄_{𝔱′,𝔰}^{vir,∗}/S^{1}; ℤ) be the classes given by Definition 4.1 and equation (2.21), respectively, while π_{𝔰} is the projection (4.4). Then, on M_{𝔱′,𝔰}^{vir}/S^{1}, we have

μ_{p}(x) = −\frac{1}{4}ι^{∗}(2π_{𝔰}^{∗}μ_{𝔰} − ν)^{2} + ι^{∗}π_{X}^{∗} PD[x] ∈ H^{4}(M_{𝔱′,𝔰}^{vir}/S^{1}; ℝ),
μ_{p}(h) = \frac{1}{2}ι^{∗}⟨c_{1}(𝔰) − c_{1}(𝔱), h⟩(2π_{𝔰}^{∗}μ_{𝔰} − ν) + ι^{∗}π_{X}^{∗} PD[h] ∈ H^{2}(M_{𝔱′,𝔰}^{vir}/S^{1}; ℝ). (4.24)

Proof. If β ∈ H_{•}(X; ℝ), then PD[Δ]/β = PD[β] by [28, Theorem 30.6]. Thus,

(π_{X} × id_{X})^{∗} PD[Δ]/β = π_{X}^{∗} PD[β].

The assertions now follow from Corollary 4.9 and standard computations (see, for example, [12, §5.1.4]).

Finally, we note that the results of §4.1, bearing on cohomology classes on a neighborhood M_{𝔰} × X, will extend (with appropriate modifications, though these do not cause undue difficulty) to the case of a neighborhood of M_{𝔰} × Sym^{ℓ}(X) when ℓ > 1. For example, one must replace the diagonal Δ ⊂ X × X appearing in Lemma 4.3 with an incidence subset,

{(x, x) ∈ Sym^{ℓ}(X) × X : x ∈ |x|},

where |x| ⊂ X is the support of a point x ∈ Sym^{ℓ}(X).

4.2. Euler class of the obstruction bundle. We first compute the Euler class of the background component Ῡ_{𝔱′,𝔰}^{s}/S^{1} → M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} of the obstruction bundle defined in (3.64).

Lemma 4.11. Continue the hypotheses of Lemma 4.3. Let ν be the class in Definition 4.1. Let Ῡ_{𝔱′,𝔰}^{s}/S^{1} → M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} be the extended Seiberg-Witten obstruction bundle (3.64), and let r_{Ξ} denote its complex rank. Then

e(Ῡ_{𝔱′,𝔰}^{s}/S^{1}) = (−ν)^{r_{Ξ}} ∈ H^{2r_{Ξ}}(M̄_{𝔱′,𝔰}^{vir,∗}/S^{1}; ℤ). (4.25)

Proof. The S^{1} quotient of the bundle Ῡ_{𝔱′,𝔰}^{s} by the action described at the end of §3.6.1 can be written as

Ῡ_{𝔱′,𝔰}^{s}/S^{1} ≅ M̄_{𝔱′,𝔰}^{vir,∗} ×_{(S^{1},−1)} ℂ^{r_{Ξ}}, (4.26)

with the factor of negative one appearing because the action is diagonal, as explained in Lemma 4.6. Note that the circle action described in Lemma 3.6 is twice the action in the definition of ν (see Definition 4.1), so the weight two action on ℂ^{r_{Ξ}} described at the end of §3.6.1 becomes a weight one action here. Equation (4.25) then follows immediately from equation (4.26) and the definition of ν.

We now compute the Euler class of the instanton component (3.68) of the obstruction bundle when ℓ(𝔱′, 𝔰) = 1; more work is required to extend this calculation to the case ℓ(𝔱′, 𝔰) > 1. For the proof of the following lemma, it is convenient to define

(4.27) ∂^{i}M_{𝔱′,𝔰}^{vir} = Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} ∂Ḡl_{𝔱}(δ) ⊂ M_{𝔱′,𝔰}^{vir},

so that L_{𝔱′,𝔰}^{vir,i} = ∂^{i}M_{𝔱′,𝔰}^{vir}/S^{1}.

Lemma 4.12. Let ν be the class in Definition 4.1. Then the Euler class of the instanton obstruction bundle, Υ_{𝔱′,𝔰}^{i}/S^{1} → M_{𝔱′,𝔰}^{vir}/S^{1} defined in (3.68), is given as an element of rational cohomology by

e(Υ_{𝔱′,𝔰}^{i}/S^{1}) = \frac{1}{2} (π_{X}^{∗}c_{1}(𝔱) − ν) ∈ H^{2}(M_{𝔱′,𝔰}^{vir}/S^{1}; ℚ). (4.28)

Proof. Because M_{𝔱′,𝔰}^{vir}/S^{1} retracts onto ∂^{i}M_{𝔱′,𝔰}^{vir}/S^{1} = L_{𝔱′,𝔰}^{vir,i}, as one can see from the definition (3.72) of L_{𝔱′,𝔰}^{vir,i}, it suffices to compute the Euler class of the restriction of e(Υ_{𝔱′,𝔰}^{i}/S^{1}) to ∂^{i}M_{𝔱′,𝔰}^{vir}/S^{1}. It is easier to compute the Euler class of the bundle (Υ_{𝔱′,𝔰}^{i}/S^{1})^{⊗2}. The observation (following from (3.66)) that

(Coker D)^{⊗2} ≅ SU(2) ×_{{±id}} (ℂ ⊗ ℂ) = SO(3) × ℂ,

Lemma 3.7, and the description of Υ_{𝔱′,𝔰}^{i} in (3.68) imply that

(Υ_{𝔱′,𝔰}^{i})^{⊗2}|_{∂^{i}M_{𝔱′,𝔰}^{vir}} ≅ Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} Fr_{ℂℓ(T^{∗}X)}(V) ×_{Spin^{u}(4)} (SO(3) × ℂ).

By the description of the circle action on the bundle Υ_{𝔱′,𝔰}^{i} prior to its definition (3.68), the preceding isomorphism is circle-equivariant if the circle acts on the factor Ñ_{𝔱,𝔰}(ε) by the action (2.37) and with weight two on the fiber ℂ and trivially on the remaining factors; the weight of the circle action on the fiber ℂ is two because Υ_{𝔱′,𝔰}^{i} appears in the tensor-product square on the left-hand side. The action of Spin^{u}(4) on the vector bundle (Coker D)^{⊗2} → M_{1}^{s,♮}(S^{4}, δ) is given, for (M̂, ζ) ∈ SO(3) × ℂ = (Coker D)^{⊗2} and Ũ ∈ Spin^{u}(4), by

(M̂, ζ) ↦ (Ad(Ũ)M̂, det^{u}(Ũ)^{−1}ζ), (4.29)

where the homomorphisms det^{u} : Spin^{u}(4) → S^{1} and Ad = (Ad^{u}, Ad^{𝔤}) : Spin^{u}(4) → SO(4) × SO(3) are defined in (3.11) and (3.13), respectively. To see this, observe that

1. The action of Spin^{u}(4) on Coker D covers the action of SO(4) × SO(3) on M_{1}^{s,♮}(S^{4}, δ),

2. The central S^{1} in Spin^{u}(4) acts on Coker D by scalar multiplication with weight negative one.

Property (1) implies that the action of Spin^{u}(4) on the component M_{1}^{s,♮}(S^{4}, δ) of

(Coker D)^{⊗2} = M_{1}^{s,♮}(S^{4}, δ) × ℂ

is given by the projection Ad : Spin^{u} → SO(4) × SO(3) and the action of SO(4) × SO(3) on M_{1}^{s,♮}(S^{4}, δ). Property (2) implies that the central S^{1} acts on (Coker D)^{⊗2} by scalar multiplication on the fibers with weight negative two, just as in the definition of the homomorphism det^{u} before equation (3.11). Any group action satisfying the above two properties will differ from the action (4.29) by a representation of Spin(4) ⊂ Spin^{u}(4) on ℂ. However, by [7, Proposition 5.1] there are no such non-trivial representations. Hence, these two properties characterize the above group action. Equations (3.14) and (3.12) yield the bundle isomorphisms,

Fr_{ℂℓ(T^{∗}X)}(V) ×_{(Spin^{u}(4),Ad)} SO(3) ≅ Fr(T^{∗}X) ×_{X} Fr(𝔤_{V}) ×_{SO(4)×SO(3)} SO(3)
= ∂Ḡl_{𝔱}(δ),
Fr_{ℂℓ(T^{∗}X)}(V) ×_{(Spin^{u}(4),det^{u})} ℂ ≅ det^{\frac{1}{2}} (V^{+}),

and so

Fr_{ℂℓ(T^{∗}X)}(V) ×_{Spin^{u}(4)} (SO(3) × ℂ) ≅ ∂Ḡl_{𝔱}(δ) ×_{X} det^{\frac{1}{2}} (V^{+}).

The preceding isomorphism is G_{𝔰}-equivariant, where G_{𝔰} acts on Fr_{ℂℓ(T^{∗}X)}(V) as in the description of the instanton obstruction bundle (3.68), if G_{𝔰} acts trivially on det^{\frac{1}{2}} (V^{+}) and on ∂Ḡl_{𝔱}(δ) by the standard action on Fr(𝔤_{V}). (The action is trivial on det^{\frac{1}{2}} (V^{+}) because elements of ϱ(G_{𝔰}) ⊂ G_{𝔱} act on the fiber of Fr_{ℂℓ(T^{∗}X)}(V) by elements of Spin^{u}(4) which are in the kernel of the homomorphism det^{u} : Spin^{u}(4) → S^{1}.) Therefore,

(4.30) (Υ_{𝔱′,𝔰}^{i})^{⊗2}|_{∂^{i}M_{𝔱′,𝔰}^{vir}} ≅ Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}} ∂Ḡl_{𝔱}(δ) ×_{X} det^{\frac{1}{2}} (V^{+}).

The bundle isomorphism (4.30) is circle-equivariant and so, from the definition (4.27) of ∂^{i}M_{𝔱′,𝔰}^{vir}, we see that the isomorphism (4.30) descends to an isomorphism

(4.31) (Υ_{𝔱′,𝔰}^{i}/S^{1})^{⊗2}|_{∂^{i}M_{𝔱′,𝔰}^{vir}/S^{1}} ≅ (∂^{i}M_{𝔱′,𝔰}^{vir} ×_{X} det^{\frac{1}{2}} (V^{+})) /S^{1},

where the circle acts with weight two on the fibers of det^{\frac{1}{2}} (V^{+}) and the action (2.37) on the factor Ñ_{𝔱′,𝔰}(ε) in ∂^{i}M_{𝔱′,𝔰}^{vir}. By the equivalence of the circle actions on M_{𝔱′,𝔰}^{vir} given by Lemma 3.6 and because the circle action of Lemma 3.6 is twice the action in the Definition 4.1, the circle quotient on the right-hand side of (4.31) is equivalent to one where the circle acts diagonally by scalar multiplication on the fibers of det^{\frac{1}{2}} (V^{+}) and by the action of Definition 4.1 on ∂^{i}M_{𝔱′,𝔰}^{vir} ⊂ M_{𝔱′,𝔰}^{vir}. The desired equation (4.28) for the Euler class of Υ_{𝔱′,𝔰}^{i}/S^{1} then follows from the isomorphism (4.31), the preceding description of the circle quotient on the right-hand side of (4.31), and Lemma 4.6.

5. Intersection numbers and cohomology

In this section we prove the topological results necessary to compute the intersection number

# (V̄(z) ∩ W̄^{η} ∩ L̄_{𝔱′,𝔰}) (5.1)

where L̄_{𝔱′,𝔰} is the link of M_{𝔰} × Sym^{ℓ}(X) in M̄_{𝔱′}, with ℓ(𝔱′, 𝔰) = 1 and deg(z) + 2(η + 1) = dim M_{𝔱′}.

We solve this problem in two stages. In §5.1 we prove that the intersection number (5.1) is equal to the pairing of certain cohomology classes with a homology class [L̄_{𝔱′,𝔰}^{vir}] which can be understood as a fundamental class of L̄_{𝔱′,𝔰}^{vir}; see Proposition 5.2. In §5.2 we compute enough of the cohomology ring of L̄_{𝔱′,𝔰}^{vir} to allow us to compute this pairing.

Since the gluing map γ : M̄_{𝔱′,𝔰}^{vir} → C̄_{𝔱′} is a circle-equivariant embedding of stratified spaces (a homeomorphism preserving strata and restricting to a diffeomorphism on smooth strata), when no confusion can arise we shall follow the convention stated in the introduction to §4. Thus, we shall not explicitly distinguish in this section between cohomology classes on C̄_{𝔱′} and their pullbacks to M̄_{𝔱′,𝔰}^{vir} or between cycles or geometric representatives in C̄_{𝔱′} and their pre-images in M̄_{𝔱′,𝔰}^{vir}. For example, we shall simply write V(β) and μ_{p}(β) for the pre-image γ^{−1}(V(β)) or pullback γ^{∗}μ_{p}(β) in M_{𝔱′,𝔰}^{vir} of the corresponding geometric representative or cohomology class on γ(M_{𝔱′,𝔰}^{vir}) ⊂ C_{𝔱′}^{∗,0}.

It is worth mentioning where we use the various properties of the gluing maps and obstruction sections described in Theorem 3.8, as we shall occasionally exploit these properties without comment later this section. The fact that γ is a homeomorphism from M̄_{𝔱′,𝔰}^{vir} into C̄_{𝔱′} ensures that the image of γ contains an open neighborhood of V̄(z) ∩ W̄ ∩ L̄_{𝔱′,𝔰} in M̄_{𝔱′}. The gluing map is a smooth embedding of the top stratum M_{𝔱′,𝔰}^{vir} into C_{𝔱′}^{∗,0} and, when orientations are properly taken into account, orientation-preserving; thus, the pre-images in M_{𝔱′,𝔰}^{vir} of oriented, transverse intersections in M_{𝔱′} are again appropriately oriented, and transverse. We use continuity of γ on M̄_{𝔱′,𝔰}^{vir} in arguments involving extensions of certain cocycles on L_{𝔱′,𝔰} to L̄_{𝔱′,𝔰}; see the introduction to §5.1.5 for precise statements.

Transversality and smoothness of the obstruction section χ is used in Lemma 5.9, where we use the fact that γ(χ^{−1}(0)) is a submanifold of M_{𝔱′} of the appropriate ‘multiplicity’. Continuity of the section χ_{s} on M̄_{𝔱′,𝔰}^{vir} is used in the proofs of Lemmas 5.17 and 5.18 and in writing the decomposition (5.44) of a certain relative Euler class.

5.1. Duality. The goal of this subsection is to prove Proposition 5.2.

Definition 5.1. Let ē_{s} and ē_{i} be the extensions of the Euler classes e(Υ_{𝔱′,𝔰}^{s}/S^{1}) and e(Υ_{𝔱′,𝔰}^{i}/S^{1}) from M_{𝔱′,𝔰}^{vir}/S^{1} to M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} defined by Lemmas 4.11 and 4.12 respectively, and set ē = ē_{s} ⌣ ē_{i}.

Proposition 5.2. Assume w ∈ H^{2}(X; ℤ) is such that w (mod 2) is good in the sense of Definition 2.3. Suppose (𝔱′, 𝔰) is a pair with ℓ(𝔱′, 𝔰) = 1 and w_{2}(𝔱′) ≡ w (mod 2). Let [L̄_{𝔱′,𝔰}^{vir}] ∈ H_{max}(L̄_{𝔱′,𝔰}^{vir}; ℤ) be the homology class defined in (5.9). Let z ∈ 𝔸(X) and η ∈ ℕ satisfy deg(z) + 2(η + 1) = dim M_{𝔱′}. Let μ̄_{p}(z) and μ̄_{c} be the extensions of μ_{p}(z) and μ_{c} in Definition 4.5 from M_{𝔱′,𝔰}^{vir}/S^{1} to M̄_{𝔱′,𝔰}^{vir,∗}/S^{1}. Give L_{𝔱′,𝔰} the standard orientation described in §3.9. Then,

# (V̄(z) ∩ W̄^{η} ∩ L̄_{𝔱′,𝔰}) = ⟨μ̄_{p}(z) ⌣ μ̄_{c}^{η} ⌣ ē, [L̄_{𝔱′,𝔰}^{vir}]⟩ . (5.2)

The corresponding result [18, Lemma 3.31] for the level-zero Seiberg-Witten stratum, M_{𝔰} ⊂ M_{𝔱}/S^{1}, followed trivially from the definition of a geometric representative (see [18, Definition 3.4] or [31, p. 588]) because the link of M_{𝔰} ⊂ M_{𝔱}/S^{1} is a smooth, compact manifold without boundary whose fundamental class can be represented by a smooth cycle intersecting the geometric representatives transversely.

By contrast, the link of the level-one Seiberg-Witten stratum,

(M_{𝔰} × X) ∩ M̄_{𝔱′}/S^{1} ⊂ M̄_{𝔱′}/S^{1},

can have non-empty intersection with a lower level of M̄_{𝔱′}/S^{1}. This raises two difficulties which prevent an immediate translation of the level-zero argument in [18] to the level-one case here:

1. The obstruction section χ does not vanish transversely on the lower level L_{𝔱′,𝔰}^{low} of L̄_{𝔱′,𝔰}^{vir} = L_{𝔱′,𝔰}^{vir} ⊔ L_{𝔱′,𝔰}^{low} (see (3.78)), so it is not immediately obvious that the link L̄_{𝔱′,𝔰} = χ^{−1}(0) ∩ L̄_{𝔱′,𝔰}^{vir} defines a homology class.

2. It is not obvious that the closure V̄(β) in the compactification M̄_{𝔱′,𝔰}^{vir}/S^{1} of the geometric representative V(β) in M_{𝔱′,𝔰}^{vir}/S^{1} defines a geometric representative for the cohomology class μ̄_{p}(β). (See Definition 5.6 for a review of the concept of a geometric representative for a cohomology class.)

The intersection number on the left-hand side of (5.2) is a count with sign of the points in the transverse intersection of the geometric representatives and the zero-locus, L_{𝔱′,𝔰}, of the section χ of the obstruction bundle over L_{𝔱′,𝔰}^{vir}. We emphasize that this intersection is contained in the top stratum of L̄_{𝔱′,𝔰}^{vir}; see Lemma 5.4. As discussed in the following paragraphs, the geometric representatives and the obstruction section define a cohomology class on L_{𝔱′,𝔰}^{vir} with compact support in the manifold-with-boundary L_{𝔱′,𝔰}^{vir,i} (defined in (3.72)). This compactly supported cohomology class is Poincaré dual to the intersection

V̄(z) ∩ W̄^{η} ∩ L̄_{𝔱′,𝔰}. (5.3)

Therefore, the intersection number (5.2) is equal to the pairing of this compactly supported cohomology class with the relative fundamental class [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}] of L_{𝔱′,𝔰}^{vir,i}.

The first step in defining the compactly supported cohomology class is to define a representative of the cohomology class μ_{p}(z) ⌣ μ_{c}^{η} with specified support. In §5.1.3 we observe that the geometric representatives V(β) and W define singular cocycles c_{β} and c_{W}, respectively, in the cohomology classes μ_{p}(β) and μ_{c}. The cocycle c_{β} has support on V(β) ∩ M_{𝔱′𝔰}^{vir}/S^{1}, in the sense that c_{β} restricts to zero on any singular chain not intersecting V(β). Therefore, c_{β} defines a relative cohomology class, [c_{β}], which maps to μ_{p}(β) under the homomorphism

H^{deg(β)}(M_{𝔱′𝔰}^{vir}/S^{1}, M_{𝔱′𝔰}^{vir}/S^{1} − V(β) ∩ M_{𝔱′𝔰}^{vir}/S^{1}; ℝ) → H^{deg(β)}(M_{𝔱′𝔰}^{vir}/S^{1}; ℝ),

given by the exact sequence of the pair. We say a relative cohomology class [c] ∈ H^{•}(X, A; ℝ) is a representative of μ ∈ H^{•}(X; ℝ) if ȷ_{A}^{∗}[c] = μ, where ȷ_{A} : (X, ∅) → (X, A) is the inclusion map. Thus, [c_{β}] is a representative of μ_{p}(β). Similar comments apply to the support of the cocycle c_{W}. We will write c(z, η) for the cup-product of the cocycles defined by the geometric representatives appearing in the left-hand side of equation (5.2). The cocycle c(z, η) defines a relative cohomology class, [c(z, η)], supported on the intersection of the geometric representatives.

The relative Euler class [29] of the obstruction bundle and section, e(Υ_{𝔱′,𝔰}/S^{1}, χ), has support on the zero-locus in M_{𝔱′,𝔰}^{vir} of χ. In §5.1.4 (see (5.20)), we show that the cup-product [c(z, η)] ⌣ e(Υ_{𝔱′,𝔰}/S^{1}, χ) has compact support in the top level L_{𝔱′,𝔰}^{vir} of L̄_{𝔱′,𝔰}^{vir}. Note that because the geometric representatives can intersect the lower strata of L̄_{𝔱′,𝔰}^{vir}, the class [c(z, η)] does not have compact support in L_{𝔱′,𝔰}^{vir,i}; it is only the cup-product [c(z, η)] ⌣ e(Υ_{𝔱′,𝔰}/S^{1}, χ) which has this compact support. In Lemma 5.9, we then prove that this cup-product is Poincaré dual to the intersection (5.3) and hence the intersection number (5.1) is equal to the pairing

(5.4) ⟨[c(z, η)] ⌣ e(Υ_{𝔱′,𝔰}/S^{1}, χ), [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}]⟩ .

Computing the pairing of the relative classes in (5.4) directly does not seem practical. The representatives [c_{β}] of μ_{p}(β), [c_{W}] of μ_{c}, and e(Υ_{𝔱′,𝔰}/S^{1}, χ) of e(Υ_{𝔱′,𝔰}/S^{1}) in the cup-product in (5.4) are elements of different cohomology rings (compare (5.15) and (5.18)), so it is not possible to compute their product using only the algebraic structure of the cohomology ring H^{•}(L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}; ℝ). We overcome this difficulty by replacing the pairing (5.4) with a pairing with relative cohomology classes in H^{max}(L̄_{𝔱′,𝔰}^{vir}, L̄_{𝔱′,𝔰}^{vir,s}; ℝ). There is an isomorphism

ȷ^{∗} : H^{max}(L̄_{𝔱′,𝔰}^{vir}, L̄_{𝔱′,𝔰}^{vir,s}; ℝ) ≅ H^{max}(L̄_{𝔱′,𝔰}^{vir}; ℝ),

because L̄_{𝔱′,𝔰}^{vir,s} retracts onto the codimension-four subspace L_{𝔱′,𝔰}^{low}. Thus, a pairing with relative cohomology classes in H^{•}(L̄_{𝔱′,𝔰}^{vir}, L̄_{𝔱′,𝔰}^{vir,s}; ℝ) will depend only on the image of the relative cohomology classes under ȷ^{∗}, and thus only on the absolute cohomology classes which these relative cohomology classes represent. These absolute cohomology classes are all elements of the ring H^{•}(L̄_{𝔱′,𝔰}^{vir}; ℝ), allowing us to compute products in the algebra of that single ring.

To replace the pairing (5.4) with one involving the compactification, we construct extensions of the cocycles c_{β} and c_{W} from L_{𝔱′,𝔰}^{vir} to L̄_{𝔱′,𝔰}^{vir} in §5.1.5. (This extension process for cocycles is simpler than the corresponding process for geometric representatives.) We find exact cocycles δθ_{β} and δθ_{W} such that c_{β} + δθ_{β} = ι^{∗}c̄_{β} and c_{W} + δθ_{W} = ι^{∗}c̄_{W}, where c̄_{β} and c̄_{W} are cocycles on L̄_{𝔱′,𝔰}^{vir} and ι is the inclusion map (4.2). By keeping track of the support of δθ_{β} and δθ_{W}, we can prove that replacing c_{β} and c_{W} with ι^{∗}c̄_{β} and ι^{∗}c̄_{W} does not change the pairing (5.4). The construction of c̄_{β} and c̄_{W} implies that they represent the cohomology classes on the right-hand-side of identity (5.2), namely μ̄_{p}(β) and μ̄_{c}.

Having applied this extension method to the cocycle c(z, η), we next apply it to the relative Euler class in the pairing (5.4). In §5.1.6, we describe a homotopy of the obstruction section χ which does not change the pairing (5.4). We then recognize the relative Euler class of the obstruction bundle and this homotoped obstruction section as the restriction of a relative Euler class on L̄_{𝔱′,𝔰}^{vir}.

These extensions accomplish the goal of replacing the relative cohomology classes in (5.4) with relative cohomology classes in H^{•}(L̄_{𝔱′,𝔰}^{vir}, L̄_{𝔱′,𝔰}^{vir}; ℝ). Proposition 5.2 then follows, by the argument sketched above, from Lemma 5.9 and our computations in §5.1.5 of the cohomology classes which the extended cocycles represent.

The preceding argument contains, albeit implicitly, solutions to the two previously mentioned problems preventing a direct translation of [18, Corollary 3.11]. The homotopy of the obstruction section should define a deformation of L̄_{𝔱′,𝔰} in L̄_{𝔱′,𝔰}^{vir} to a subspace whose intersection with the lower strata of L̄_{𝔱′,𝔰}^{vir} is sufficiently regular to define a homology class. Similarly, changing the cocycle c_{β} to c_{β} +δθ_{β} should be equivalent to changing the geometric representative V(β) (by a cobordism) to a geometric representative whose closure is a geometric representative for μ̄_{p}(β). We believe that the cohomological formulations given here are easier to construct than these deformations of L̄_{𝔱′,𝔰} and the geometric representatives.

Employing the techniques of virtual fundamental classes developed in [8], [35], [26], [46] [47], and [49] to construct a homology class representing L̄_{𝔱′,𝔰} — and thus eliminating the first of the two problems discussed above — would not simplify the proof of Proposition 5.2. While those techniques are well developed for the moduli space of pseudoholomorphic curves, applying that theory to the moduli spaces in this article would require additional discussion. For example, we would have to construct an extension of the obstruction bundle Υ_{𝔱′,𝔰}/S^{1} over an appropriate compactification of M_{𝔱′,𝔰}^{vir}/S^{1}; this would require further work as the instanton component Υ_{𝔱′,𝔰}^{i}/S^{1} does not extend from M_{𝔱′,𝔰}^{vir}/S^{1} to the relatively simple ‘cone’ compactification M̄_{𝔱′,𝔰}^{vir}/S^{1} in (3.55) which we currently employ. In contrast, the method used here — involving a relative Euler class and a homotopy of the obstruction section rather than constructing a homology class for L̄_{𝔱′,𝔰} — does not require any elaborate technical apparatus. Moreover, defining a virtual fundamental class would only solve the first of the two previously mentioned problems — defining a fundamental class for L̄_{𝔱′,𝔰}; it would still be necessary to construct the extensions of the cocycles c_{β} and c_{W} and to prove that they represent the cohomology classes μ̄_{p}(β) and μ̄_{c}. As the construction of the extended cocycles c̄_{β} and c̄_{W} and the proof that they represent the cohomology classes μ̄_{p}(β) and μ̄_{c} make up the bulk of the proof of Proposition 5.2, a construction of a fundamental class for L̄_{𝔱′,𝔰} would not shorten this section appreciably.

5.1.1. The fundamental class of the virtual link. We begin by defining the homology class [L̄_{𝔱′,𝔰}^{vir}] ∈ H_{max}(L̄_{𝔱′,𝔰}^{vir}; ℤ) referred to in Proposition 5.2. We refer to this homology class as the ‘fundamental class’ of the virtual link, although a precise definition of a fundamental class for a stratified space will not be necessary for this article. The existence and uniqueness of this class is an easy application of the exact sequence of the pair (L̄_{𝔱′,𝔰}^{vir}, L̄_{𝔱′,𝔰}^{vir,s}), where L̄_{𝔱′,𝔰}^{vir,s} is the neighborhood (3.72) of the lower level of L̄_{𝔱′,𝔰}^{vir}. The definition of [L̄_{𝔱′,𝔰}^{vir}] will extend from the case of ℓ(𝔱′, 𝔰) = 1 to the general case of ℓ(𝔱′, 𝔰) ≥ 1 as it only requires the existence of a neighborhood of L_{𝔱′,𝔰}^{low} in L̄_{𝔱′,𝔰}^{vir} which retracts onto L_{𝔱′,𝔰}^{low}. We note, however, that the construction of this neighborhood is more difficult in the general case.

By definition (3.72) of L̄_{𝔱′,𝔰}^{vir,s} there is a deformation retraction of L̄_{𝔱′,𝔰}^{vir,s} onto L_{𝔱′,𝔰}^{low} (see (3.77)) induced by the deformation retraction Ḡl_{𝔱′}(δ) → X (given by shrinking the scale to zero). We have an inclusion map of pairs:

(5.5) ȷ : (L̄_{𝔱′,𝔰}^{vir}, ∅) → (L̄_{𝔱′,𝔰}^{vir}, L̄_{𝔱′,𝔰}^{vir,s}) .

Because L̄_{𝔱′,𝔰}^{vir,s} retracts onto L_{𝔱′,𝔰}^{low} and because L_{𝔱′,𝔰}^{low} has codimension four in L̄_{𝔱′,𝔰}^{vir}, the inclusion (5.5) induces an isomorphism,

(5.6) ȷ_{∗} : H_{max}(L_{𝔱′,𝔰}^{vir}; ℤ) ≅ H_{max}(L_{𝔱′,𝔰}^{vir}, L_{𝔱′,𝔰}^{vir,s}; ℤ).

Let

(5.7) [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}] ∈ H_{max}(L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}; ℤ)

be the relative fundamental class [50, p. 303] of the manifold-with-boundary L_{𝔱′,𝔰}^{vir,i}. By the construction (3.72) of L_{𝔱′,𝔰}^{vir,i} and L̄_{𝔱′,𝔰}^{vir,s}, the boundary of L_{𝔱′,𝔰}^{vir,i} lies in L_{𝔱′,𝔰}^{vir,s} and so there is an inclusion map

(5.8)  j̄_{L^{i}} : (L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}) → (L̄_{𝔱′,𝔰}^{vir}, L̄_{𝔱′,𝔰}^{vir,s}) .

Using the isomorphism (5.6), we then define [L̄_{𝔱′,𝔰}^{vir}] to be the unique homology class satisfying

(5.9) (j̄_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}] = ȷ_{∗}[L̄_{𝔱′,𝔰}^{vir}].

5.1.2. Deforming the geometric representatives. The intersection of the geometric representatives used to define the Donaldson invariants in [31] with the lower strata can only be easily understood on the set of triples [A, Φ, x] where the sets of points representing x ∈ Sym^{•}(X) do not intersect the ‘suitable neighborhoods’ [31] used to define the geometric representatives. In this section we prove that the intersection number on the right-hand side of (5.2) is equal to one defined by geometric representatives where suitable neighborhoods are replaced by tubular neighborhoods, simplifying our cohomological computations in §5.1.5.

For β ∈ H_{•}(X; ℝ), let T_{β} be a smooth submanifold of X with fundamental class [T_{β}] = β. The geometric representative V(β) was defined in [18, §3.2] by pulling back a geometric representative from the quotient space of connections over a ‘suitable’ neighborhood U_{β} of T_{β}. Recall that a suitable neighborhood of T_{β} was defined in [18, Definition 3.8] or [31, p. 589] as a smoothing of the union of a tubular neighborhood of T_{β} and a set of loops {γ_{i}}, where the loops γ_{i} generate H_{1}(X; ℤ/2ℤ), are mutually disjoint, and are transverse to T_{β}. We need in [18] to use a suitable neighborhood rather than a tubular neighborhood because if H^{1}(X; ℤ/2ℤ) ≠ 0, there could be a point [A, 0] ∈ M_{𝔱′} such that the restriction of the induced SO(3) connection Â on 𝔤_{V} to a tubular neighborhood of T_{β} would be reducible [31, p. 586] even if Â is not globally reducible. We note that the assumption that w (mod 2) is good (as defined prior to (2.14)) implies that there are no reducible, zero-section pairs in M̄_{𝔱′}; Lemmas 5.3, 5.4, and 5.5 rely on this constraint on w. The following lemma shows that when defining geometric representatives near the strata of reducible pairs (but not zero-section pairs) it suffices to use tubular rather than suitable neighborhoods of T_{β}:

Lemma 5.3. Assume w ∈ H^{2}(X; ℤ) is such that w (mod 2) is good. Given a Riemannian metric on X and a pair (𝔱′, 𝔰) with ℓ(𝔱′, 𝔰) = 1 and w_{2}(𝔱′) ≡ w (mod 2), there are positive constants ε_{0} and δ_{0} such that for all positive ε ≤ ε_{0} and δ ≤ δ_{0} used in the definition of M_{𝔱′,𝔰}^{vir}, the following holds. For β ∈ H_{•}(X; ℝ), let T_{β} be the submanifold of X with [T_{β}] = β and let ν(β) be a tubular neighborhood of T_{β}. Then, for any [A, Φ, x] ∈ γ(M̄_{𝔱′,𝔰}^{vir,∗}), the restriction of Â to ν(β) is not reducible.

Proof. Because w_{2}(𝔱′) = w_{2}(𝔱) is good, there are no zero-section pairs in M_{𝔰}. Thus, if the parameters ε and δ in the definition of M_{𝔱′,𝔰}^{vir} are sufficiently small, then the neighborhood γ(M̄_{𝔱′,𝔰}^{vir,∗}) is disjoint from M̄_{κ}^{w}. Hence, for any point [A, Φ, x] ∈ γ(M̄_{𝔱′,𝔰}^{vir,∗}), the section Φ is not identically zero and Â is not a reducible connection. By [18, Theorem 3.10] or [16, Theorem 5.10], the restriction of Â to any open subspace of X cannot be reducible.

Lemma 5.3 implies that there is a geometric representative, V′(β) for μ_{p}(β), pulled back from the quotient space of irreducible SO(3) connections, B^{∗}(ν(β)), over the tubular neighborhood ν(β). This geometric representative is then constructed by the same methods used to construct V(β) in [31]. The only difference between V(β) and V′(β) is that V(β) is pulled back from B^{∗}(U_{β}) rather than B^{∗}(ν(β)).

Henceforth, we assume that the z ∈ 𝔸(X) considered in Proposition 5.2 is a monomial, z = β_{1}β_{2} ⋯ β_{m}. We can then choose the submanifold T_{β_{i}} defining the geometric representative V′(β_{i}), the points x_{1}, …, x_{η} defining the geometric representatives W, and their tubular neighborhoods so that for any point x ∈ X we have

∑_{{β_{i}: x∈ν′(β_{i})}} deg(β_{i}) + ∑_{{x_{j}: x∈ν′(x_{j})}} 4 ≤ 4, (5.10)

where ν′(β_{i}) is a tubular neighborhood of T_{β_{i}} with ν(β_{i}) ⋐ ν′(β_{i}); compare [31, Equation (2.7)]. Let V′(z) = ∩_{i}V′(β_{i}) and let V̄′(z) denote the Uhlenbeck closure of V′(z). The following lemma shows we can define an intersection number of these new geometric representatives with L̄_{𝔱′,𝔰}.

Lemma 5.4. Assume w ∈ H^{2}(X; ℤ) is such that w (mod 2) is good. Suppose (𝔱′, 𝔰) is a pair with ℓ(𝔱′, 𝔰) = 1 and w_{2}(𝔱′) ≡ w (mod 2). Then the following hold:

1. The intersection, V̄′(z) ∩ W̄^{η} ∩ L̄_{𝔱′,𝔰}, is contained in the top stratum L_{𝔱′,𝔰} of L̄_{𝔱′,𝔰} ⊂ M̄_{𝔱′}/S^{1} and is disjoint from the edge (3.76).

2. The following intersection is empty,

V̄′(z) ∩ W̄^{η} ∩ γ (χ_{s}^{−1}(0) ∩ L_{𝔱′,𝔰}^{low}) = ∅.

Proof. Inequality (5.10) and the dimension-counting argument used in the proof of [18, Corollary 3.18] imply that the intersection V̄′(z) ∩ W̄^{η} is disjoint from the lower level, (M̄_{𝔱′} − M_{𝔱′}) ∩ γ(M̄_{𝔱′,𝔰}^{vir,∗})/S^{1}. The first assertion then follows from the proof of Lemma 3.9.

The image γ(χ_{s}^{−1}(0) ∩ L_{𝔱′,𝔰}^{low}) is the intersection of γ(L̄_{𝔱′,𝔰}^{vir}) with the union of the lower levels, (M̄_{𝔱′} − M_{𝔱′})/S^{1}. Therefore, the second assertion follows from the first and the fact that γ preserves strata on M̄_{𝔱′,𝔰}^{vir}.

We now prove that replacing V̄(z) with V̄′(z) does not change the intersection number in (5.2).

Lemma 5.5. Assume w ∈ H^{2}(X; ℤ) is such that w (mod 2) is good. Suppose (𝔱′, 𝔰) is a pair with ℓ(𝔱′, 𝔰) = 1 and w_{2}(𝔱′) ≡ w (mod 2). Then, for z ∈ 𝔸(X) and η ∈ ℕ as in Proposition 5.2, we have:

# (V̄(z) ∩ W̄^{η} ∩ L̄_{𝔱′,𝔰}) = # (V̄′(z) ∩ W̄^{η} ∩ L̄_{𝔱′,𝔰}) . (5.11)

Proof. Both sides of (5.11) are linear in z, so we may assume without loss of generality that z = β_{1} ⋯ β_{m} for β_{i} ∈ H_{•}(X; ℝ).

First, we note that the intersection numbers on both sides of (5.11) do not change if we decrease the parameter δ defining L_{𝔱′,𝔰}^{vir} (to another positive generic value) because of the obvious cobordism defined by this change of parameter. Thus, in the proof we may assume that the parameter δ is as small as desired.

There is a cobordism

H ⊂ γ(M_{𝔱′,𝔰}^{vir}/S^{1}) × [0, 1],

with boundaries given by

(V(β) ∩ γ(M_{𝔱′,𝔰}^{vir}/S^{1})) × {0} and (V′(β) ∩ γ(M_{𝔱′,𝔰}^{vir}/S^{1})) × {1}.

Such a cobordism exists because V(β) and V′(β) are defined by pullbacks (by the appropriate restriction maps) of zero loci of sections of the same line bundle, for β ∈ H_{2}(X; ℝ), or by the degeneracy locus of sections of the same vector bundle for β ∈ H_{0}(X). If z = z_{1}β_{1}, where z_{1} ∈ 𝔸(X) and β_{1} ∈ H_{•}(X; ℝ), we replace V̄(z) with V̄(z_{1})∩V̄′(β_{1}) in the intersection (5.11) as follows. By perturbing the cobordism H, we can assume that H is transverse to

(V̄(z_{1}) ∩ W̄^{η} ∩ L̄_{𝔱′,𝔰} ∩ γ(L_{𝔱′,𝔰}^{vir,i})) × [0, 1]. (5.12)

The dimension-counting arguments in the proof of Corollary 3.18 in [18] show that the closure H̄ in γ(M̄_{𝔱′,𝔰}^{vir}/S^{1}) × [0, 1] of the cobordism H will not intersect

(V̄(z_{1}) ∩ W̄^{η} ∩ (L̄_{𝔱′,𝔰} − L_{𝔱′,𝔰})) × [0, 1].

Then, for a sufficiently small parameter δ, the space H̄ will not intersect

(V̄(z_{1}) ∩ W̄^{η} ∩ L̄_{𝔱′,𝔰} ∩ γ(L̄_{𝔱′,𝔰}^{vir,s})) × [0, 1].

Therefore the intersection H̄ ∩ (V̄(z_{1}) ∩ W̄^{η} ∩ L̄_{𝔱′,𝔰} × [0, 1]) is contained in the transverse intersection of H with the space (5.12). Hence, this last intersection is a family of smooth, compact, oriented one-dimensional submanifolds with one boundary given by the set of points in the intersection on the left-hand side of (5.11) and the other boundary given by the set of points in the intersection

V̄(z_{1}) ∩ V̄′(β_{1}) ∩ W̄^{η} ∩ L̄_{𝔱′,𝔰}.

We now repeat this process with each β_{i} in the product z until V̄(z) has been replaced by V̄′(z). This completes the proof.

From this point onwards, we shall work with the geometric representative V̄′(z) defined by tubular rather than suitable neighborhoods. By choosing generic parameters ε and δ in the definition of L̄_{𝔱′,𝔰}^{vir} (see (3.72)), we can also ensure that these geometric representatives are transverse to L_{𝔱′,𝔰}^{vir,i}. To simplify notation, we shall omit the primes, writing V̄(z) for V̄′(z), as the original geometric representatives defined by suitable neighborhoods will not reappear in the remainder of this article.

5.1.3. Geometric representatives and cocycles. We now define cocycles dual to the geometric representatives V(β) and W. We first recall the following definition:

Definition 5.6. [31, p 588]. Let Z be a smoothly stratified space. A geometric representative for a real cohomology class μ of dimension d on Z is a closed, smoothly stratified subspace V of Z together with a real coefficient q, the multiplicity, satisfying

1. The intersection Z_{0} ∩ V of V with the top stratum Z_{0} of Z has codimension d in Z_{0} and has an oriented normal bundle.

2. The intersection of V with all strata of Z other than the top stratum has codimension 2 or more in V.

3. The pairing of μ with a homology class h of dimension d is obtained by choosing a smooth singular cycle σ representing h whose intersection with all strata of V has codimension dim Z_{0} − d in that stratum of V, and then taking q times the count (with signs) of the intersection points between the cycle and the top stratum of V:

⟨μ, h⟩ = q · #((Z_{0} ∩ V) ∩ σ).

The intersection of the geometric representative with the top stratum with real coefficient, (V ∩ Z_{0}, q), defines a singular cocycle c representing the restriction of the cohomology class μ to Z_{0} with the properties described in the following lemma.

Lemma 5.7. Let Z_{0} be a smooth manifold and let (V, q) be a geometric representative for a real cohomology class μ on Z_{0}. Then, there is a singular cocycle c on Z_{0} which represents the cohomology class μ and which vanishes when restricted to Z_{0} − V.

Proof. For any smooth manifold W, let Δ_{•}^{∞}(W) denote the chain complex of smooth singular chains [6, p. 291] and let S_{∞}^{•}(W; ℝ) = Hom(Δ_{•}^{∞}(W), ℝ) be the complex of smooth singular cochains. For any smooth submanifold Y ⊂ W, we define the complex of smooth singular, relative cochains by

S_{∞}^{•}(W, Y; ℝ) = Hom(Δ_{•}^{∞}(W)/Δ_{•}^{∞}(Y), ℝ).

We will write H_{∞}^{•}(W; ℝ) and H_{∞}^{•}(W, Y; ℝ) for the homology of the complexes S_{∞}^{•}(W; ℝ) and S_{∞}^{•}(W, Y; ℝ), respectively. Thus, H_{∞}^{•}(W; ℝ) is the smooth singular cohomology of W and H_{∞}^{•}(W, Y; ℝ) is the smooth singular, relative cohomology of (W, Y). By the de Rham theorem (see the discussion in [6, p. 291]), there is a functorial isomorphism H^{p}(W; ℝ) ≅ H_{∞}^{•}(W; ℝ). By applying the Five Lemma to the long exact sequences of the pair (W, Y) in singular and smooth singular cohomology, we obtain an isomorphism H^{•}(W, Y; ℝ) ≅ H_{∞}^{•}(W, Y; ℝ).

We first define a smooth singular cocycle, that is, a closed c′ ∈ S_{∞}^{•}(Z_{0}; ℝ), which represents the cohomology class μ. For any smooth singular chain σ of dimension equal to that of μ, set

⟨c′, σ⟩ = q · #((V ∩ Z_{0}) ∩ σ′), (5.13) where σ′ is any smooth singular chain in Z_{0} which is homologous in Z_{0} to σ and transverse to V. By definition of V, if σ is a smooth singular cycle in Z_{0} which represents a homology class h, we then have

⟨c′, σ⟩ = ⟨μ, h⟩,

so c′ is a smooth singular cocycle representing the cohomology class μ on Z_{0}. (By the de Rham isomorphism, it does not matter whether we consider μ as a singular cohomology class or a smooth singular cohomology class.) Because the restriction of c′ to Z_{0} −V vanishes, the cocycle c′ defines an element [c′] of the smooth singular, relative cohomology H_{∞}^{•}(Z_{0}, Z_{0} − V; ℝ) which maps to μ under the homomorphism H_{∞}^{•}(Z_{0}, Z_{0} −V; ℝ) → H_{∞}^{•}(Z_{0}; ℝ). Let α ∈ H^{•}(Z_{0}, Z_{0} − V; ℝ) be the element of relative singular cohomology given by the image of [c′] under the isomorphism H_{∞}^{•}(Z_{0}, Z_{0} −V; ℝ) ≅ H^{•}(Z_{0}, Z_{0} −V; ℝ) and let ĉ ∈ S^{•}(Z_{0}, Z_{0} −V; ℝ) be any representative of α. Then the element c ∈ S^{•}(Z_{0}; ℝ) given by the image of ĉ under the homomorphism S^{•}(Z_{0}, Z_{0} − V; ℝ) → S^{•}(Z_{0}; ℝ) will satisfy the conclusion of the lemma.

For β ∈ H_{•}(X; ℝ), we let c_{β} be the cocycle on M_{𝔱′,𝔰}^{vir}/S^{1} representing μ_{p}(β), defined by Lemma 5.7 and the geometric representative V(β); we let c_{W} be the cocycle on M_{𝔱′,𝔰}^{vir}/S^{1} representing μ_{c}, defined as in Lemma 5.7, by the geometric representative W. Note that by considering the smooth manifold Z_{0} in Lemma 5.7 to be the image of M_{𝔱′,𝔰}^{vir}/S^{1} under the composition of the gluing map and the restriction map r_{ν(β)}, we can assume the cocycle c_{β} is pulled back from B^{∗}(ν(β)).

5.1.4. Relative Euler classes and intersection numbers. We next prove that the intersection number in equation (5.2) is equal to a pairing of a relative cohomology class with the relative fundamental class [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}] of L_{𝔱′,𝔰}^{vir,i}.

Definition 5.8. If e(V) is the Euler class of a real-rank r, oriented vector bundle V a CW complex over M and χ : U ⊂ M → V is a continuous, local section of V, then the relative Euler class [29] of the pair (V, χ) is a relative cohomology class,

e(V, χ) ∈ H^{r}(M, U − χ^{−1}(0); ℤ),

satisfying the relation

j_{χ}^{∗}e(V, χ) = e(V), (5.14)

defined by the inclusion map:

j_{χ} : (M, ∅) → (M, U − χ^{−1}(0)).

The relative Euler class e(Υ_{𝔱′,𝔰}/S^{1}, χ) of the obstruction bundle Υ_{𝔱′,𝔰}/S^{1} and obstruction section χ of Theorem 3.8 thus defines a relative cohomology class

(5.15) e(Υ_{𝔱′,𝔰}/S^{1}, χ) ∈ H^{2r_{Ξ}+2}(L_{𝔱′,𝔰}^{vir}, L_{𝔱′,𝔰}^{vir} − χ^{−1}(0); ℤ).

Recall that r_{Ξ} is the complex rank of the background component of the obstruction bundle, Ῡ_{𝔱′,𝔰}^{s}/S^{1}, defined in §3.6.1. For z = β_{1} ⋯ β_{m} ∈ 𝔸(X), define a subspace

K(z, η) = V(z) ∩ W^{η} ∩ L_{𝔱′,𝔰}^{vir} ⊂ L_{𝔱′,𝔰}^{vir},

with closure

(5.16)  K̄(z, η) = V̄(z) ∩ W̄^{η} ∩ L̄_{𝔱′,𝔰}^{vir} ⊂ L̄_{𝔱′,𝔰}^{vir}.

The cocycle on M_{𝔱′,𝔰}^{vir},

c(z, η) := c_{β_{1}} ⌣ ⋯ ⌣ c_{β_{m}} ⌣ c_{W}^{η}, (5.17)

58 PAUL M. N. FEEHAN AND THOMAS G. LENESS

vanishes on the complement of K(z, η) in M_{𝔱′,𝔰}^{vir}, and so defines a relative cohomology class (5.18) [c(z, η)] ∈ H^{deg(z)+2η}(L_{𝔱′,𝔰}^{vir}, L_{𝔱′,𝔰}^{vir} − K(z, η); ℝ). By Lemma 5.4 and the continuity of the gluing map γ, the following intersection is empty:

K̄(z, η) ∩ L_{𝔱′,𝔰}^{low} ∩ χ^{−1}(0) = ∅.

Therefore, noting that the link definition (3.72) gives

L̄_{𝔱′,𝔰}^{vir,s} = L_{𝔱′,𝔰}^{vir,s} ⊔ (∂N_{𝔱,𝔰}(ε)/S^{1} × X) ,

and that L̄_{𝔱′,𝔰}^{vir,s} retracts onto the lower level ∂N_{𝔱,𝔰}(ε)/S^{1} × X of L̄_{𝔱′,𝔰}^{vir,s} by shrinking the instanton scale δ, we can also arrange that the following intersection is empty by taking δ to be sufficiently small: K(z, η) ∩ L_{𝔱′,𝔰}^{vir,s} ∩ χ^{−1}(0) = ∅. (5.19) Because cup product of relative cohomology is a map [50, p. 251],

H^{i}(Y, A) ⊗ H^{j}(Y, B) → H^{i+j}(Y, A ∪ B),

and (5.19) yields

(L_{𝔱′,𝔰}^{vir} − χ^{−1}(0)) ∪ (L_{𝔱′,𝔰}^{vir} − K(z, η)) ⊃ L_{𝔱′,𝔰}^{vir,s},

we see from (5.15) and (5.18) that the cup product [c(z, η)] ⌣ e(Υ_{𝔱′,𝔰}/S^{1}, χ) vanishes on L_{𝔱′,𝔰}^{vir,s} and therefore we have

(5.20) [c(z, η)] ⌣ e(Υ_{𝔱′,𝔰}/S^{1}, χ) ∈ H^{max} (L_{𝔱′,𝔰}^{vir}, L_{𝔱′,𝔰}^{vir,s}; ℝ) . The inclusion (5.21) ι : L_{𝔱′,𝔰}^{vir} → L̄_{𝔱′,𝔰}^{vir}, induces an isomorphism of relative homology by excision, (5.22) ι_{∗} : H_{•} (L_{𝔱′,𝔰}^{vir}, L_{𝔱′,𝔰}^{vir,s}; ℝ) ≅ H_{•} (L̄_{𝔱′,𝔰}^{vir}, L̄_{𝔱′,𝔰}^{vir,s}; ℝ) . Hence, we can express the intersection number in (5.2) as a pairing of relative homology and cohomology classes: Lemma 5.9. If [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}] is the relative fundamental class of L_{𝔱′,𝔰}^{vir,i}, if j̄_{L^{i}} is the inclusion map (5.8), and if ι_{∗} is the excision isomorphism (5.22), then

# (V̄(z) ∩ W̄^{η} ∩ L̄_{𝔱′,𝔰})

= ⟨[c(z, η)] ⌣ e (Υ_{𝔱′,𝔰}/S^{1}, χ|_{L_{𝔱′,𝔰}^{vir,s}}) , ι_{∗}^{−1}(j̄_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}]⟩ . (5.23)

Proof. Let

ȷ_{L^{i}} : (L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}) → (L_{𝔱′,𝔰}^{vir}, L_{𝔱′,𝔰}^{vir,s})

be the inclusion map of pairs. The equality j̄_{L^{i}} = ι ∘ ȷ_{L^{i}} implies that ι_{∗}^{−1}(j̄_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}] = (ȷ_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}]

is the image of the relative fundamental class of L_{𝔱′,𝔰}^{vir,i} under the inclusion ȷ_{L^{i}}. By (5.19), the intersection on the left-hand-side of (5.23) is a finite collection of points in the interior of L_{𝔱′,𝔰}^{vir,i} (with the multiplicities of the geometric representatives). The obstruction section χ vanishes transversely on L_{𝔱′,𝔰}^{vir,i} by Theorem 3.8 and the assumption that the parameters ε and δ in the definition of the link are generic, so the relative fundamental class of the manifold-with-boundary L_{𝔱′,𝔰} − L_{𝔱′,𝔰}^{vir,s} is given by

(5.24) e (Υ_{𝔱′,𝔰}/S^{1}, χ|_{L_{𝔱′,𝔰}^{vir,s}}) ∩ (ȷ_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}] .

The geometric representatives intersect L_{𝔱′,𝔰} − L_{𝔱′,𝔰}^{vir,s} transversely so, by the definition of a geometric representative and the cocycle c(z, η), the intersection number on the left-hand-side of (5.23) is given by evaluating [c(z, η)] on the relative fundamental class (5.24), yielding (5.23).

5.1.5. Extending the cocycles. To rewrite the pairing in identity (5.23) as a pairing of absolute cohomology classes via the isomorphism ȷ_{∗} in (5.6), we must first eliminate the excision isomorphism ι_{∗}. For this reason, we now show how the cocycles c_{β} and c_{W} are, modulo exact cocycles, the restrictions of cocycles on L̄_{𝔱′,𝔰}^{vir}. Because the cocycle c_{β} on L_{𝔱′,𝔰}^{vir} represents the cohomology class μ_{p}(β), which extends to μ̄_{p}(β) on L̄_{𝔱′,𝔰}^{vir}, by Definition 4.5, we can always find cocycles in the class μ_{p}(β) which extend to L̄_{𝔱′,𝔰}^{vir} and represent μ̄_{p}(β). However, we emphasize that the changes of the cocycles c_{β} and c_{W} which we construct here have the property that they do not alter the pairing in (5.23), which is false for arbitrary choices of cocycles in the cohomology classes μ_{p}(β) or μ_{c}.

For any subset U ⊂ X, we define

I(U) = L̄_{𝔱′,𝔰}^{vir} ∩ γ^{−1} ({[A, Φ, x] : |x| ∩ U ≠ ∅}) , (5.25)

where |x| ⊂ X denotes the support of x ∈ Sym^{ℓ}(X); since ℓ = 1 here, the set on the right-hand side above is simply {[A, Φ, x] : x ∈ U}. Because the cocycles c_{β} and c_{W} are pulled back from B^{∗}(ν(β)) and C_{𝔱′}^{∗,0}(ν(x))/S^{1}, respectively, by the composition of the gluing map and the restriction map, the cocycles c_{β} and c_{W} naturally extend over L̄_{𝔱′,𝔰}^{vir} − I(ν(β)) and L̄_{𝔱′,𝔰}^{vir} − I(ν(x)), respectively. (Note that we use the continuity of the gluing map with respect to Uhlenbeck limits here.) To construct the extension of c_{β} to L̄_{𝔱′,𝔰}^{vir}, we first show in Lemma 5.12 that the cohomology class represented by c_{β} on L̄_{𝔱′,𝔰}^{vir} − I(ν(β)) is equal to the restriction of μ̄_{p}(β) to this subspace. The proof of Lemma 5.12 uses some cohomological computations appearing in Lemmas 5.10 and 5.11 and the continuity of the gluing map mentioned above. Then, in Lemma 5.14 we describe the perturbation, in the form of an exact cocycle, necessary to extend c_{β} from L̄_{𝔱′,𝔰}^{vir} − I(ν(β)) to a cocycle c̄_{β} on L̄_{𝔱′,𝔰}^{vir}. The construction of c̄_{β} will show that it represents the cohomology class μ̄_{p}(β).

We begin by proving the following lemma about the cohomology of the link. While the proofs of Lemmas 5.10 and 5.11 apply only to the link of the stratum M_{𝔰} × Sym^{ℓ}(X) when ℓ = 1, the results should be true for any ℓ ≥ 1. However, the proofs for larger ℓ may be difficult so it may be easier to prove Lemma 5.12 by a direct, if tedious, analysis of the bundles discussed there.

Lemma 5.10. If π_{X} : L_{𝔱′,𝔰}^{vir} → X is the projection map, then the induced map on cohomology is injective:

π_{X}^{∗} : H^{•}(X; ℝ) → H^{•}(L_{𝔱′,𝔰}^{vir}; ℝ).

Proof. The definition of L_{𝔱′,𝔰}^{vir} in (3.75) and the existence of a retraction from the uncompactified gluing data space Gl_{𝔱}(δ) to the boundary ∂Ḡl_{𝔱}(δ) ⊂ Gl_{𝔱}(δ) imply that L_{𝔱′,𝔰}^{vir} deformationretracts to L_{𝔱′,𝔰}^{vir,i}. In turn, L_{𝔱′,𝔰}^{vir,i} deformation-retracts to the space BL_{𝔱′,𝔰}^{vir,i} defined in (3.83). These retractions commute with π_{X}, so it suffices to prove the lemma for the restriction of the projection π_{X} to BL_{𝔱′,𝔰}^{vir,i}. This restriction of π_{X} can be written as the composition of the projections

π_{1} : M_{𝔰} × ∂Ḡl_{𝔱}(δ)/S^{1} → M_{𝔰} × X and M_{𝔰} × X → X.

The projection M_{𝔰} × X → X induces an injective homomorphism on cohomology, so we need only verify that π_{1} induces an injective homomorphism on cohomology. The map π_{1} is the product of the identity on M_{𝔰} and the projection

∂Ḡl_{𝔱}(δ)/S^{1} → X.

As will be discussed the proof of Lemma 5.24, the space ∂Ḡl_{𝔱}(δ)/S^{1} can be identified with the projectivization of a complex rank-two vector bundle F → X. By [5, Equation (20.7)], the projection map ℙ(F) → X induces an injective homomorphism on cohomology. This proves the lemma.

The following lemma will be used to identify the cohomology classes which the extensions of the cocycles represent; the result should be true for the links of ideal Seiberg-Witten moduli spaces M_{𝔰} × Sym^{ℓ}(X) defined in [21] for ℓ ≥ 1. The difficulties in extending the proof of Lemma 5.11 to the case ℓ > 1 lie first in the use of Lemma 5.10 (which is only proven for ℓ = 1) and secondly in applying the Thom isomorphism theorem (as is done below, before (5.29)) to the analogue of ν(β) in Sym^{ℓ}(X).

Lemma 5.11. For β ∈ H_{•}(X; ℝ), let T_{β} be a smooth submanifold of X with [T_{β}] = β and let ν(β) be a tubular neighborhood of T_{β} in X. Then for p = deg(β), the following direct sum of restriction maps is injective:

(5.26) H^{p}(L̄_{𝔱′,𝔰}^{vir} − I(ν(β)); ℝ) → H^{p}(L_{𝔱′,𝔰}^{vir}; ℝ) ⊕ H^{p}(L̄_{𝔱′,𝔰}^{vir} − π_{X}^{−1}(ν(β)); ℝ).

Proof. To prove the homomorphism (5.26) is injective, we first observe that it appears in the Meyer-Vietoris sequence for the open cover

L̄_{𝔱′,𝔰}^{vir} − I(ν(β)) = L_{𝔱′,𝔰}^{vir} ∪ (L̄_{𝔱′,𝔰}^{vir} − π_{X}^{−1}(ν(β))) .

The intersection of these two open sets is L_{𝔱′,𝔰}^{vir} − π_{X}^{−1}(ν(β)). Hence, the homomorphism (5.26) will be injective if the restriction map

(5.27) H^{p−1}(L_{𝔱′,𝔰}^{vir}; ℝ) → H^{p−1}(L_{𝔱′,𝔰}^{vir} − π_{X}^{−1}(ν(β)); ℝ)

is surjective. As noted in the proof of Lemma 5.10, there is a deformation retraction from L_{𝔱′,𝔰}^{vir} to BL_{𝔱′,𝔰}^{vir,i} which commutes with π_{X}. Thus, proving the map (5.27) is surjective is equivalent to proving that the following restriction map is surjective:

(5.28) H^{p−1}(BL_{𝔱′,𝔰}^{vir,i}; ℝ) → H^{p−1}(BL_{𝔱′,𝔰}^{vir,i} − π_{X}^{−1}(ν(β)); ℝ),

The surjectivity of the map (5.28) follows from a discussion of the following diagram in which the vertical maps come from the long exact sequences of the pairs:

H^{p−1}(X − ν(β); ℝ) ⟶^{π_{X}^{∗}} H^{p−1}(BL_{𝔱′,𝔰}^{vir,i} − π_{X}^{−1}(ν(β)); ℝ)
δ_{X,β}^{∗} ↓  ^{δ_{L,β}^{∗}} ↓
H^{p}(X, X − ν(β); ℝ) ⟶^{π_{X}^{∗}} H^{p}(BL_{𝔱′,𝔰}^{vir,i}, BL_{𝔱′,𝔰}^{vir,i} − π_{X}^{−1}(ν(β)); ℝ)
ȷ_{X,β}^{∗} ↓  ^{ȷ_{L,β}^{∗}} ↓
H^{p}(X; ℝ)  ⟶^{π_{X}^{∗}}  H^{p}(BL_{𝔱′,𝔰}^{vir,i}; ℝ)

Because π_{X}^{−1}(T_{β}) is a smooth, codimension-p submanifold of BL_{𝔱′,𝔰}^{vir,i}, with the Thom class of its normal bundle given by the pullback of the Thom class π_{X}^{∗} Th(β) of the normal bundle of T_{β} in X, we see that the relative cohomology

(5.29) H^{p}(BL_{𝔱′,𝔰}^{vir,i}, BL_{𝔱′,𝔰}^{vir,i} − π_{X}^{−1}(ν(β)); ℝ),

is generated by π_{X}^{∗} Th(β). By Lemma 5.10, the following class is non-zero:

ȷ_{L,β}^{∗}π_{X}^{∗} Th(β) = π_{X}^{∗}ȷ_{X,β}^{∗} Th(β) = π_{X}^{∗} PD[β].

Therefore, the generator π_{X}^{∗} Th(β) of the relative cohomology (5.29) is not in the image of δ_{L,β}^{∗}, implying that δ_{L,β}^{∗} is the zero map. Hence, the map (5.28) is surjective. As discussed previously, this implies that the map (5.26) is injective.

Lemma 5.11 gives the following identity between the cohomology class represented by the cocycle c_{β} and the restriction of the cohomology class μ̄_{p}(β):

Lemma 5.12. For β ∈ H_{•}(X; ℝ), let c_{β} be the cocycle on L̄_{𝔱′,𝔰}^{vir} − I(ν(β)) defined at the beginning of §5.1.5 and consider the inclusion map

(5.30) ι_{β} : L̄_{𝔱′,𝔰}^{vir} − I(ν(β)) → L̄_{𝔱′,𝔰}^{vir}.

Let [c_{β}] ∈ H^{deg(β)}(L̄_{𝔱′,𝔰}^{vir}−I(ν(β)); ℝ) denote the cohomology class which c_{β} represents and let μ̄_{p}(β) ∈ H^{deg(β)}(L̄_{𝔱′,𝔰}^{vir}; ℝ) be the restriction from M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} to L̄_{𝔱′,𝔰}^{vir} of the class in Definition 4.5. Then,

[c_{β}] = ι_{β}^{∗}μ̄_{p}(β). (5.31)

Proof. By the definition of c_{β} in §5.1.3, the restrictions of the cohomology classes in (5.31) to L_{𝔱′,𝔰}^{vir} coincide. The identity (5.31) will then follow from Lemma 5.11 if we can prove that the difference of the two cohomology classes is in the kernel of the restriction map given by the second component of the map (5.26):

ι_{1}^{∗} : H^{deg(β)}(L̄_{𝔱′,𝔰}^{vir} − I(ν(β)); ℝ) → H^{deg(β)}(L̄_{𝔱′,𝔰}^{vir} − π_{X}^{−1}(ν(β)); ℝ).

First, we describe the image of μ̄_{p}(β) under this map. Because there is an inclusion

(L̄_{𝔱′,𝔰}^{vir} − π_{X}^{−1}(ν(β))) × T_{β} → (L̄_{𝔱′,𝔰}^{vir} × X) − (π_{X} × id_{X})^{−1}(Δ),

the restriction of (π_{X} × id_{X})^{∗} PD[Δ]/β to L̄_{𝔱′,𝔰}^{vir} − π_{X}^{−1}(ν(β)) vanishes. The expression for μ̄_{p}(β) in Definition 4.5 then implies that

ι_{1}^{∗}ι_{β}^{∗}μ̄_{p}(β) = −\frac{1}{4}ι_{1}^{∗}ι_{β}^{∗}(p_{1}(𝔽_{𝔱′,𝔰}^{vir})/β). (5.32)

We now consider the restriction of c_{β}. Because the geometric representative V(β) is pulled back from the quotient space B^{∗}(ν(β)) of irreducible SO(3) connections over ν(β) by the map

r_{ν(β)} ∘ γ : L̄_{𝔱′,𝔰}^{vir} − I(ν(β)) → B^{∗}(ν(β)),

the cocycle c_{β} is pulled back from a cocycle c_{T,β} on B^{∗}(ν(β)). By the construction [31, pp. 588-592] of the geometric representatives, the cocycle c_{T,β} represents the cohomology class −\frac{1}{4}p_{1}(𝔽_{β})/β defined by the universal SO(3) bundle:

𝔽_{β} := A^{∗}(ν(β)) ×_{G} 𝔤_{V′}|_{T_{β}} → B^{∗}(ν(β)) × T_{β}

We conclude that

[ι_{1}^{∗}c_{β}] = −\frac{1}{4}ι_{1}^{∗}(r_{ν(β)} ∘ γ)^{∗}(p_{1}(𝔽_{β})/β). (5.33)

62 PAUL M. N. FEEHAN AND THOMAS G. LENESS

To compare the cohomology classes (5.32) and (5.33), we compare the bundles 𝔽_{β} and 𝔽̄_{𝔱′,𝔰}^{vir,∗}. If ι_{T} : T_{β} → X is the inclusion map, then there is an isomorphism ((ι_{β} ∘ ι_{1}) × ι_{T})^{∗}𝔽̄_{𝔱′,𝔰}^{vir,∗} ≅ ((r_{ν(β)} ∘ γ ∘ ∘ι_{1}) × id_{T_{β}})^{∗}𝔽_{β}. (5.34) (The existence of this isomorphism follows from the method used to obtain the corresponding isomorphism in Lemma 4.3.) The isomorphism (5.34) then implies that

ι_{1}^{∗}(r_{ν(β)} ∘ γ)^{∗}(p_{1}(𝔽_{β})/β) = ι_{1}^{∗}ι_{β}^{∗}(p_{1}(𝔽̄_{𝔱′,𝔰}^{vir,∗})/β)

which, together with the identities (5.32) and (5.33), yields the desired result, (5.31).

A similar argument gives the Lemma 5.13. Let c_{W} be the cocycle defined on L_{𝔱′,𝔰}^{vir} − I(ν(x)) which represents μ_{c} when restricted to L_{𝔱′,𝔰}^{vir}. Consider the inclusion map (5.35) ι_{W} : L̄_{𝔱′,𝔰}^{vir} − I(ν(x)) → L̄_{𝔱′,𝔰}^{vir}. If μ̄_{c} ∈ H^{2}(L̄_{𝔱′,𝔰}^{vir}; ℝ) denotes the restriction from M̄_{𝔱′,𝔰}^{vir,∗} to L̄_{𝔱′,𝔰}^{vir} of the class in Definition 4.5, then [c_{W}] = ι_{W}^{∗} μ̄_{c}. (5.36)

We now construct the modification of the cocycle c_{β} on L_{𝔱′,𝔰}^{vir} necessary to extend it to a cocycle on L̄_{𝔱′,𝔰}^{vir}. For any space A, let S^{p}(A; ℝ) be the module of real, singular p-cochains on A. For any pair (A, B), let S^{p}(A, B; ℝ) be the module of real, singular p-cochains on A, vanishing on p-chains in B. Lemma 5.14. For β ∈ H_{•}(X; ℝ), let U(β) ⊂ L̄_{𝔱′,𝔰}^{vir} be any open neighborhood of I(ν(β) satisfying I(ν(β)) ⋐ U(β). Let ι_{β} be the inclusion map (5.30). Then, there is a cochain (5.37) θ_{β} ∈ S^{deg(β)−1}(L̄_{𝔱′,𝔰}^{vir} − I(ν(β)), L̄_{𝔱′,𝔰}^{vir} − U(β); ℝ), and a cocycle c̄_{β} representing the class μ̄_{p}(β) ∈ H^{•}(L̄_{𝔱′,𝔰}^{vir}; ℝ) in Definition 4.5 such that

ι_{β}^{∗}c̄_{β} = c_{β} + δθ_{β}.

Proof. Let c̄′_{β} be any cocycle in the cohomology class μ̄_{p}(β). By Lemma 5.12, there is a cochain θ_{0} of degree deg(β) − 1 on L_{𝔱′,𝔰}^{vir} − I(ν(β)) such that ι_{β}^{∗}c̄′_{β} = c_{β} + δθ_{0}. (5.38) Because I(ν(β)) ⋐ U(β), there is an open subspace U′(β) ⊂ L̄_{𝔱′,𝔰}^{vir} with

I(ν(β)) ⊂ U′(β) ⋐ U(β).

Hence, the following intersection is empty:

(L̄_{𝔱′,𝔰}^{vir} − U(β)) ∩ U′(β) = ∅.

This implies that the map

S^{p} (L̄_{𝔱′,𝔰}^{vir} − I(ν(β)), L̄_{𝔱′,𝔰}^{vir} − U(β); ℝ) ⊕ S^{p} (L̄_{𝔱′,𝔰}^{vir} − I(ν(β)), U′(β); ℝ)

↓
S^{p}(L̄_{𝔱′,𝔰}^{vir} − I(ν(β)), (L̄_{𝔱′,𝔰}^{vir} − U(β)) ∩ U′(β); ℝ)

which is surjective because the pairs appearing in this diagram are excisive couples; see [50, Theorem 4.6.3] and [50, p. 218], is actually a map to the space of absolute cochains,

S^{p}(L̄_{𝔱′,𝔰}^{vir} − I(ν(β)), ∅; ℝ).

Thus, we can write θ_{0} = θ_{β} + θ_{p}, where

θ_{β} ∈ S^{deg(β)−1} (L̄_{𝔱′,𝔰}^{vir} − I(ν(β)), L̄_{𝔱′,𝔰}^{vir} − U(β); ℝ) ,
θ_{p} ∈ S^{deg(β)−1} (L̄_{𝔱′,𝔰}^{vir} − I(ν(β)), U′(β); ℝ) .

Because θ_{p} is supported on the complement of U′(β) in L̄_{𝔱′,𝔰}^{vir}−I(ν(β)), we see that θ_{p} defines a cochain of degree deg(β) − 1 on L̄_{𝔱′,𝔰}^{vir} (by extending θ_{p} by zero) and if we set

c̄_{β} = c̄′_{β} − δθ_{p},

then c̄_{β} also represents the cohomology class μ̄_{p}(β) and equation (5.38) yields

ι_{β}^{∗}c̄_{β} = ι_{β}^{∗}c̄′_{β} − δθ_{p}
= c_{β} + δ(θ_{β} + θ_{p}) − δθ_{p}
= c_{β} + δθ_{β}.

This completes the proof.

A similar argument yields:

Lemma 5.15. Let U(x) ⊂ L̄_{𝔱′,𝔰}^{vir} be any open neighborhood of I(ν(x) satisfying I(ν(x)) ⋐ U(x). Let ι_{W} be the inclusion map (5.35). Then, there is a cochain

(5.39) θ_{W} ∈ S^{1}(L̄_{𝔱′,𝔰}^{vir} − I(ν(x)), L̄_{𝔱′,𝔰}^{vir} − U(x); ℝ)

and a cocycle c̄_{W} in the cohomology class μ̄_{c} ∈ H^{2}(L̄_{𝔱′,𝔰}^{vir}; ℤ) in Definition 4.5 such that

ι_{W}^{∗} c̄_{W} = c_{W} + δθ_{W}.

The construction of c̄_{β} and c̄_{W} in Lemmas 5.14 and 5.15 yields the following useful corollary:

Corollary 5.16. For β ∈ H_{•}(X; ℝ), let c̄_{β} and c̄_{W} be the cocycles defined in Lemmas 5.14 and 5.15 respectively. Then, as elements of H^{•}(L̄_{𝔱′,𝔰}^{vir}; ℝ),

[c̄_{β}] = μ̄_{p}(β) and [c̄_{W}] = μ̄_{c}. (5.40)

5.1.6. Eliminating the excision isomorphism. The next step in the proof of Proposition 5.2 is to eliminate the excision isomorphism ι_{∗} from the pairing in (5.23). We do this by presenting the relative cohomology classes as restrictions (that is, the image under ι^{∗}) of relative cohomology classes on L̄_{𝔱′,𝔰}^{vir} and using the adjoint relation between ι_{∗} and ι^{∗}.

The relative Euler class is not the restriction of a relative cohomology class on L̄_{𝔱′,𝔰}^{vir} because the obstruction bundle Υ_{𝔱′,𝔰}/S^{1} does not extend over L_{𝔱′,𝔰}^{low}. We use the decomposition Υ_{𝔱′,𝔰} = Υ_{𝔱′,𝔰}^{s} ⊕ Υ_{𝔱′,𝔰}^{i} of the obstruction bundle, with background component Υ_{𝔱′,𝔰}^{s}/S^{1} which extends over L_{𝔱′,𝔰}^{low} (yielding Ῡ_{𝔱′,𝔰}^{s}/S^{1} in (3.64)), and an instanton component (which does not extend), Υ_{𝔱′,𝔰}^{i}/S^{1}, to overcome this difficulty. We observe in Lemma 5.17 that we can take advantage of this decomposition of Υ_{𝔱′,𝔰}/S^{1} to perturb the section χ without changing the intersection number in (5.23). Recall from (5.7) and (5.22) that the relative homology class in the pairing (5.41) below is contained in H_{max}(L_{𝔱′,𝔰}^{vir}, L_{𝔱′,𝔰}^{vir,s}; ℤ).

Lemma 5.17. Let χ = χ_{s} ⊕ χ_{i} be the obstruction section provided by Theorem 3.8. Then the cup product in (5.41) is a cohomology class in H^{max}(L_{𝔱′,𝔰}^{vir}, L_{𝔱′,𝔰}^{vir,s}; ℤ) and so the following pairing is well-defined:

(5.41) ⟨[c(z, η)] ⌣ e (Υ_{𝔱′,𝔰}/S^{1}, χ_{s}|_{L_{𝔱′,𝔰}^{vir,s}}) , ι_{∗}^{−1}(j̄_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}]⟩ .

Moreover, we can replace the section χ_{s} in the definition of the relative Euler class in the pairing (5.41) with the section χ, so the pairing (5.41) is equal to the pairing on the right-hand side of equation (5.23):

⟨[c(z, η)] ⌣ e (Υ_{𝔱′,𝔰}/S^{1}, χ|_{L_{𝔱′,𝔰}^{vir,s}}) , ι_{∗}^{−1}(j̄_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}]⟩
= ⟨[c(z, η)] ⌣ e (Υ_{𝔱′,𝔰}/S^{1}, χ_{s}|_{L_{𝔱′,𝔰}^{vir,s}}) , ι_{∗}^{−1}(j̄_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}]⟩ .

(5.42)

Proof. Because χ^{−1}(0) ⊂ χ_{s}^{−1}(0), there is an inclusion map of pairs:

ȷ_{s} : (L_{𝔱′,𝔰}^{vir}, L_{𝔱′,𝔰}^{vir,s} − χ_{s}^{−1}(0)) → (L_{𝔱′,𝔰}^{vir}, L_{𝔱′,𝔰}^{vir,s} − χ^{−1}(0)) .

The sections χ and χ_{s} are homotopic through non-vanishing sections on L_{𝔱′,𝔰}^{vir,s}−χ_{s}^{−1}(0) (by the homotopy χ_{s} ⊕ tχ_{i}, for t ∈ [0, 1]) and so the relative Euler classes are equal,

(5.43) ȷ_{s}^{∗}e (Υ_{𝔱′,𝔰}/S^{1}, χ|_{L_{𝔱′,𝔰}^{vir,s}}) = e (Υ_{𝔱′,𝔰}/S^{1}, χ_{s}|_{L_{𝔱′,𝔰}^{vir,s}})

as elements of

H^{2r_{Ξ}+2} (L_{𝔱′,𝔰}^{vir}, L_{𝔱′,𝔰}^{vir,s} − χ_{s}^{−1}(0); ℤ) .

For δ sufficiently small, the intersection K̄(z, η) ∩ L̄_{𝔱′,𝔰}^{vir,s} ∩ χ_{s}^{−1}(0) is empty by Lemma 5.4 and the continuity of the gluing map and obstruction section χ_{s} with respect to Uhlenbeck limits. Because [c(z, η)] is supported on K(z, η) while the relative Euler class in (5.43) is supported on the complement of χ_{s}^{−1}(0), the cup-product in (5.41) thus defines an element of

H^{max} (L_{𝔱′,𝔰}^{vir}, L_{𝔱′,𝔰}^{vir,s}; ℝ) ,

(compare the argument giving (5.20)), so the pairing (5.41) is well-defined. We can then write

⟨[c(z, η)] ⌣ e (Υ_{𝔱′,𝔰}/S^{1}, χ|_{L_{𝔱′,𝔰}^{vir,s}}) , ι_{∗}^{−1}(j̄_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}]⟩
= ⟨[c(z, η)] ⌣ ȷ_{s}^{∗}e (Υ_{𝔱′,𝔰}/S^{1}, χ|_{L_{𝔱′,𝔰}^{vir,s}}) , ι_{∗}^{−1}(j̄_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}]⟩
= ⟨[c(z, η)] ⌣ e (Υ_{𝔱′,𝔰}/S^{1}, χ_{s}|_{L_{𝔱′,𝔰}^{vir,s}}) , ι_{∗}^{−1}(j̄_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}]⟩ ,

using equation (5.43) in the last line. This proves the desired identity (5.42).

The advantage of the relative Euler class e(Υ_{𝔱′,𝔰}/S^{1}, χ_{s}) over e(Υ_{𝔱′,𝔰}/S^{1}, χ) is that it can be written as a cup-product,

(5.44) e (Υ_{𝔱′,𝔰}/S^{1}, χ_{s}|_{L_{𝔱′,𝔰}^{vir,s}}) = ι^{∗}e (Ῡ_{𝔱′,𝔰}^{s}/S^{1}, χ_{s}|L̄_{𝔱′,𝔰}^{vir,s}) ⌣ e (Υ_{𝔱′,𝔰}^{i}/S^{1}) ,

where ι is the inclusion map (5.21). Note that the continuity of the obstruction map χ_{s} with respect to Uhlenbeck limits is required to define the first relative Euler class appearing on the right-hand-side of (5.44) as a continuous section is needed to define a relative Euler class. The relative cohomology class and the absolute cohomology class on the right-hand-side of equation (5.44) are both restrictions of classes on L̄_{𝔱′,𝔰}^{vir}. We now show that we can also replace the relative cohomology class [c(z, η)] with one which extends over L̄_{𝔱′,𝔰}^{vir} without changing the pairing (5.41).

Let z = β_{1} ⋯ β_{m} ∈ 𝔸(X) be the monomial appearing in Proposition 5.2. We replace c_{W}^{η} with

c_{W,1} ⌣ ⋯ ⌣ c_{W,η},

as it will be necessary to distinguish between the factors of this cup-product, which are defined via specific choices described below. Let ν′(β_{i}) and ν′(x_{j}), j = 1, …, η, be tubular neighborhoods of T_{β_{i}} and the point x_{j} defining c_{W,j}, respectively. Assume these neighborhoods satisfy both ν(β_{i}) ⋐ ν′(β_{i}) and ν(x_{j}) ⋐ ν′(x_{j}) and the intersection condition (5.10). Therefore, we can find open subspaces U(β_{i}) and U(x_{j}) in L̄_{𝔱′,𝔰}^{vir} such that

I(ν(β)) ⋐ U(β) and I(ν(x_{j})) ⋐ U(x_{j}),

and such that the intersection

U(β_{i_{1}}) ∩ ⋯ ∩ U(β_{i_{k}}) ∩ U(x_{j_{1}}) ∩ ⋯ ∩ U(x_{j_{r}}), (5.45)

is empty unless

∑_{p=1}^{k} deg(β_{i_{p}}) + 4r ≤ 4. (5.46)

Then let

ι_{β_{i}}^{∗}c̄_{β_{i}} = c_{β_{i}} + δθ_{β_{i}} and ι_{W}^{∗} c̄_{W,j} = c_{W,j} + δθ_{W,j}, (5.47)

be the cocycles constructed in Lemmas 5.14 and 5.15 respectively, where c̄_{β_{i}} and c̄_{W,j} are cocycles on L̄_{𝔱′,𝔰}^{vir}, δθ_{β_{i}} is supported in U(β_{i}), and δθ_{W,j} is supported in U(x_{j}). Define

c̄(z, η) = c̄_{β_{1}} ⌣ ⋯ ⌣ c̄_{β_{m}} ⌣ c̄_{W,1} ⌣ ⋯ ⌣ c̄_{W,η}, (5.48)

so ι^{∗}c̄(z, η) = c(z, η). We now show that we can replace c(z, η) with ι^{∗}c̄(z, η) without changing the pairing (5.41).

Lemma 5.18. Let z = β_{1} ⋯ β_{m} ∈ 𝔸(X) be the monomial appearing in Proposition 5.2. Let ι be the inclusion map (5.21). Then we can replace the class [c(z, η)] in the pairing (5.41) by the class ι^{∗}[c̄(z, η)]:

⟨[c(z, η)] ⌣ e (Υ_{𝔱′,𝔰}/S^{1}, χ_{s}|_{L_{𝔱′,𝔰}^{vir,s}}) , ι_{∗}^{−1}(j̄_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}]⟩
= ⟨ι^{∗}[c̄(z, η)] ⌣ e (Υ_{𝔱′,𝔰}/S^{1}, χ_{s}|_{L_{𝔱′,𝔰}^{vir,s}}) , ι_{∗}^{−1}(j̄_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}]⟩ .

(5.49)

Proof. By construction and (5.47), the cocycles c(z, η) and ι^{∗}c̄(z, η) differ by cocycles of the form (up to a re-ordering which will be seen to be irrelevant)

c(z′, η − r) ⌣ δθ_{i_{1}} ⌣ ⋯ ⌣ δθ_{i_{k}} ⌣ δθ_{j_{1}} ⌣ ⋯ ⌣ δθ_{j_{r}}, (5.50)

where z = z′β_{i_{1}} ⋯ β_{i_{k}}. Note that we assume k > 0 or r > 0, so we actually have a difference term in (5.50) and not just ι^{∗}c̄(z, η). Because the intersection (5.45) is empty unless the condition (5.46) holds and because δθ_{β_{i}} and δθ_{W,j} have support in U(β_{i}) and U(x_{j}) respectively, the term (5.50) vanishes unless condition (5.46) holds. Consequently,

deg(z′) + 2(η − r) ≥ deg(z) + 2η − (∑_{μ=1}^{k} deg(β_{i}) + 4r)
≥ dim M_{𝔱′} − 6
= dim M_{𝔱}
= dim (χ_{𝔰}^{−1}(0) ∩ L_{𝔱′,𝔰}^{low}) + 2.

(5.51)

The inequality (5.51) implies that the following intersection is empty:

K̄(z′, η − r) ∩ L_{𝔱′,𝔰}^{low} ∩ χ_{s}^{−1}(0) = ∅.

Hence, because of the continuity of the gluing map and of the obstruction section χ_{s} on L̄_{𝔱′,𝔰}^{vir,s} and by the same argument used to establish (5.19), the following intersection is also empty for δ sufficiently small:

K(z′, η − r) ∩ L_{𝔱′,𝔰}^{vir,s} ∩ χ_{s}^{−1}(0) = ∅.

By the reasoning which gave (5.20), the preceding equality implies that the cup-product

[c(z′, η − r)] ⌣ e (Ῡ_{𝔱′,𝔰}^{s}/S^{1}, χ_{s}|_{L_{𝔱′,𝔰}^{vir,s}})

vanishes on L_{𝔱′,𝔰}^{vir,s} for δ sufficiently small. If we assume that the neighborhoods U(β_{i}) and U(x_{j}) are contained in L_{𝔱′,𝔰}^{vir,s} for all i and j, the assumption that k > 0 or r > 0 implies that the difference term (5.50) is supported in L_{𝔱′,𝔰}^{vir,s} and hence the cup-product of the difference term with the relative Euler class vanishes. Therefore identity (5.49) holds.

We now have all the ingredients we need to conclude the proof of Proposition 5.2 at our disposal:

Proof of Proposition 5.2. Equations (5.23), (5.42), (5.44), and (5.49) yield the identity:

# (V̄(z) ∩ W̄^{η} ∩ L̄_{𝔱′,𝔰})
= ⟨ι^{∗} ([c̄(z, η)] ⌣ e (Ῡ_{𝔱′,𝔰}^{s}/S^{1}, χ_{s}|L̄_{𝔱′,𝔰}^{vir,s}) ⌣ ē_{i}) , ι_{∗}^{−1}(j̄_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}]⟩
= ⟨[c̄(z, η)] ⌣ e (Ῡ_{𝔱′,𝔰}^{s}/S^{1}, χ_{s}|L̄_{𝔱′,𝔰}^{vir,s}) ⌣ ē_{i}, (j̄_{L^{i}})_{∗} [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}]⟩ .

(5.52)

Because c̄(z, η) is a cocycle representing the cohomology class μ̄_{p}(z) ⌣ μ̄_{c}^{η} on M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} by Corollary 5.16 and because of the relation (5.14) between relative and absolute Euler classes and the definition of ē_{s} given by Lemma 4.11, we see that

\begin{matrix}ȷ^{∗} ([c̄(z, η)] ⌣ e (Ῡ_{𝔱′,𝔰}^{s}/S^{1}, χ_{s}|L̄_{𝔱′,𝔰}^{vir,s}) ⌣ ē_{i}) \\ = μ̄_{p}(z) ⌣ μ̄_{c}^{η} ⌣ ē_{s} ⌣ ē_{i}, \end{matrix} (5.53)

where ȷ is the inclusion map (5.5). Combining the expression (5.52) for the intersection number, equation (5.53), and the definition of [L̄_{𝔱′,𝔰}^{vir}] in (5.9) then completes the proof of Proposition 5.2.

5.2. Cohomological results. In this section we prove some results about the cohomology ring of L_{𝔱′,𝔰}^{vir} which we shall need to compute the right-hand side of equation (5.2) and thus complete the proof of Theorem 6.1. In Lemma 5.19, we compute the Poincaré dual of the submanifold of L_{𝔱′,𝔰}^{vir},

BL_{𝔱′,𝔰}^{vir,i} = M̃_{𝔰} ×_{G_{𝔰}×S^{1}} ∂Ḡl_{𝔱}(δ)
⊂ Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}×S^{1}} ∂Ḡl_{𝔱}(δ) = L_{𝔱′,𝔰}^{vir,i} ⊂ L_{𝔱′,𝔰}^{vir},

(5.54)

and in Propositions 5.20 and 5.21 we give a formula for division by this Poincaré dual. Dividing the cohomology class in the pairing on the right-side of identity (5.2) by the Poincaré dual of the fundamental class of BL_{𝔱′,𝔰}^{vir,i} reduces the pairing with [L̄_{𝔱′,𝔰}^{vir}] to one with [BL_{𝔱′,𝔰}^{vir,i}]. Because, as noted in equation (3.83), the space BL_{𝔱′,𝔰}^{vir,i} is a product

BL_{𝔱′,𝔰}^{vir,i} = M_{𝔰} × ∂Ḡl_{𝔱}(δ)/S^{1}, (5.55)

the result of pairing with the fundamental class [BL_{𝔱′,𝔰}^{vir,i}] is equal to a product of pairings with [M_{𝔰}] and with [∂Ḡl_{𝔱}(δ)/S^{1}]. The former gives the Seiberg-Witten invariant, while the latter has been previously computed in [34] (see Lemma 5.24).

A similar technique would work for the link of M_{𝔰} × Sym^{ℓ}(X), for ℓ ≥ 2, although the computations of Lemma 5.24 for the analogue of the space ∂Ḡl_{𝔱}(δ)/S^{1} when ℓ ≥ 3 appear very difficult, though the case of ℓ = 2 can be deduced from the work of [34].

Both the computation of the Poincaré dual of BL_{𝔱′,𝔰}^{vir,i} and our formula for division by this Poincaré dual use equivariant cohomology, which we now briefly review.

As customary, ES^{1} → BS^{1} denotes the universal circle bundle, where BS^{1} is homotopy equivalent to ℂℙ^{∞}. For any pair of topological spaces (X, Y) with a circle action,

H_{S^{1}}^{•}(X; ℤ) = H^{•}(ES^{1} ×_{S^{1}}X; ℤ),
H_{S^{1}}^{•}(X, Y; ℤ) = H^{•}(ES^{1} ×_{S^{1}}X, ES^{1} ×_{S^{1}}Y; ℤ),

are the equivariant cohomology of X and relative equivariant cohomology of (X, Y), respectively [1, §2].

Lemma 5.19. There is a continuous map

π_{N,ν} : M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} → N_{𝔱,𝔰} ×_{S^{1}} ES^{1}, (5.56)

such that if Th_{S^{1}}(N_{𝔱,𝔰}) ∈ H_{S^{1}}^{•}(N_{𝔱,𝔰}, N_{𝔱,𝔰}^{0}; ℤ) is the Thom class [37, Theorem 9.1], [5, §6] of the bundle

N_{𝔱,𝔰} ×_{S^{1}} ES^{1} → M_{𝔰} × BS^{1}, (5.57)

where N_{𝔱,𝔰}^{0} denotes the complement of the zero-section, and we have an inclusion map,

(5.58) j_{BL} : (L̄_{𝔱′,𝔰}^{vir}, ∅) → (L̄_{𝔱′,𝔰}^{vir}, L̄_{𝔱′,𝔰}^{vir} − BL_{𝔱′,𝔰}^{vir,i}) ,

then the following hold:

1. For any ω ∈ H^{d_{s}(𝔰)+6}(L̄_{𝔱′,𝔰}^{vir}; ℤ), where dim BL_{𝔱′,𝔰}^{vir,i} = d_{s}(𝔰) + 6, we have

(5.59) ⟨ω ⌣ j_{BL}^{∗}π_{N,ν}^{∗} Th_{S^{1}}(N_{𝔱,𝔰}), [L̄_{𝔱′,𝔰}^{vir}]⟩ = ⟨ω, [BL_{𝔱′,𝔰}^{vir,i}]⟩ .

2. If c ∈ H^{2}(BS^{1}; ℤ) is the universal first Chern class and

π_{B} : N_{𝔱,𝔰} ×_{S^{1}} ES^{1} → BS^{1},

is the projection and ν ∈ H^{2}(M̄_{𝔱′,𝔰}^{vir}/S^{1}; ℤ) is the cohomology class of Definition 4.1, then

π_{N,ν}^{∗}π_{B}^{∗}c = ν.

3. If μ_{𝔰} ∈ H^{2}(M_{𝔰}; ℤ) is as defined in (2.21) and

π_{M} : N_{𝔱,𝔰} ×_{S^{1}} ES^{1} → M_{𝔰},

is the projection, then

π_{N,ν}^{∗}π_{M}^{∗}μ_{𝔰} = π_{𝔰}^{∗}μ_{𝔰}.

Proof. We define a bundle map π_{N} by the obvious projection so that the following diagram commutes:

\begin{matrix}Ñ_{𝔱,𝔰} ×_{G_{𝔰}} Ḡl_{𝔱}(δ) & ⟶^{π_{N}} & N_{𝔱,𝔰} \\ ↓ & & ↓ \\ M̃_{𝔰} ×_{G_{𝔰}} Ḡl_{𝔱}(δ) & ⟶^{π_{𝔰}} & M_{𝔰} \end{matrix} (5.60)

By construction, the projection π_{N} is S^{1}-equivariant where the circle acts by scalar multiplication on the fibers of N_{𝔱,𝔰} and by the action in Definition 4.1 defining ν on the fibers of the bundle on the left-hand-side of the diagram. Let ι_{ν} be the classifying map for the circle bundle defining ν, covered by the bundle map ι̃_{ν}, so that the following diagram commutes:

M̄_{𝔱′,𝔰}^{vir,∗} ⟶^{ι̃_{ν}} ES^{1}
↓  ↓
M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} ⟶^{ι_{ν}} BS^{1}

(5.61)

By construction of the map ι_{ν}, we have ι_{ν}^{∗}c = ν. Then, because the maps π_{N} and ι̃_{ν} are S^{1}-equivariant, the product π_{N} × ι̃_{ν} defines the map π_{N,ν} on the circle quotients:

π_{N,ν} : M̄_{𝔱′,𝔰}^{vir,∗}/S^{1} → N_{𝔱,𝔰} ×_{S^{1}} ES^{1} .

Observe that the intersection of the pre-image under π_{N,ν} of the zero-section of the bundle (5.57) with L̄_{𝔱′,𝔰}^{vir} is the base space (5.54):

(5.62)  L̄_{𝔱′,𝔰}^{vir} ∩ π_{N,ν}^{−1}(M_{𝔰} × BS^{1}) = BL_{𝔱′,𝔰}^{vir,i}.

Hence, the map π_{N,ν} induces a map on relative cohomology,

π_{N,ν}^{∗} : H_{S^{1}}^{•}(N_{𝔱,𝔰}, N_{𝔱,𝔰}^{0}; ℤ) → H^{•}(L̄_{𝔱′,𝔰}^{vir}, L̄_{𝔱′,𝔰}^{vir} − BL_{𝔱′,𝔰}^{vir,i}; ℤ),

where N_{𝔱,𝔰}^{0} denotes the complement of the zero-section. Because the restriction of π_{N,ν} to L_{𝔱′,𝔰}^{vir,i} defines a bundle map,

Ñ_{𝔱,𝔰}(ε) ×_{G_{𝔰}×S^{1}} ∂Ḡl_{𝔱}(δ) ⟶^{π_{N,ν}} N_{𝔱,𝔰} ×_{S^{1}} ES^{1}
π_{Gl}↓  ^{π_{N} ×π_{B}} ↓
M̃_{𝔰} ×_{G_{𝔰}×S^{1}} ∂Ḡl_{𝔱}(δ) ⟶^{π_{𝔰}×ι_{ν}} M_{𝔰} × BS^{1}

(5.63)

the Thom class of the normal bundle of BL_{𝔱′,𝔰}^{vir,i} in L_{𝔱′,𝔰}^{vir,i} is given by π_{N,ν}^{∗} Th_{S^{1}}(N_{𝔱,𝔰}). The relative fundamental class of the manifold with boundary L_{𝔱′,𝔰}^{vir,i} is equal to [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}], as noted in (5.7). Hence, because the Thom class of the normal bundle of a submanifold is equal to the Poincaré dual of that submanifold [6, p. 371], the fundamental class of BL_{𝔱′,𝔰}^{vir,i} is given by

(5.64) [BL_{𝔱′,𝔰}^{vir,i}] = j_{1}^{∗}π_{N,ν}^{∗} Th_{S^{1}}(N_{𝔱,𝔰}) ∩ [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}] ,

where [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}] is defined following (5.6) and

j_{1} : (L̄_{𝔱′,𝔰}^{vir}, L̄_{𝔱′,𝔰}^{vir,s}) → (L̄_{𝔱′,𝔰}^{vir}, L̄_{𝔱′,𝔰}^{vir} − BL_{𝔱′,𝔰}^{vir,i})

is the inclusion of pairs. Observe that

j_{BL} = j_{1} ∘ j, (5.65)

where j : (L̄_{𝔱′,𝔰}^{vir}, ∅) → (L̄_{𝔱′,𝔰}^{vir}, L̄_{𝔱′,𝔰}^{vir,s}) is the inclusion (5.5). Thus, for any ω ∈ H^{•}(L̄_{𝔱′,𝔰}^{vir}; ℤ), we have

⟨ω ⌣ j_{BL}^{∗}π_{N,ν}^{∗} Th_{S^{1}}(N_{𝔱,𝔰}), [L̄_{𝔱′,𝔰}^{vir}]⟩
= ⟨j^{∗} (ω ⌣ j_{1}^{∗}π_{N,ν}^{∗} Th_{S^{1}}(N_{𝔱,𝔰})) , [L̄_{𝔱′,𝔰}^{vir}]⟩ (By (5.65) & [50, Statement 5.6.8])
= ⟨ω ⌣ j_{1}^{∗}π_{N,ν}^{∗} Th_{S^{1}}(N_{𝔱,𝔰}), j_{∗}[L̄_{𝔱′,𝔰}^{vir}]⟩
= ⟨ω, j_{1}^{∗}π_{N,ν}^{∗} Th_{S^{1}}(N_{𝔱,𝔰}) ∩ [L_{𝔱′,𝔰}^{vir,i}, ∂L_{𝔱′,𝔰}^{vir,i}]⟩ (By Equation (5.9))
= ⟨ω, [BL_{𝔱′,𝔰}^{vir,i}]⟩ , (By Equation (5.64))

which proves equation (5.59) and thus Assertion (1).

Because the diagram (5.61) commutes, we observe that π_{B} ∘ π_{N,ν} = ι_{ν}; this observation and the fact that ι_{ν}^{∗}c = ν proves Assertion (2).

Because the diagram (5.60) commutes, we have π_{M} ∘ π_{N,ν} = π_{𝔰} and this proves Assertion (3), completing the proof of the lemma.

We next give an explicit formula for division by an equivariant Thom class in terms of the Segre classes of the vector bundle; this division process is also referred to as equivariant localization — see [4, Theorem 7.13]. Recall that the total Segre class s(N) = s_{0}(N) + s_{1}(N) + ⋯, where s_{i}(N) ∈ H^{2i}(M; ℤ), of a complex vector bundle N over a topological space M is the formal inverse of the total Chern class c(N) = 1 + c_{1}(N) + c_{2}(N) + ⋯, where c_{i}(N) ∈ H^{2i}(M; ℤ), so that s(N)c(N) = 1 [24, p. 69], [18, Lemma 4.10].

Proposition 5.20. Let π_{M} : N → M be a complex vector bundle of rank r over a topological space M with a fundamental class of dimension d. Suppose π_{B} : ES^{1} ×_{S^{1}}N → BS^{1} and π_{M} : ES^{1} ×_{S^{1}}N → M are the projection maps, where the circle acts diagonally on N × ES^{1} by scalar multiplication on the fibers of N and by the standard action on ES^{1}. Let N^{0} denote the complement of the zero-section and let Th_{S^{1}}(N) ∈ H_{S^{1}}^{2r}(N, N^{0}; ℤ) be the Thom class of

π_{B} × π_{M} : ES^{1} ×_{S^{1}}N → BS^{1} ×M. (5.66)

If c ∈ H^{2}(BS^{1}; ℤ) denotes the universal first Chern class, 0 ≤ k ≤ [d/2] is an integer, and α ∈ H^{d−2k}(M; ℤ), then for integers m ≥ k + r, one has

π_{B}^{∗}c^{m} ⌣ π_{M}^{∗}α = ι_{n}^{∗} Th_{S^{1}}(N) ⌣ (∑_{j=0}^{k}(−1)^{r+j}π_{B}^{∗}c^{m−r−j} ⌣ π_{M}^{∗}(α ⌣ s_{j}(N))) , (5.67)

where ι_{N} : (N, ∅) → (N, N^{0}) is the inclusion of pairs, and the pushforward formula,

π_{B,∗}(c^{m} ⌣ π_{M}^{∗}α) = (−1)^{k+r}c^{m−r−k}⟨α ⌣ s_{k}(N), [M]⟩. (5.68)

Proof. By the splitting principle we may suppose that N = ⊕_{i=1}^{r}L_{i}, with y_{i} = c_{1}(L_{i}), and so Lemma 4.6 implies that

j^{∗} Th_{S^{1}}(N) = ∏_{i=1}^{r}(−π_{B}^{∗}c + π_{M}^{∗}y_{i}) = ∑_{i=0}^{r}(−1)^{r−i}π_{B}^{∗}c^{r−i}π_{M}^{∗}c_{i}(N). (5.69)

The negative sign above arises because the circle action is diagonal, as explained in Lemma 4.6. The bundle N has total Segre class s(N) = ∏_{i=1}^{r}(1 + y_{i})^{−1}, and thus

\frac{1}{Th_{S^{1}}(N)} = ∏_{i=1}^{r} \frac{1}{(−π_{B}^{∗}c + π_{M}^{∗}y_{i})}

= (−π_{B}^{∗}c)^{−r} ∏_{i=1}^{r} \frac{1}{(1 − π_{M}^{∗}y_{i}/π_{B}^{∗}c)}

= ∑_{j=0}^{∞}(−π_{B}^{∗}c)^{−r−j}π_{M}^{∗}s_{j}(N).

This expression for the formal inverse of Th_{S^{1}}(N) yields

\frac{π_{B}^{∗}c^{m} ⌣ π_{M}^{∗}α}{Th_{S^{1}}(N)} = ∑_{j=0}^{k}(−1)^{r+j}π_{B}^{∗}c^{m−r−j} ⌣ π_{M}^{∗}(α ⌣ s_{j}(N)), (5.70)

and equation (5.67) follows. Because the pushforward π_{B,∗} is the composition of division by the Thom class Th_{S^{1}}(N) and integration over M [5, §6], we have

π_{B,∗}(π_{B}^{∗}c^{m} ⌣ π_{M}^{∗}α) = (\frac{π_{B}^{∗}c^{m} ⌣ π_{M}^{∗}α}{Th_{S^{1}}(N)}) /[M]

= (∑_{j=0}^{k}(−1)^{r+j}π_{B}^{∗}c^{m−r−j} ⌣ π_{M}^{∗}(α ⌣ s_{j}(N))) /[M],

which yields equation (5.68), since M has dimension d.

Lemmas 4.2, 4.10, 4.11, and 4.12 show how to express the cohomology classes in (5.2) in terms of the cohomology classes ν, μ_{𝔰}, and classes pulled back from X. The next proposition reduces the pairing in (5.2) of products of these latter classes with [L_{𝔱′,𝔰}^{vir}] to a pairing with the fundamental class of the submanifold (5.54).

Proposition 5.21. Let d_{s} = dim M_{𝔰} and let r_{N} denote the rank of the complex vector bundle N_{𝔱,𝔰} over M_{𝔰}. Assume b_{1}(X) = 0 and b_{2}^{+}(X) is odd. For integers 0 ≤ i ≤ d = \frac{1}{2}d_{s} and 0 ≤ k ≤ 2 and any class α ∈ H^{2k}(X; ℤ), we have

⟨ν^{d+r_{N} +3−k−i} ⌣ π_{𝔰}^{∗}μ_{𝔰}^{i} ⌣ π_{X}^{∗}α, [L̄_{𝔱′,𝔰}^{vir}]⟩

= ∑_{j=0}^{d−i}(−1)^{r_{N} +j} ⟨ν^{d+3−i−j−k} ⌣ π_{𝔰}^{∗}(μ_{𝔰}^{i} ⌣ s_{j}(N_{𝔱,𝔰})) ⌣ π_{X}^{∗}α, [BL_{𝔱′,𝔰}^{vir,i}]⟩ ,

(5.71) where ν ∈ H^{2}(L̄_{𝔱′,𝔰}^{vir}; ℤ) is the cohomology class of Definition 4.1, μ_{𝔰} ∈ H^{2}(M_{𝔰}; ℤ) is the restriction of the cohomology class defined in (2.21), and π_{X} : L̄_{𝔱′,𝔰}^{vir} → X and π_{𝔰} : L̄_{𝔱′,𝔰}^{vir} → M_{𝔰} are the restrictions of the projections defined in (4.4).

Proof. Applying the equality ν = ι_{ν}^{∗}c from Lemma 5.19 and using the abbreviations A = d + r_{N} + 3 − k, T = Th_{S^{1}}(N_{𝔱,𝔰}), and s_{j} = s_{j}(N_{𝔱,𝔰}) yields

⟨ν^{A−i} ⌣ π_{𝔰}^{∗}μ_{𝔰}^{i} ⌣ π_{X}^{∗}α, [L̄_{𝔱′,𝔰}^{vir}]⟩

= ⟨π_{N,ν}^{∗} (π_{B}^{∗}c^{A−i} ⌣ π_{𝔰}^{∗}μ_{𝔰}^{i}) ⌣ π_{X}^{∗}α, [L̄_{𝔱′,𝔰}^{vir}]⟩

= ∑_{j=0}^{d−i}(−1)^{r_{N} +j} ⟨π_{N,ν}^{∗} (π_{B}^{∗}c^{A−i−j} ⌣ π_{𝔰}^{∗}(μ_{𝔰}^{i} ⌣ s_{j}) ⌣ j^{∗}T ) ⌣ π_{X}^{∗}α, [L̄_{𝔱′,𝔰}^{vir}]⟩

(By Equation (5.67))

= ∑_{j=0}^{d−i}(−1)^{r_{N} +j} ⟨(j^{∗}π_{N,ν}^{∗}T) ⌣ ν^{A−r_{N} −i−j} ⌣ π_{𝔰}^{∗}(μ_{𝔰}^{i} ⌣ s_{j}) ⌣ π_{X}^{∗}α, [L̄_{𝔱′,𝔰}^{vir}]⟩

= ∑_{j=0}^{d−i}(−1)^{r_{N} +j} ⟨ν^{A−r_{N} −i−j} ⌣ π_{𝔰}^{∗}(μ_{𝔰}^{i} ⌣ s_{j}) ⌣ π_{X}^{∗}α, [BL_{𝔱′,𝔰}^{vir,i}]⟩

(5.72)

where the last equality follows from equation (5.59). This proves the desired formula (5.71).

The next step is to use the product decomposition (5.55) of BL_{𝔱′,𝔰}^{vir,i} to reduce the pairing on the right-hand side of equation (5.71) to a sum of products of pairings with [M_{𝔰}] and pairings with [∂Ḡl_{𝔱}(δ)/S^{1}]. To accomplish this we must write the cohomology classes appearing in (5.71), namely μ_{𝔰}, ν, and π_{X}^{∗}α, as pullbacks of cohomology classes on the factors M_{𝔰} and ∂Ḡl_{𝔱}(δ)/S^{1} of the product; of these three cohomology classes, only ν is not obviously such a pullback.

Definition 5.22. Let ν_{𝔱} ∈ H^{2}(∂Ḡl_{𝔱}(δ)/S^{1}; ℝ) be the first Chern class of the circle bundle

∂Ḡl_{𝔱}(δ) → ∂Ḡl_{𝔱}(δ)/S^{1},

where the circle action is described in the paragraph preceding equation (3.56). We shall also let ν_{𝔱} denote the pullback of this class to the product M_{𝔰} × ∂Ḡl_{𝔱}(δ)/S^{1}, where it is the first Chern class of the circle bundle

M_{𝔰} × ∂Ḡl_{𝔱}(δ) → M_{𝔰} × ∂Ḡl_{𝔱}(δ)/S^{1}.

In the following lemma, we obtain the desired decomposition of ν by comparing this class with ν_{𝔱}:

Lemma 5.23. Let ν, ν_{𝔱} be the first Chern classes in Definitions 4.1 and 5.22, respectively. Let π_{𝔰} : M_{𝔰} × ∂Ḡl_{𝔱}(δ)/S^{1} → M_{𝔰} be the projection. Then the restriction of ν to M_{𝔰} × ∂Ḡl_{𝔱}(δ)/S^{1} is given by

ν|_{M_{𝔰}×∂Ḡl_{𝔱}(δ)/S^{1}} = ν_{𝔱} + 2c_{1}(𝕃_{𝔰}), (5.73)

and if b_{1}(X) = 0, this simplifies to

ν|_{M_{𝔰}×∂Ḡl_{𝔱}(δ)/S^{1}} = ν_{𝔱} + 2μ_{𝔰}.

Proof. To prove the lemma, we must compare the second bundle in Definition 5.22 with the circle bundle

M̃_{𝔰} ×_{G_{𝔰}} ∂Ḡl_{𝔱}(δ) → M̃_{𝔰} ×_{G_{𝔰}×S^{1}} ∂Ḡl_{𝔱}(δ). (5.74)

The circle action in (5.74) is trivial on M̃_{𝔰} and given by the action described prior to (3.56) on ∂Ḡl_{𝔱}(δ) ⊂ Ḡl_{𝔱}(δ); observe that the action of the group G_{𝔰} in the bundle (5.74) is defined in (3.56), for s ∈ G_{𝔰}, (B, Ψ) ∈ M̃_{𝔰}, and g ∈ ∂Ḡl_{𝔱}(δ), by

(s, (B, Ψ, g)) ↦ (s_{∗}(B, Ψ), s^{−2}g).

Let 𝕃_{𝔰}^{1} be the unit sphere of the bundle (2.19), so

𝕃_{𝔰}^{1} = M̃_{𝔰} ×_{G_{𝔰}} (X × S^{1}),

with the action (2.20) of G_{𝔰}. Consider the bundle

(𝕃_{𝔰}^{1} ×_{X} ∂Ḡl_{𝔱}(δ)) /S^{1} → M_{𝔰} × ∂Ḡl_{𝔱}(δ)/S^{1}, (5.75)

where the action of e^{iγ} ∈ S^{1} is given for [B, Ψ, x, e^{iθ}] ∈ 𝕃_{𝔰}^{1}|_{x} and g ∈ ∂Ḡl_{𝔱}(δ)|_{x} by

([B, Ψ, x, e^{iθ}], g) ↦ ([B, Ψ, x, e^{i(θ+γ)}], e^{−2iγ}g) .

By Lemma 4.6, the first Chern class of the bundle (5.75) is ν_{𝔱} + 2c_{1}(𝕃_{𝔰}). To prove (5.73), it suffices to show that the bundles (5.75) and (5.74) are isomorphic. Because G_{𝔰} acts anti-diagonally in the definition of 𝕃_{𝔰}, the map

(𝕃_{𝔰}^{1} ×_{X} ∂Ḡl_{𝔱}(δ)) /S^{1} → M̃_{𝔰} ×_{G_{𝔰}} ∂Ḡl_{𝔱}(δ),
[[B, Ψ, x, e^{iθ}], g] ↦ [(B, Ψ), e^{2iθ}g]

is a well-defined bundle isomorphism. The final statement of the lemma follows from the fact that c_{1}(𝕃_{𝔰}) = μ [17, Lemma 2.24].

As discussed in the paragraph preceding Definition 5.22, the results of Lemma 5.23 show how the relevant cohomology classes on BL_{𝔱′,𝔰}^{vir,i} pull back from either M_{𝔰} or ∂Ḡl_{𝔱}(δ)/S^{1}. Pairing the appropriate cohomology class with the fundamental class [M_{𝔰}] yields the Seiberg-Witten invariant. The required pairings with [∂Ḡl_{𝔱}(δ)/S^{1}] are given by the following lemma:

Lemma 5.24. Let ν_{𝔱} be the characteristic class in Definition 5.22. Let ∂Ḡl_{𝔱}(δ)/S^{1} have the standard orientation as defined in the paragraph following (3.84). If x ∈ H_{0}(X; ℤ) is the positive generator and h ∈ H_{2}(X; ℝ), then

⟨ν_{𝔱} ⌣ π_{X}^{∗} PD[x], [∂Ḡl_{𝔱}(δ)/S^{1}]⟩ = 2,
⟨ν_{𝔱}^{2} ⌣ π_{X}^{∗} PD[h], [∂Ḡl_{𝔱}(δ)/S^{1}]⟩ = −4⟨(c_{1}(𝔰) − c_{1}(𝔱)) ⌣ PD[h], [X]⟩,
⟨ν_{𝔱}^{3}, [∂Ḡl_{𝔱}(δ)/S^{1}]⟩ = 6(c_{1}(𝔰) − c_{1}(𝔱))^{2} + 2c_{1}^{2}(X).

(5.76)

Proof. In [34, §3], a rank-two, Hermitian vector bundle, F → X′, where π : X′ → X is a finite-degree cover is constructed with F → π^{∗} Gl_{𝔱} a branched cover of degree negative two. (Similar bundles are constructed in [54, 42, 33].) The circle action on F induced by the action of ℂ^{∗} and the inclusion S^{1} ⊂ ℂ^{∗} maps to twice the circle action on π^{∗} Gl_{X}. This implies that the map on quotients, ℙ(F) → π^{∗}∂Ḡl_{𝔱}(δ)/S^{1}, has degree negative one and that the class ν_{𝔱} pulls back to 2h where h is the first Chern class of the circle bundle F/ℝ^{∗} → ℙ(F). Pulling back the cohomology classes in (5.76) to ℙ(F) and applying the computation of the Segre classes of F in [34, §3] then yield the formulas in (5.76).

Using Poincaré duality, we can use Lemma 5.24 to compute further pairings needed in our proof of Theorem 6.1. For example, if α ∈ H^{2}(X; ℝ), then

(5.77) ⟨ν_{𝔱} ⌣ π_{X}^{∗}(α ⌣ PD[h]), [∂Ḡl_{𝔱}(δ)/S^{1}]⟩ = ⟨ν_{𝔱} ⌣ π_{X}^{∗} PD[x], [∂Ḡl_{𝔱}(δ)/S^{1}]⟩⟨α, h⟩,

using

α ⌣ PD[h] = ⟨α ⌣ PD[h], [X]⟩ PD[x]
= ⟨α, PD[h] ∩ [X]⟩ PD[x]
= ⟨α, h⟩ PD[x],

by Poincaré duality.

Remark 5.25. In comparing the formulas of this and related articles with computations of wall-crossing formulas for Donaldson invariants, the cohomology class ξ = c_{1}(Q_{ξ}) ∈ H^{2}(X; ℤ) of [30, 34] defining a reduction 𝔤_{V} ≅ \underline{ℝ} ⊕ L is equal to the cohomology class c_{1}(L) = c_{1}(𝔱) − c_{1}(𝔰) = c_{1}(𝔱′) − c_{1}(𝔰) appearing in this article and in [17, 18, 14].

6. Intersection with link of level-one Seiberg-Witten moduli space

In this section we prove Theorem 1.1 by computing a formula, given in Theorem 6.1, for intersection number (5.1) and applying this formula to the cobordism sum (2.52). We divide this task into three parts. Section 6.1 is devoted to the proof of Theorem 6.1. Because the Donaldson invariants are defined using a moduli space of anti-self-dual connections on X#\overline{ℂℙ}^{2} (see (2.47)), we present a blow-up formula for the link intersection numbers in §6.2. Finally, in §6.3, we apply the results of [14] and Theorem 6.1 to the cobordism formula (2.52) to prove Theorem 1.1.

6.1. Algebraic computations and completion of proof of Theorem 6.1. With the results of §5, the proof of the following theorem is essentially algebraic. Recall that a spin^{u} structure over X splits, 𝔱 = 𝔰 ⊕ 𝔰′, if and only if (c_{1}(𝔱) − c_{1}(𝔰))^{2} = p_{1}(𝔱) [18, Lemma 3.32]. Hence, a Seiberg-Witten stratum M_{𝔰} × Sym^{ℓ}(X) is contained in the level M_{𝔱_{ℓ}} × Sym^{ℓ}(X) of the space of ideal SO(3) monopoles enclosing M̄_{𝔱′} if and only if (c_{1}(𝔱′)−c_{1}(𝔰))^{2} = p_{1}(𝔱′)+4ℓ; in this situation, L_{𝔱′,𝔰} is the link (3.79) of the stratum M_{𝔰}×Sym^{ℓ}(X) in M̄_{𝔱′}/S^{1}.

Theorem 6.1. Let X be a four-manifold with b_{1}(X) = 0, odd b_{2}^{+}(X) ≥ 1. Suppose X has a generic Riemannian metric and a spin^{u} structure 𝔱′, where w_{2}(𝔱′) is good in the sense of Definition 2.3. Let 𝔰 be a spin^{c} structure over X for which (c_{1}(𝔱′) − c_{1}(𝔰))^{2} = p_{1}(𝔱′) + 4. Let δ, m, and η be non-negative integers satisfying

0 ≤ m ≤ [δ/2] and 2(δ + η) = dim(M_{𝔱′}/S^{1}) − 1. (6.1)

If x ∈ H_{0}(X; ℤ) is the positive generator, h ∈ H_{2}(X; ℝ), and L_{𝔱′,𝔰} has the standard orientation defined prior to Lemma 3.13, then

\begin{matrix}# (V̄(h^{δ−2m}x^{m}) ∩ W̄^{η} ∩ L_{𝔱′,𝔰}) \\ = (−1)^{m+1+d_{s}(𝔰)/2}2^{−δ}2^{d_{s}(𝔰)/2}P_{d_{s}(𝔰)/2}^{a,b}(0)⟨μ_{𝔰}^{d_{s}(𝔰)/2}, [M_{𝔰}]⟩ (a_{0}⟨c_{1}(𝔰) − c_{1}(𝔱′), h⟩^{δ−2m} \\ + b_{0}⟨c_{1}(𝔰) − c_{1}(𝔱′), h⟩^{δ−2m−1}⟨c_{1}(𝔱′), h⟩ \\ + a_{1}⟨c_{1}(𝔰) − c_{1}(𝔱′), h⟩^{δ−2m−2}Q_{X}(h, h)) , \end{matrix} (6.2)

where all terms on the right which would have a negative exponent are omitted,

a_{0} = 3(c_{1}(𝔰) − c_{1}(𝔱′))^{2} + c_{1}^{2}(X) + 2(c_{1}(𝔰) − c_{1}(𝔱′)) · c_{1}(𝔱′) + 4(δ − 2m) − 4m,
b_{0} = 2(δ − 2m)\frac{P_{d_{s}(𝔰)/2}^{a−1,b+1}(0)}{P_{d_{s}(𝔰)/2}^{a,b}(0)},

a_{1} = 4\binom{δ − 2m}{2},

and P_{d_{s}(𝔰)/2}^{a,b}(0) is the constant coefficient of the Jacobi polynomial (1.7) with

\begin{aligned}a & = η − \frac{1}{2}d_{s}(𝔰) + 1, \\ b & = \frac{1}{2}(2δ − d_{a}(𝔱′) − d_{s}(𝔰)) − \frac{1}{4}(χ + σ), \end{aligned} (6.3)

where d_{a}(𝔱′) is given by equation (2.11). If d_{s}(𝔰) = 0, then P_{d_{s}(𝔰)/2}^{a,b}(0) = 1.

Proof. According to Proposition 5.2, the intersection number on the left-hand side of equation (6.2) is equal to

\begin{aligned}P : & = # (V̄(h^{δ−2m}x^{m}) ∩ W̄^{η} ∩ L_{𝔱′,𝔰}) \\ & = ⟨γ^{∗}μ̄_{p}(h^{δ−2m}x^{m}) ⌣ γ^{∗}μ̄_{c}^{η} ⌣ ē_{s} ⌣ ē_{i}, [L̄_{𝔱′,𝔰}^{vir}]⟩, \end{aligned} (6.4)

where L_{𝔱′,𝔰}^{vir} has the standard orientation defined in §3.9. Writing

d = \frac{1}{2}d_{s}(𝔰) and 𝔟 = ⟨c_{1}(𝔰) − c_{1}(𝔱′), h⟩, (6.5)

we apply Lemmas 4.10, 4.11 and 4.12 to expand the cup product in the pairing (6.4) to give

P = ⟨(\frac{1}{2}𝔟(2μ_{𝔰} − ν) + PD[h])^{δ−2m} ⌣ (−\frac{1}{4}(2μ_{𝔰} − ν)^{2} + PD[x])^{m}
⌣ (−ν)^{η+r_{Ξ}} (\frac{1}{2}c_{1}(𝔱′) − \frac{1}{2}ν) , [L̄_{𝔱′,𝔰}^{vir}]⟩ ,

(6.6)

where, for notational simplicity, we omit explicit mention of pullbacks. Denoting

C = (−1)^{m+η+r_{Ξ}+1}2^{−δ−1}, (6.7)

using the fact that the classes PD[x], PD[h], c_{1}(𝔱′) are pulled back from X, and multiplying out, we see that equation (6.6) becomes

P = C ⟨𝔟^{δ−2m}(2μ_{𝔰} − ν)^{δ}ν^{η+r_{Ξ}+1} − 𝔟^{δ−2m}(2μ_{𝔰} − ν)^{δ}ν^{η+r_{Ξ}}c_{1}(𝔱′)

+ 2(δ − 2m)𝔟^{δ−2m−1}(2μ_{𝔰} − ν)^{δ−1}ν^{η+r_{Ξ}+1} PD[h]

− 2(δ − 2m)𝔟^{δ−2m−1}(2μ_{𝔰} − ν)^{δ−1}ν^{η+r_{Ξ}} PD[h]c_{1}(𝔱′)

+ 4\binom{δ − 2m}{2}𝔟^{δ−2m−2}(2μ_{𝔰} − ν)^{δ−2}ν^{η+r_{Ξ}+1}Q_{X}(h, h) PD[x]

−4m𝔟^{δ−2m}(2μ_{𝔰} − ν)^{δ−2}ν^{η+r_{Ξ}+1} PD[x], [L̄_{𝔱′,𝔰}^{vir}]⟩ .

(6.8)

If r_{N} denotes the rank of the complex vector bundle N_{𝔱,𝔰} → M_{𝔰}, then (see equations (2.32) and (2.34))

δ + η = \frac{1}{2} dim(M_{𝔱′}/S^{1}) − 1
= \frac{1}{2} dim(M_{𝔱}/S^{1}) + 2 (By Equation (2.11))
= d + r_{N} − r_{Ξ} + 2.

(6.9)

The identity (6.9) and the fact that μ_{𝔰}^{i} = 0 when i > d, because the class μ_{𝔰} is pulled back from M_{𝔰}, imply that equation (6.8) can be rewritten as

P = (−1)^{δ}C ∑_{i=0}^{d} ⟨𝔟^{δ−2m}\binom{δ}{i}2^{i}(−1)^{i}μ_{𝔰}^{i}ν^{d+r_{N} +3−i}

− 𝔟^{δ−2m}\binom{δ}{i}2^{i}(−1)^{i}μ_{𝔰}^{i}ν^{d+r_{N} +2−i}c_{1}(𝔱′)

− 2(δ − 2m)𝔟^{δ−2m−1}\binom{δ − 1}{i}2^{i}(−1)^{i}μ_{𝔰}^{i}ν^{d+r_{N} +2−i} PD[h]

+ 2(δ − 2m)𝔟^{δ−2m−1}\binom{δ − 1}{i}2^{i}(−1)^{i}μ_{𝔰}^{i}ν^{d+r_{N} +1−i} PD[h]c_{1}(𝔱′)

+ 4\binom{δ − 2m}{2}𝔟^{δ−2m−2}\binom{δ − 2}{i}2^{i}(−1)^{i}μ_{𝔰}^{i}ν^{d+r_{N} +1−i}Q_{X}(h, h) PD[x]

−4m𝔟^{δ−2m}\binom{δ − 2}{i}2^{i}(−1)^{i}μ_{𝔰}^{i}ν^{d+r_{N} +1−i} PD[x], [L̄_{𝔱′,𝔰}^{vir}]⟩ ,

(6.10)

where it is understood that binomial coefficients \binom{δ}{i} are by definition zero when i > δ. By applying the formula (5.71) for division by the Poincaré dual of [BL_{𝔱′,𝔰}^{vir,i}] = [M_{𝔰}×∂Ḡl_{𝔱}(δ)/S^{1}], we see that equation (6.10) yields

P = (−1)^{δ+r_{N}} C ∑_{i=0}^{d} ∑_{j=0}^{d−i} ⟨𝔟^{δ−2m}\binom{δ}{i}2^{i}(−1)^{i+j}ν^{d+3−i−j}μ_{𝔰}^{i}s_{j}(N_{𝔱,𝔰})

− 𝔟^{δ−2m}\binom{δ}{i}2^{i}(−1)^{i+j}ν^{d+2−i−j}μ_{𝔰}^{i}s_{j}(N_{𝔱,𝔰})c_{1}(𝔱′)

− 2(δ − 2m)𝔟^{δ−2m−1}\binom{δ − 1}{i}2^{i}(−1)^{i+j}ν^{d+2−i−j}μ_{𝔰}^{i}s_{j}(N_{𝔱,𝔰}) PD[h]

+ 2(δ − 2m)𝔟^{δ−2m−1}\binom{δ − 1}{i}2^{i}(−1)^{i+j}ν^{d+1−i−j}μ_{𝔰}^{i}s_{j}(N_{𝔱,𝔰}) PD[h]c_{1}(𝔱′)

+ 4\binom{δ − 2m}{2}𝔟^{δ−2m−2}\binom{δ − 2}{i}2^{i}(−1)^{i+j}ν^{d+1−i−j}μ_{𝔰}^{i}s_{j}(N_{𝔱,𝔰})Q_{X}(h, h) PD[x]

−4m𝔟^{δ−2m}\binom{δ − 2}{i}2^{i}(−1)^{i+j}ν^{d+1−i−j}μ_{𝔰}^{i}s_{j}(N_{𝔱,𝔰}) PD[x], [M_{𝔰} × ∂Ḡl_{𝔱}(δ)/S^{1}]⟩ .

(6.11)

We then make the substitution ν = 2μ_{𝔰} + ν_{𝔱} from equation (5.73) and expand the powers of ν in (6.11) as binomial sums. For α ∈ H^{2k}(X; ℝ), the equality

⟨ν_{𝔱}^{d+3−k−i−j−ℓ} ⌣ μ_{𝔰}^{i+ℓ} ⌣ s_{j}(N_{𝔱,𝔰}) ⌣ π_{X}^{∗}α, [M_{𝔰} × ∂Ḡl_{𝔱}(δ)/S^{1}]⟩

= \begin{cases}⟨μ_{𝔰}^{d−j} ⌣ s_{j}(N_{𝔱,𝔰}), [M_{𝔰}]⟩⟨ν_{𝔱}^{3−k} ⌣ π_{X}^{∗}α, [∂Ḡl_{𝔱}(δ)/S^{1}]⟩, & if ℓ = d − i − j, \\ 0, & if ℓ ≠ d − i − j, \end{cases}

(6.12) implies that only one term from each of the binomial expansions of the powers of ν in equation (6.11) will not vanish. Therefore, equation (6.11) yields

P = (−1)^{δ+r_{N}} C ∑_{i=0}^{d} ∑_{j=0}^{d−i} S_{j}⟨μ_{𝔰}^{d}, [M_{𝔰}]⟩ ⟨𝔟^{δ−2m}\binom{δ}{i}\binom{d + 3 − i − j}{d − i − j}(−1)^{i+j}2^{d−j}ν_{𝔱}^{3}

− 𝔟^{δ−2m}\binom{δ}{i}\binom{d + 2 − i − j}{d − i − j}(−1)^{i+j}2^{d−j}ν_{𝔱}^{2}c_{1}(𝔱′)

− 2(δ − 2m)𝔟^{δ−2m−1}\binom{δ − 1}{i}\binom{d + 2 − i − j}{d − i − j}(−1)^{i+j}2^{d−j}ν_{𝔱}^{2} PD[h]

+ 2(δ − 2m)𝔟^{δ−2m−1}\binom{δ − 1}{i}\binom{d + 1 − i − j}{d − i − j}(−1)^{i+j}2^{d−j}ν_{𝔱} PD[h]c_{1}(𝔱′)

+ 4\binom{δ − 2m}{2}𝔟^{δ−2m−2}\binom{δ − 2}{i}\binom{d + 1 − i − j}{d − i − j}(−1)^{i+j}2^{d−j}Q_{X}(h, h)ν_{𝔱} PD[x]

−4m𝔟^{δ−2m}\binom{δ − 2}{i}\binom{d + 1 − i − j}{d − i − j}(−1)^{i+j}2^{d−j}ν_{𝔱} PD[x], [∂Ḡl_{𝔱}(δ)/S^{1}]⟩ ,

(6.13)

where we use Lemma 2.4 to see that

⟨μ_{𝔰}^{d−j}s_{j}(N_{𝔱,𝔰}), [M_{𝔰}]⟩ = S_{j}⟨μ_{𝔰}^{d}, [M_{𝔰}]⟩,

and the constants S_{j} are defined by

S_{j} = ∑_{k=0}^{j} 2^{k}\binom{−n′_{s}}{k}\binom{−n′′_{s}}{j − k}. (6.14)

See [18, Lemma 4.16] for identities involving binomial coefficients and extensions of the definition of \binom{n}{p} to the case n ≤ 0. Before applying Lemma 5.24 to compute each of the pairings with [∂Ḡl_{𝔱}(δ)/S^{1}] on the right-hand side of equation (6.13), we simplify the combinatorial factors appearing in each term in (6.13) using the identities, for v = 0, …, 3,

∑_{i=0}^{d} ∑_{j=0}^{d−i}(−1)^{i+j}2^{d−j}\binom{δ − v}{i}\binom{d + 3 − v − i − j}{d − i − j}S_{j}

= ∑_{i=0}^{d} ∑_{j=0}^{d−i} ∑_{k=0}^{j}(−1)^{i+j}2^{d−j+k}\binom{δ − v}{i}\binom{d + 3 − v − i − j}{d − i − j}\binom{−n′_{s}}{k}\binom{−n′′_{s}}{j − k}

(By Equation (6.14))

= 2^{d}P_{d}^{a,b}(0) (By Equation (6.19)),

(6.15)

where P_{d}^{a,b}(ζ) is the Jacobi polynomial (1.7), with constants

a = 3 + n′_{s} + n′′_{s} − δ and b = δ − n′_{s} − 4 − d. (6.16)

Because the constant on the left-hand side of equation (6.15) is independent of v, we can factor it out of all the terms in equation (6.13) except from the fourth term (the one containing PD[h]c_{1}(𝔱′)), where the binomial factors can be rewritten as

\binom{(δ + 1) − 2}{i}\binom{d + 1 − i − j}{d − i − j},

and thus, by replacing δ by δ + 1 in the definition of a and b in (6.16), the combinatorial expression from the fourth term of equation (6.13) corresponding to equation (6.15) will give the Jacobi polynomial P_{d}^{a−1,b+1}(0). Hence, by substituting (6.15), equation (6.13) takes the shape

P = (−1)^{δ+r_{N}} C2^{d}P_{d}^{a,b}(0)⟨μ_{𝔰}^{d}, [M_{𝔰}]⟩ ⟨𝔟^{δ−2m}(ν_{𝔱}^{3} − ν_{𝔱}^{2}c_{1}(𝔱′))

− 2(δ − 2m)𝔟^{δ−2m−1}ν_{𝔱}^{2} PD[h] + 2(δ − 2m)\frac{P_{d}^{a−1,b+1}(0)}{P_{d}^{a,b}(0)}𝔟^{δ−2m−1}ν_{𝔱} PD[h]c_{1}(𝔱′)

+4\binom{δ − 2m}{2}𝔟^{δ−2m−2}Q_{X}(h, h)ν_{𝔱} PD[x] − 4m𝔟^{δ−2m}ν_{𝔱} PD[x], [∂Ḡl_{𝔱}(δ)/S^{1}]⟩ .

(6.17)

Lemma 5.24 and equation (5.77) provide formulae for the pairings:

⟨ν_{𝔱} PD[x], [∂Ḡl_{𝔱}(δ)/S^{1}]⟩,
⟨ν_{𝔱} PD[h]c_{1}(𝔱′), [∂Ḡl_{𝔱}(δ)/S^{1}]⟩,
⟨ν_{𝔱}^{2} PD[h], [∂Ḡl_{𝔱}(δ)/S^{1}]⟩,
⟨ν_{𝔱}^{3}, [∂Ḡl_{𝔱}(δ)/S^{1}]⟩.

Hence, Lemma 5.24 and the definition (6.7) of C imply that equation (6.17) can be written as

\begin{aligned}P & = (−1)^{δ+η+r_{N} +r_{Ξ}+m+1}2^{d−δ−1}P_{d}^{a,b}(0)⟨μ_{𝔰}^{d}, [M_{𝔰}]⟩ \\ & × (𝔟^{δ−2m} (6(c_{1}(𝔰) − c_{1}(𝔱′))^{2} + 2c_{1}^{2}(X) \\ & +4(c_{1}(𝔰) − c_{1}(𝔱′)) · c_{1}(𝔱′) + 8(δ − 2m) − 8m) \\ & + 4(δ − 2m)𝔟^{δ−2m−1} \frac{P_{d}^{a−1,b+1}(0)}{P_{d}^{a,b}(0)}⟨c_{1}(𝔱′), h⟩ \\ & +8𝔟^{δ−2m−2}\binom{δ − 2m}{2}Q_{X}(h, h)) . \end{aligned} (6.18)

We write the power of 2 as 2^{d_{s}(𝔰)/2}2^{−δ}2^{−1}, since d = d_{s}(𝔰)/2 by definition (6.5), with the factor 2^{−1} being absorbed into the coefficients a_{0}, b_{0}, a_{1}, whose resulting formulae agree by inspection with those following equation (6.2).

Equation (6.9) implies that the power of (−1) in equation (6.18) simplifies to

(−1)^{d_{s}(𝔰)/2+m+1},

which agrees with the power of (−1) in (6.2).

Finally, we simplify the combinatorial coefficients P_{d}^{a,b}(0) in (6.18). Because

d + n′_{s}(𝔱, 𝔰) + n′′_{s}(𝔱, 𝔰) = \frac{1}{2} dim M_{𝔱} (By Equation (2.32))

= \frac{1}{2} dim M_{𝔱′} − 3 (By Equation (2.11)) = δ + η − 2 (By Equation (6.1)),

we have, for a as defined in (6.16),

n′′_{s} + n′_{s} − δ + 3 = η − d + 1
= a,

agreeing with the claimed formula for a in (6.3). Then, using

n′_{s} = \frac{1}{2}d_{a}(𝔱) + \frac{1}{4}(χ + σ) (By Equation (2.33))
= \frac{1}{2}d_{a}(𝔱′) − 4 + \frac{1}{4}(χ + σ),

and d_{s}(𝔰) = 2d, we have, for b as defined in (6.16),

δ − 4 − n′_{s} − d = \frac{1}{2}(2δ − d_{a}(𝔱′) − d_{s}(𝔰)) − \frac{1}{4}(χ + σ)
= b,

agreeing with the claimed formula for b in (6.3). This completes the proof of Theorem 6.1.

It remains to prove the combinatorial identity (6.15).

Lemma 6.2. For integers A, M, N, d with d ≥ 0, and v = 0, …, 3, and P_{d}^{a,b}(ζ) the Jacobi polynomial (1.7), we have

2^{d}P_{d}^{a,b}(0) = ∑_{i=0}^{d} ∑_{j=0}^{d−i} ∑_{k=0}^{j}(−1)^{i+j}2^{d−j+k}\binom{A − v}{i}\binom{d + 3 − v − i − j}{d − i − j}\binom{M}{k}\binom{N}{j − k}, (6.19)

where the constants a and b are given by

a = 3 − N − A − M and b = A + M − 4 − d.

Proof. For r ∈ ℝ and ℓ ∈ ℕ it is convenient to define

(r)_{ℓ} := r(r + 1) ⋯ (r + ℓ − 1).

We then recall the following identities (see [18, Lemma 4.16]):

\binom{r}{ℓ} = \frac{(−1)^{ℓ}(−r)_{ℓ}}{ℓ!} and (r)_{ℓ} = (−1)^{ℓ}(1 − r − ℓ)_{ℓ}. (6.20)

The preceding identities yield

\begin{aligned}\binom{d + 3 − v − i − j}{d − i − j} & = \frac{(−1)^{d−i−j}(−d − 3 + i + j + v)_{d−i−j}}{(d − i − j)!} \\ & = \frac{(4 − v)_{d−i−j}}{(d − i − j)!} \\ & = (−1)^{d−i−j}\binom{v − 4}{d − i − j}. \end{aligned} (6.21)

Let S denote right-hand-side of equation (6.19) and observe that, using equation (6.21), we can write S in the form

S = ∑_{i=0}^{d} ∑_{j=0}^{d−i} ∑_{k=0}^{j}(−1)^{d}2^{d−j+k}\binom{A − v}{i}\binom{v − 4}{d − i − j}\binom{M}{k}\binom{N}{j − k}. (6.22)

With the substitution u = j − k, equation (6.22) becomes

\begin{aligned}S & = (−2)^{d} ∑_{i=0}^{d} ∑_{u=0}^{d−i} ∑_{j=u}^{d−i} 2^{−u}\binom{A − v}{i}\binom{v − 4}{d − i − j}\binom{M}{j − u}\binom{N}{u} \\ & = (−2)^{d} ∑_{i=0}^{d} ∑_{u=0}^{d−i} 2^{−u}\binom{A − v}{i}\binom{N}{u} ∑_{j′=0}^{d−i−u} \binom{v − 4}{d − i − u − j′}\binom{M}{j′} \end{aligned} (6.23)

where we have set j′ = j − u. Applying the Vandermonde convolution identity [18, Lemma 4.16 (5)] to the sum over j′ in the right-hand side of (6.23), we see that

S = (−2)^{d} ∑_{i=0}^{d} ∑_{u=0}^{d−i} 2^{−u}\binom{A − v}{i}\binom{N}{u}\binom{v + M − 4}{d − i − u}

= (−2)^{d} ∑_{u=0}^{d} 2^{−u}\binom{N}{u} ∑_{i=0}^{d−u} \binom{A − v}{i}\binom{v + M − 4}{d − i − u} (Reversing summation order)

= (−2)^{d} ∑_{u=0}^{d} 2^{−u}\binom{N}{u}\binom{A + M − 4}{d − u} (By [18, Lemma 4.16(5)]).

(6.24)

We now express this sum in terms of the hypergeometric function [27, §9.10]

2F_{1}(a, b; c; ζ) = ∑_{k=0}^{∞} \frac{(a)_{k}(b)_{k}}{(c)_{k}k!} ζ^{k}, ζ ∈ ℂ.

Using the first identity in (6.20), equation (6.24) yields

S = (−2)^{d} ∑_{u=0}^{d} 2^{−u} \frac{(−1)^{u}(−N)_{u}}{u!} \frac{(−1)^{d−u}(4 − A − M)_{d−u}}{(d − u)!} . (6.25)

Then applying the identities (see [18, Lemma 4.16])

(a)_{d−u} = \frac{(−1)^{u}(a)_{d}}{(1 − a − d)_{u}} and (d − u)! = \frac{d!}{(−1)^{u}(−d)_{u}} (6.26)

to equation (6.25) gives

S = 2^{d} ∑_{u=0}^{d} 2^{−u} \frac{(−N)_{u}(−1)^{u}(4 − A − M)_{d}(−1)^{u}(−d)_{u}}{u!(A + M − 3 − d)_{u}d!}

= \frac{2^{d}(4 − A − M)_{d}}{d!} ∑_{r=0}^{d} \frac{(−d)_{u}(−N)_{u}}{(A + M − 3 − d)_{u}u!}2^{−u}

= \frac{2^{d}(4 − A − M)_{d}}{d!}_{2}F_{1}(−d, −N; A + M − 3 − d; \frac{1}{2})

= \frac{(−2)^{d}(4 − A − M)_{d}}{(A + M − 3 − d)_{d}} P_{d}^{3−N−A−M,A+M−4−d}(0) (By [18, Equation (4.40)]).

(6.27)

Finally, we note that

\frac{(−2)^{d}(4 − A − M)_{d}}{(A + M − 3 − d)_{d}} = 2^{d}

by the second identity in (6.20). This completes the proof of the lemma.

6.2. The blow-up formula for level-one Seiberg-Witten link pairings. To apply Theorem 6.1 and express the link pairing (6.2) in terms of Seiberg-Witten invariants of X and thus express, via (2.52), the Donaldson invariants of X in terms of Seiberg-Witten invariants of X, we must apply the SO(3)-monopole cobordism to the blow-up, X̃ = X#\overline{ℂℙ}^{2}, of X in view of our definitions (2.47) and (2.23) of the four-manifold invariants. Our definitions of the gauge-theoretic four-manifold invariants incorporate the blow-up formula because the classes w + PD[e] (mod 2) and c_{1}(𝔰) ± PD[e] − Λ (mod 2) are always good in the sense of Definition 2.3, for arbitrary w, c_{1}(𝔰), and Λ.

Following the discussion in [18, §4.5], every spin^{c} structure on X̃ is given by 𝔰#𝔰_{2k−1}, the result of splicing a spin^{c} structure 𝔰 on X and the spin^{c} structure 𝔰_{2k−1} on \overline{ℂℙ}^{2} with c_{1}(𝔰_{2k−1}) = (2k − 1) PD[e], k ∈ ℤ. Then,

c_{1}(𝔰#𝔰_{2k−1}) = c_{1}(𝔰) + (2k − 1) PD[e]. (6.28)

The dimensions of the Seiberg-Witten moduli spaces are related by

d_{s}(𝔰#𝔰_{2k−1}) = d_{s}(𝔰) − k(k − 1). (6.29)

For the cases k = 1 or 0, we denote 𝔰^{+} = 𝔰#𝔰_{1} and 𝔰^{−} = 𝔰#𝔰_{−1}, so that

c_{1}(𝔰^{±}) = c_{1}(𝔰) ± PD[e]. (6.30)

If d_{s}(𝔰#𝔰_{2k−1}) ≥ 0, then SW_{X}(𝔰#𝔰_{2k−1}) = SW_{X}(𝔰) according to the blow-up formula [22, Theorem 1.4], [43, Theorem 3.2] (see the statement given in [18, Theorem 4.20]).

Recall from [18, Lemma 4.19] that given a spin^{u} structure 𝔱 on X, there is a spin^{u} structure 𝔱̃ on X̃ such that

c_{1}(𝔱̃) = c_{1}(𝔱), p_{1}(𝔱̃) = p_{1}(𝔱) − 1, w_{2}(𝔱̃) ≡ w_{2}(𝔱) + PD[e] (mod 2). (6.31)

We then have the following extension of Lemma 4.19 in [18].

Lemma 6.3. Continue the notation of the preceding paragraph. Then the following hold for each integer ℓ ≥ 0:

1. There is a bijection between the • Set of strata ι(M_{𝔰}(X)) × Sym^{ℓ}(X) in IM_{𝔱}(X), and the

• Set of families of strata ι(M_{𝔰_{k^{±}}}(X)) × Sym^{ℓ−j}(X) in IM𝔱̃(X̃), where 𝔰_{k^{±}} =

𝔰#𝔰_{2k^{±}−1} and j, k^{±} ∈ ℤ_{≥0} obey j = k^{±}(k^{±} − 1) ≤ min{ℓ, d_{s}(𝔰)}.

2. The preceding correspondence further restricts to a bijection between the • Set of strata in level ℓ with SW_{X}(𝔰) ≠ 0, and the

• Set of families of strata in levels ℓ − j with SW_{X}(𝔰_{k^{±}}) ≠ 0, where j, k^{±} ∈ ℤ_{≥0}

obey j = k^{±}(k^{±} − 1) ≤ min{ℓ, d_{s}(𝔰)}.

3. If X has SW-simple type then so has X̃ and the preceding correspondence further

restricts to a bijection between the • Set of strata in level ℓ with SW_{X}(𝔰) ≠ 0, and the

• Set of pairs of strata in level ℓ with SW_{X}(𝔰^{±}) ≠ 0.

Proof. From [18, Lemma 3.32] there is a splitting 𝔱_{ℓ} = 𝔰 ⊕ 𝔰′ if and only if (c_{1}(𝔰) − c_{1}(𝔱))^{2} = p_{1}(𝔱) + 4ℓ, recalling that 𝔱_{ℓ} is a spin^{u} structure with p_{1}(𝔱_{ℓ}) = p_{1}(𝔱) + 4ℓ, c_{1}(𝔱_{ℓ}) = c_{1}(𝔱), and w_{2}(𝔱_{ℓ}) = w_{2}(𝔱). Hence, M_{𝔰} is contained in the level M_{𝔱_{ℓ}} × Sym^{ℓ}(X) if and only if

ℓ = ℓ(𝔱, 𝔰) := \frac{1}{4}((c_{1}(𝔰) − c_{1}(𝔱))^{2} − p_{1}(𝔱)). (6.32)

Substituting (6.28) and (6.31) into (6.32), we see that

ℓ(𝔱̃, 𝔰#𝔰_{2k−1}) = ℓ(𝔱, 𝔰) − k(k − 1) ≤ ℓ(𝔱, 𝔰).

Assertion (1) follows from the preceding formula relating levels and the relation (6.29) between the dimensions of the Seiberg-Witten moduli spaces. Assertions (2) and (3) follow from the blow-up formula [22, 43] and the definition of SW-simple type.

Remark 6.4. If ℓ = 0 or 1 in Cases (1) or (2) of Lemma 6.3, then we must have j = 0, k^{+} = 1, k^{−} = 0, and ℓ(𝔱, 𝔰) = ℓ(𝔱̃, 𝔰^{±}). Hence, for ℓ ∈ {0, 1} and 𝔰 ∈ Spin^{c}(X), there is an injective map from the

• Set of strata, M_{𝔰} × Sym^{ℓ}(X), in IM_{𝔱} with SW_{X}(𝔰) ≠ 0, to the

• Set of pairs of strata, M_{𝔰^{±}} × Sym^{ℓ}(X̃), in IM𝔱̃ with SW_{X}(𝔰^{±}) ≠ 0.

Furthermore, if all 𝔰 ∈ Spin^{c}(X) with ℓ(𝔱, 𝔰) ≥ 2 have SW_{X}(𝔰) = 0, then the preceding map is bijective and all 𝔰̃ ∈ Spin^{c}(X̃) with ℓ(𝔱̃, 𝔰̃) ≥ 2 have SWX̃(𝔰̃) = 0.

Proposition 6.5. Continue the hypotheses of Theorem 6.1 leading to equation (6.2), except we omit the requirement that M_{𝔰} contains no zero-section pairs. Let the links L_{𝔱̃′,𝔰^{±}} have the standard orientation. Then, for k even,

\begin{matrix}(−1)^{o𝔱̃′(w+PD[e],𝔰^{+})} # (V̄(h^{δ−2m−k}e^{k+1}x^{m}) ∩ W̄^{η} ∩ L_{𝔱̃′,𝔰^{+}}) \\ + & (−1)^{o𝔱̃′(w+PD[e],𝔰^{−})} # (V̄(h^{δ−2m−k}e^{k+1}x^{m}) ∩ W̄^{η} ∩ L_{𝔱̃′,𝔰^{−}}) \\ = (−1)^{o_{𝔱}(w,𝔰)+m+1−d_{s}(𝔰)/2}2^{−δ}2^{d_{s}(𝔰)/2}P_{d_{s}(𝔰)/2}^{a,b}(0)SW_{X}(𝔰) \\ × (ã_{0}⟨c_{1}(𝔰) − c_{1}(𝔱′), h⟩^{δ−2m−k} \\ + b̃_{0}⟨c_{1}(𝔰) − c_{1}(𝔱′), h⟩^{δ−2m−k−1}⟨c_{1}(𝔱′), h⟩ \\ +ã_{1}⟨c_{1}(𝔰) − c_{1}(𝔱′), h⟩^{δ−2m−k−2}Q_{X}(h, h)) , \end{matrix} (6.33)

where the sign change factor o_{𝔱}(w, 𝔰) is defined by (3.91), the coefficients ã_{0}, b̃_{0}, ã_{1} are related to those of Theorem 6.1 by

ã_{0} = a_{0} − 4\binom{k + 1}{2}, b̃_{0} = b_{0} − 2k \frac{P_{d_{s}(𝔰)/2}^{a−1,b+1}(0)}{P_{d_{s}(𝔰)/2}^{a,b}(0)}, and ã_{1} = 4\binom{δ − 2m − k}{2},

and P_{d_{s}(𝔰)/2}^{a,b}(0) is given by (1.7). If k is odd, the sum on the left-hand side of (6.33) is zero.

Proof. From [18, Equation (4.55)] we see that o𝔱̃′(w + PD[e], 𝔰^{+}) = o_{𝔱′}(w, 𝔰) − 1 and that o𝔱̃′(w + PD[e], 𝔰^{−}) = o_{𝔱′}(w, 𝔰), so the left-hand side of (6.33) is (−1)^{o_{𝔱′}(w,𝔰)−1} times

# (V̄(h^{δ−2m−k}e^{k+1}x^{m}) ∩ W̄^{η} ∩ L_{𝔱̃′,𝔰^{+}}) − # (V̄(h^{δ−2m−k}e^{k+1}x^{m}) ∩ W̄^{η} ∩ L_{𝔱̃′,𝔰^{−}}) , (6.34)

where the links L_{𝔱̃′,𝔰^{±}} have the standard orientation. Suppose h_{i} = h, for 0 ≤ i ≤ δ −2m−k and h_{i} = e for δ − 2m − k < i ≤ δ + 1 − 2m. Applying the polarization identity [23, p. 396] to the formula (6.2) to compute the pairings

# (V̄(h^{δ−2m−k}e^{k+1}x^{m}) ∩ W̄^{η} ∩ L_{𝔱̃′,𝔰^{±}}) ,

and noting that dim M𝔱̃′(X̃) = dim M_{𝔱′}(X) + 2 by equation (2.11), gives (using d = d_{s}(𝔰)/2 = d_{s}(𝔰^{±})/2 for brevity)

\begin{matrix}# (V̄(h^{δ−2m−k}e^{k+1}x^{m}) ∩ W̄^{η} ∩ L_{𝔱̃′,𝔰^{±}}) \\ = (−1)^{m+1+d}2^{−δ−1}2^{d}P_{d}^{a^{±},b^{±}}(0)⟨μ_{𝔰^{±}}^{d}, M_{𝔰^{±}}(X̃)⟩ (a_{0}^{±} ∏_{i=1}^{δ+1−2m}⟨c_{1}(𝔰^{±}) − c_{1}(𝔱̃′), h_{i}⟩ \\ + 2\frac{P_{d}^{a^{±}−1,b^{±}+1}(0)}{P_{d}^{a^{±},b^{±}}(0)} ∑_{j=1}^{δ+1−2m} ∏_{\begin{matrix}i=1 \\ i≠j \end{matrix}}^{δ+1−2m}⟨c_{1}(𝔰^{±}) − c_{1}(𝔱̃′), h_{i}⟩⟨c_{1}(𝔱̃′), h_{j}⟩ \\ +4 ∑_{1≤j<l≤δ+1−2m} ∏_{\begin{matrix}i=1 \\ i≠j,l \end{matrix}}^{δ+1−2m}⟨c_{1}(𝔰^{±}) − c_{1}(𝔱̃′), h_{i}⟩QX̃(h_{j}, h_{l})) , \end{matrix} (6.35)

where, using c_{1}(𝔰^{±}) = c_{1}(𝔰) ± PD[e] and c_{1}^{2}(X̃) = c_{1}^{2}(X) − 1, and c_{1}(𝔱̃′) = c_{1}(𝔱′),

\begin{aligned}a_{0}^{±} & = 3 (c_{1}(𝔰^{±}) − c_{1}(𝔱̃′))^{2} + c_{1}^{2}(X̃) + 2 (c_{1}(𝔰^{±}) − c_{1}(𝔱̃′)) · c_{1}(𝔱̃′) \\ & + 4(δ + 1 − 2m) − 4m \\ & = 3(c_{1}(𝔰) − c_{1}(𝔱′))^{2} − 3 + c_{1}^{2}(X) − 1 + 2(c_{1}(𝔰) − c_{1}(𝔱′)) · c_{1}(𝔱′) \\ & + 4(δ − 2m) − 4m + 4 \\ & = 3(c_{1}(𝔰) − c_{1}(𝔱′))^{2} + c_{1}^{2}(X) + 2(c_{1}(𝔰) − c_{1}(𝔱′)) · c_{1}(𝔱′) \\ & + 4(δ − 2m) − 4m, \end{aligned} (6.36)

and where a^{±}, b^{±} are given by

\begin{aligned}a^{±} & = η − d + 1 \\ & = a, \\ b^{±} & = \frac{1}{2}(2(δ + 1) − d_{a}(𝔱̃′) − d_{s}(𝔰^{±})) − \frac{1}{4}(χ(X̃) + σ(X̃)) \\ & = \frac{1}{2}(2δ − d_{a}(𝔱′) − d_{s}(𝔰)) − \frac{1}{4}(χ + σ) \\ & = b. \end{aligned} (6.37)

Note that the factor of (δ + 1 − 2m) in the coefficient b_{0}/2 and the coefficient a_{1}/4 in equation (6.2) are absorbed by the application of the polarization identity [23, p. 396]. Using c_{1}(𝔱̃′) = c_{1}(𝔱′) and the identities

QX̃(e, e) = −1,
QX̃(e, h) = 0,
⟨c_{1}(𝔰^{±}) − c_{1}(𝔱̃′), e⟩ = ∓1,
⟨c_{1}(𝔰^{±}) − c_{1}(𝔱̃′), h⟩ = ⟨c_{1}(𝔰) − c_{1}(𝔱′), h⟩,
⟨c_{1}(𝔱̃′), e⟩ = 0,
⟨c_{1}(𝔱̃′), h⟩ = ⟨c_{1}(𝔱′), h⟩,

we simplify the terms on the right-hand side of equation (6.35) (recall that h_{i} = h ∈ H_{2}(X; ℝ), 1 ≤ i ≤ δ − k − 2m, and h_{i} = e for δ − k − 2m < i ≤ δ + 1 − 2m) to give:

∏_{i=1}^{δ+1−2m}⟨c_{1}(𝔰^{±}) − c_{1}(𝔱̃′), h_{i}⟩ = (∓1)^{k+1}⟨c_{1}(𝔰) − c_{1}(𝔱′), h⟩^{δ−k−2m}, (6.38)

and

\begin{matrix}∑_{j=1}^{δ+1−2m} ∏_{\begin{matrix}i=1 \\ i≠j \end{matrix}}^{δ+1−2m}⟨c_{1}(𝔰^{±}) − c_{1}(𝔱̃′), h_{i}⟩⟨c_{1}(𝔱̃′), h_{j}⟩ \\ = (∓1)^{k+1}(δ − 2m − k)⟨c_{1}(𝔰) − c_{1}(𝔱′), h⟩^{δ−k−2m−1}⟨c_{1}(𝔱′), h⟩, \end{matrix} (6.39)

and

\begin{matrix}∑_{1≤j<l≤δ+1−2m} ∏_{\begin{matrix}i=1 \\ i≠j,l \end{matrix}}^{δ+1−2m}⟨c_{1}(𝔰^{±}) − c_{1}(𝔱̃′), h_{i}⟩QX̃(h_{j}, h_{l}) \\ = (∓1)^{k+1}\binom{δ − 2m − k}{2}⟨c_{1}(𝔰) − c_{1}(𝔱′), h⟩^{δ−k−2m−2}Q_{X}(h, h) \\ + (∓1)^{k−1}\binom{k + 1}{2}⟨c_{1}(𝔰) − c_{1}(𝔱′), h⟩^{δ−k−2m}QX̃(e, e). \end{matrix} (6.40)

Thus, if k + 1 is even, the terms (6.38), (6.39), and (6.40) will cancel when we substitute them into equation (6.35) and subtract the results in (6.34), so in this case the difference (6.34) and hence the pairing (6.33) are zero, as claimed. If k + 1 is odd, the terms (6.38), (6.39), and (6.40) will add when we substitute them into equation (6.35) and subtract the results in (6.34). Hence, equation (6.33) follows from (6.34) together with the preceding substitutions and substitution of equation (6.36) for the coefficient a_{0}^{±} and equation (6.37) for a^{±}, b^{±}. Note that the factor of 2 obtained by adding like terms and the coefficient 2^{−δ−1} in (6.35) yields the coefficient 2^{−δ} in (6.33) and that the factors (−1)^{k+1} = −1 and (−1)^{o_{𝔱′}(w,𝔰)−1} (mentioned before (6.34)) yield the factor (−1)^{o_{𝔱′}(w,𝔰)} appearing on the right-hand-side of (6.33).

6.3. Proofs of main theorems. We now apply the computation of Theorem 6.1 to the sum (2.52) to prove Theorem 1.5 and hence Theorem 1.1.

Proof of Theorem 1.4. We shall derive equation (1.8) from the basic cobordism identity (2.52) applied to the blow-up X̃ = X#\overline{ℂℙ}^{2}, the definition (2.47) for the Donaldson invariant, the blow-up formula (6.33) for level-one Seiberg-Witten link pairings, and the formula from [18, Equation (4.52)] for level-zero Seiberg-Witten link pairings.

Since deg(z) = 2δ obeys condition (2.46), we can choose an integer p such that p ≡ w^{2} (mod 4) and

δ = −p − \frac{3}{4}(χ + σ). (6.41)

According to the last paragraph of [17, §2.1.3], there is a spin^{u} structure 𝔱′ over X with c_{1}(𝔱′) = Λ, w_{2}(𝔱′) ≡ w (mod 2), and p_{1}(𝔱) = p. Let 𝔱̃′ be the related spin^{u} structure over X̃ = X#\overline{ℂℙ}^{2} defined prior to Lemma 6.3. From equation (2.11) we see that

\begin{aligned}n_{a}(𝔱̃′) & = \frac{1}{4}(p_{1}(𝔱̃′) + Λ^{2} − σ(X̃)) \\ & = \frac{1}{4}(p_{1}(𝔱′) + Λ^{2} − σ) \\ & = \frac{1}{4}(i(Λ) − δ) \\ & = n_{a}(𝔱′). \end{aligned} (6.42)

Therefore, the hypothesis that δ < i(Λ) is equivalent to n_{a}(𝔱̃′) = n_{a}(𝔱′) > 0 and hence the moduli space M_{κ+1/4}^{w+PD[e]}(X̃) has real codimension greater than or equal to two in M𝔱̃′(X̃). Moreover, w_{2}(𝔱̃′) = w +PD[e] (mod 2) is good and so the cobordism formula (2.52) applies.

In general, by Remark 3.36 in [18], the Seiberg-Witten stratum M_{𝔰} × Sym^{ℓ}(X) corresponding to a splitting 𝔱′_{ℓ} = 𝔰 ⊕ 𝔰′ lies in level

ℓ(𝔱, 𝔰) = \frac{1}{4}(δ − r(Λ, c_{1}(𝔰))),

of the space of ideal SO(3) monopoles IM_{𝔱′}(X), since 2δ = d_{a}(𝔱′) = dim M_{κ}^{w}(X). By hypothesis, δ = r(Λ) + 4 and r(Λ) ≤ r(Λ, c_{1}(𝔰)) for all 𝔰 with SW_{X}(𝔰) ≠ 0 (by definition (1.6) of r(Λ)), so our choice of δ implies that

0 ≤ ℓ(𝔱, 𝔰) ≤ 1,

if M_{𝔰} × Sym^{ℓ}(X) ⊂ IM_{𝔱′} and SW_{X}(𝔰) ≠ 0.

By hypothesis, X is ‘effective’ in the sense of Definition 1.3 and so the only nonzero Seiberg-Witten contributions to D_{X}^{w}(h^{δ−2m}x^{m}) arise from moduli spaces M_{𝔰}(X) with SW_{X}(𝔰) ≠ 0 contained in levels ℓ(𝔱, 𝔰) = 0 or 1 of IM_{𝔱′}(X) or, equivalently, from moduli spaces M_{𝔰^{±}}(X̃) contained in levels ℓ(𝔱̃, 𝔰^{±}) = 0 or 1 of IM𝔱̃′(X̃), again when SW_{X}(𝔰) ≠ 0 by Remark 6.4.

Therefore, the cobordism identity (2.52) applied to the moduli space M̄𝔱̃′(X̃)/S^{1} and the definition (2.47) of D_{X}^{w}(h^{δ−2m}x^{m}) give

D_{X}^{w}(h^{δ−2m}x^{m})

= −2^{1−n_{a}(𝔱′)} ∑_{𝔰∈Spin^{c}(X)}((−1)^{o𝔱̃(w+PD[e],𝔰^{+})}# (V̄(h^{δ−2m}ex^{m}) ∩ W̄^{n_{a}(𝔱)−1} ∩ L_{𝔱̃′,𝔰^{+}})

+(−1)^{o𝔱̃(w+PD[e],𝔰^{−})}# (V̄(h^{δ−2m}ex^{m}) ∩ W̄^{n_{a}(𝔱)−1} ∩ L_{𝔱̃′,𝔰^{−}})) .

(6.43)

By the remarks in the preceding paragraphs the sum over Spin^{c}(X) has potentially non-zero terms when

• ℓ(𝔱, 𝔰) = 0 or, equivalently, r(Λ, c_{1}(𝔰)) = δ = r(Λ) + 4, and

• ℓ(𝔱, 𝔰) = 1 or, equivalently, r(Λ, c_{1}(𝔰)) = δ − 4 = r(Λ).

Recall that the links L_{𝔱̃,𝔰^{±}} are empty by definition when ℓ(𝔱̃′, 𝔰^{±}) < 0. Substituting [18, Equation (4.52)] into equation (6.43) to compute the terms with ℓ(𝔱, 𝔰) = 0 and substituting the blow-up formulas (6.33) into equation (6.43) to compute the terms with ℓ(𝔱, 𝔰) = 1 yields

D_{X}^{w}(h^{δ−2m}x^{m}) = 2^{1−n_{a}(𝔱′)−δ} ∑_{\begin{matrix}𝔰∈Spin^{c}(X) \\ r(Λ,c_{1}(𝔰))=δ \end{matrix}}(−1)^{o_{𝔱}(w,c_{1}(𝔰))+m+d_{s}(𝔰)/2}

× 2^{d_{s}(𝔰)/2}P_{d_{s}(𝔰)/2}^{a−1,b}(0)SW_{X}(𝔰)⟨c_{1}(𝔰) − Λ, h⟩^{δ−2m}

+ 2^{1−n_{a}(𝔱′)−δ} ∑_{\begin{matrix}𝔰∈Spin^{c}(X) \\ r(Λ,c_{1}(𝔰))=δ−4 \end{matrix}}(−1)^{o_{𝔱}(w,c_{1}(𝔰))+m+d_{s}(𝔰)/2}

× 2^{d_{s}(𝔰)/2}P_{d_{s}(𝔰)/2}^{a,b}(0)SW_{X}(𝔰) (a_{0}⟨c_{1}(𝔰) − Λ, h⟩^{δ−2m}
+ b_{0}⟨c_{1}(𝔰) − Λ, h⟩^{δ−2m−1}⟨Λ, h⟩
+a_{1}⟨c_{1}(𝔰) − Λ, h⟩^{δ−2m−2}Q_{X}(h, h)) ,

(6.44)

where the coefficients a_{0}, b_{0}, a_{1} are as given in the statements of Theorems 6.1 and 1.5 (since they coincide with those of Proposition 6.5 when k = 0). From equation (6.42) for n_{a}(𝔱′), we see that

\begin{aligned}1 − n_{a}(𝔱′) − δ & = 1 − \frac{1}{4}(i(Λ) − δ) − δ \\ & = 1 − \frac{1}{4}i(Λ) − \frac{3}{4}δ, \end{aligned} (6.45)

and so the power of 2 in equation (6.44) matches that in equation (1.8). Finally, via [18, Equation (4.62)], we see that

m + o_{𝔱}(w, 𝔰) ≡ m + \frac{1}{2}(σ − w^{2}) + \frac{1}{2}(w^{2} + c_{1}(𝔰) · (w − Λ)) (mod 2),

and so the power of (−1) in equation (6.44) also matches that in equation (1.8). Thus, substituting the preceding formulas for powers of 2 and (−1) into equation (6.44) yields equation (1.8).

Lastly, we simplify the expressions given in Theorem 6.1 for the constants a and b in P_{d_{s}(𝔰)/2}^{a,b}(0). From equation (6.3) and the fact that d_{a}(𝔱′) = 2δ, we have

b = −\frac{1}{2}d_{s}(𝔰) − \frac{1}{4}(χ + σ).

Similarly, equation (6.1) for η, the hypothesis that δ = r(Λ) + 4, and equation (2.11) for dim M_{𝔱′} yield

η = \frac{1}{2}(dim(M_{𝔱′}/S^{1}) − 1) − δ
= \frac{1}{2}(d_{a}(𝔱′) + 2n_{a}(𝔱′) − 2) − δ
= n_{a}(𝔱′) − 1,

so equation (6.42) for n_{a}(𝔱′) then implies that

η = \frac{1}{4}(i(Λ) − δ) − 1, (6.46)

and thus equation (6.3) gives

a = \frac{1}{4}(i(Λ) − δ) − \frac{1}{2}d_{s}(𝔰).

This completes the proof of Theorem 1.4.

Proof of Theorem 1.5. As Λ ∈ B^{⊥} and X has SW-simple type by hypothesis, equations (1.5) and (1.10) imply that

i(Λ) + r(Λ) = 2c(X).

Since δ = r(Λ) + 4, this gives

i(Λ) = 2c(X) − r(Λ) = 2c(X) − δ + 4. (6.47)

In particular, r(Λ, c_{1}(𝔰)) = r(Λ) and thus ℓ(𝔱, 𝔰) = 1 for all 𝔰 ∈ Spin^{c}(X) with SW_{X}(𝔰) ≠ 0.

We simplify the expression for the power of 2 in equation (1.8), noting that we now have d_{s}(𝔰) = 0 whenever SW_{X}(𝔰) ≠ 0 since X has SW-simple type. Substituting the equation (6.47) for i(Λ) yields

1 − \frac{1}{4}i(Λ) − \frac{3}{4}δ = −\frac{1}{2}(c(X) + δ),

matching the power of 2 in equation (1.11).

The Jacobi polynomial constants P_{d_{s}(𝔰)/2}^{a,b} are equal to 1 when d_{s}(𝔰) = 0, irrespective of the values of a, b. The assertions concerning the power of (−1) and the coefficients a_{0}, b_{0}, a_{1} follow immediately from the fact that c_{1}(𝔰) ∈ B and Λ ∈ B^{⊥}. This completes the proof of Theorem 1.5.

In addition to the hypotheses of Theorem 1.5, the hypotheses of Theorem 1.1 requires that X be abundant in the sense of §1.

Lemma 6.6. [14, Lemma 2.2] If X is an abundant four-manifold, then the following hold:

1. There are classes Λ_{0}, Λ_{1} ∈ B^{⊥} with Λ_{0} ≡ Λ_{1} (mod 2) such that Λ_{0}^{2} = −(χ + σ) and Λ_{1}^{2} = 4 − (χ + σ).

2. There is a class Λ ∈ 2B^{⊥} with Λ^{2} = −(χ + σ) if −(χ + σ) ≡ 0 (mod 8) and Λ^{2} = 4 − (χ + σ) if −(χ + σ) ≡ 4 (mod 8).

Proof. For Assertion (1), the abundance condition implies that there are classes e_{1}, e_{2} ∈ B^{⊥} with e_{1} · e_{1} = e_{2} · e_{2} = 0 and e_{1} · e_{2} = 1. Let t = \frac{1}{4}(χ + σ). Then the classes Λ_{0} = e_{1} − 2te_{2} and Λ_{1} = e_{1} + (2 − 2t)e_{2} will do.

For Assertion (2), the class Λ_{2} = 2e_{1} −te_{2} will satisfy the conclusion when t ≡ 0 (mod 2), so we can take Λ = Λ_{2}, while Λ_{3} = 2e_{1} + (1 − t)e_{2} will work when t ≡ 1 (mod 2), and we can take Λ = Λ_{3}.

Remark 6.7. Although the statement of Lemma 6.6 is identical to that of Lemma 2.2 in [14], the definitions of the classes Λ_{0}, Λ_{1} here differs slightly from those of [14]. Contrary to the assertion in the proof of Lemma 2.2 in [14], the classes Λ_{0}, Λ_{1} defined in [14] do not satisfy Λ_{0} ≡ Λ_{1} (mod 2). The proof given here corrects that error in the proof of [14, Lemma 2.2].

Assertion (1) is only used in the proof of [14, Theorem 1.1] — when applying the signchange formula for Donaldson invariants — and not in the proof of [14, Theorem 1.3], which is the only part of that article where Assertion (2) is used.

In the present article, Assertion (2) is not required and we only use Assertion (1) to prove the existence of the class Λ in Theorem 1.1.

Proof of Theorem 1.1. The existence of Λ ∈ B^{⊥} and w ∈ H^{2}(X; ℤ) with Λ^{2} = 4 − (χ + σ) and w−Λ characteristic follows immediately from Assertion (1) of Lemma 6.6. Take Λ = Λ_{1} and set w = Λ + v where v is characteristic.

We shall apply Theorem 1.5 and the vanishing result in [14, Theorem 1.1]. By hypothesis, Λ ∈ B^{⊥} and so Λ^{2} ≡ 0 (mod 2) (see first paragraph of proof of Theorem 1.1 in [18, §4.6]). Since χ + σ is even, it is convenient to write Λ^{2} = 2j − (χ + σ), where j ∈ ℤ, so equations (1.5) and (1.10) would then take the form

r(Λ) = c(X) − 2j and i(Λ) = c(X) + 2j.

The constraints δ ≤ r(Λ) + 4 and δ < i(Λ) in Theorem 1.5 are thus equivalent to (see [14, Figure 1]):

δ ≤ c(X) − 2j + 4 and δ < c(X) + 2j.

Therefore, when using Theorem 1.5 to compute D_{X}^{w}(h^{δ−2m}x^{m}), we can choose δ ∈ ℕ no larger than δ = c(X), with Λ^{2} = 2 − (χ + σ) when j = 1, or Λ^{2} = 4 − (χ + σ) when j = 2; as we shall see from equation (6.53), we have D_{X}^{w}(h^{c(X)+1−2m}x^{m}) = 0, while (6.51) shows that the term in SW_{X}^{w}(h) of degree c(X) + 1 in h is also zero.

We first observe that both sides of the second identity in (1.2) vanish in sufficiently low degree. Theorem 1.1 in [14] implies that, for v characteristic,

∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (v^{2}+v·c_{1}(𝔰))}SW_{X}(𝔰)⟨c_{1}(𝔰), h⟩^{d} = 0, (6.48)

if d < c(X) − 2 or (see [14, §2]) if d ≢ c(X) (mod 2), since the series has parity (using v^{2} ≡ σ (mod 8))

\begin{aligned}−v^{2} − \frac{3}{4}(χ + σ) & ≡ −σ − \frac{3}{4}(χ + σ) (mod 8) \\ & ≡ −\frac{1}{4}(7χ + 11σ) (mod 4) \\ & = c(X). \end{aligned} (6.49)

By hypothesis of Theorem 1.1, w − Λ is characteristic and so we may write

w = v + Λ, (6.50)

for some v ≡ w_{2}(X) (mod 2). Since Λ ∈ B^{⊥}, then equation (6.48) implies that

∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+w·c_{1}(𝔰))}SW_{X}(𝔰)⟨c_{1}(𝔰), h⟩^{d} = 0, (6.51)

if d < c(X) − 2 or if d ≢ c(X) (mod 2). In particular, aside from the fact that w need not be characteristic, the vanishing result for the Seiberg-Witten series in equation (1.2) restates Theorem 1.1 in [14].

According to the hypothesis of Theorem 1.1, we have

Λ^{2} = 4 − (χ + σ), (6.52)

so r(Λ) = c(X) − 4 and i(Λ) = c(X) + 4. Because v is characteristic we have v · Λ ≡ Λ^{2} (mod 2) and as Λ^{2} ≡ 0 (mod 4), then w^{2} ≡ (v + Λ)^{2} ≡ v^{2} (mod 4). Therefore, equations (6.49) and condition (2.46) show that

D_{X}^{w}(h^{δ−2m}x^{m}) = 0, when δ ≢ c(X) (mod 4). (6.53)

(Note that the alternative solutions to the r(Λ) and i(Λ) constraints yielding δ ≤ c(X), namely those with Λ^{2} = 2 − (χ + σ), would yield δ ≡ c(X) + 2 (mod 4) and so for that choice of Λ^{2} we could choose δ no larger than c(X) − 2.) Hence, from definition (2.49), the potentially non-zero terms in the Donaldson series D_{X}^{w}(h) take the form

\frac{1}{(c(X) − 4i)!}D_{X}^{w}(h^{c(x)−4i}) and \frac{1}{2(c(X) − 4i − 2)!}D_{X}^{w}(h^{c(x)−4i−2}x).

Because r(Λ) = c(X) − 4, Theorem 1.4(a) in [18] gives

D_{X}^{w}(h^{δ−2m}x^{m}) = 0 for δ < c(X) − 4, (6.54) while Theorem 1.4(b) in [18] (with δ = c(X) − 4) yields

\begin{aligned}D_{X}^{w}(h^{c(X)−4−2m}x^{m}) & = 2^{2−c(X)}(−1)^{m+1} ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·w)} \\ & × SW_{X}(𝔰)⟨c_{1}(𝔰) − Λ, h⟩^{c(X)−4−2m}. \end{aligned} (6.55)

Replacing the terms ⟨c_{1}(𝔰)−Λ, h⟩^{c(X)−4−2m} in equation (6.55) by their binomial expansions and applying equation (6.51) gives

D_{X}^{w}(h^{c(X)−4}) = 0 = D_{X}^{w}(h^{c(X)−6}x). (6.56)

Equations (6.53), (6.54), and (6.56) then imply that the terms of D_{X}^{w}(h) of degree less than c(X) − 2 in h are zero, yielding the vanishing result for the Donaldson series stated in (1.2). From equation (6.51) the terms of SW_{X}^{w}(h) of degree less than c(X) − 2 in h are also zero.

Thus, to obtain the second identity in (1.2), it suffices to prove that

D_{X}^{w}(h^{c(X)−2}x) = 2^{3−c(X)} ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+w·c_{1}(𝔰))}SW_{X}(𝔰)⟨c_{1}(𝔰), h⟩^{c(X)−2}, (6.57)

and, noting that \frac{1}{2} \frac{c(X)!}{(c(X)−2)!} = \binom{c(X)}{2},

\begin{aligned}D_{X}^{w}(h^{c(X)}) & = 2^{2−c(X)} ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+w·c_{1}(𝔰))}SW_{X}(𝔰) \\ & × (⟨c_{1}(𝔰), h⟩^{c(X)} + \binom{c(X)}{2}⟨c_{1}(𝔰), h⟩^{c(X)−2}Q_{X}(h, h)) . \end{aligned} (6.58)

Since Λ^{2} = 4 − (χ + σ), δ = c(X), c(X) = −\frac{1}{4}(7χ + 11σ), and c_{1}^{2}(X) = 2χ + 3σ by definition (1.13), the coefficients in equation (1.11) for D_{X}^{w}(h^{δ−2m}x^{m}) (with w − Λ characteristic) become

\begin{aligned}a_{0} & = 4c_{1}^{2}(X) + Λ^{2} + 4c(X) − 12m \\ & = 4 − 12m, \\ b_{0} & = 2(c(X) − 2m), \\ a_{1} & = 4\binom{c(X) − 2m}{2}. \end{aligned} (6.59)

According to equation (6.51), when we expand the terms

(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·w)}SW_{X}(𝔰)⟨c_{1}(𝔰) − Λ, h⟩^{d}

in equation (1.11) into a binomial sum of terms of the form

(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·w)}SW_{X}(𝔰)(−1)^{k}\binom{d}{k}⟨c_{1}(𝔰), h⟩^{d−k}⟨Λ, h⟩^{k}

and sum over 𝔰 ∈ Spin^{c}(X), the sums of terms with d − k < c(X) − 2 or d − k ≢ c(X) (mod 2) will vanish. Therefore, applying equation (6.51) to the right-hand side of equation (1.11) with δ = c(X) and m = 1, only the terms of degree δ − 2 in h will be non-zero. Hence, only the term with coefficient a_{0} will be non-zero and equation (6.51) implies that it only contributes

a_{0}⟨c_{1}(𝔰), h⟩^{c(X)−2}. (6.60)

Since δ = c(X) and a_{0} = −8 by equation (6.59), the power of 2 the right-hand side of equation (1.11) reduces to −c(X) + 3. As m = 1, and a_{0} = −8, and w^{2} ≡ σ (mod 8)

(as w ≡ w_{2}(X) (mod 2)), the power of (−1) in the right-hand side of (1.11) reduces to \frac{1}{2}(w^{2} + w · c_{1}(𝔰)). Therefore, by these remarks and observation (6.60) we obtain equation (6.57) for D_{X}^{w}(h^{c(X)−2}x).

Next, applying equation (6.51) to the right-hand side of equation (1.11) with δ = c(X) and m = 0, only the terms of degree δ in h will be non-zero. In particular, equation (6.51) implies that the term in the right-hand side of equation (1.11) with coefficient a_{0} contributes only

a_{0} (⟨c_{1}(𝔰), h⟩^{c(X)} + \binom{c(X)}{2}⟨c_{1}(𝔰), h⟩^{c(X)−2}⟨Λ, h⟩^{2}) . (6.61)

Similarly the term on the right in equation (1.11) with coefficient b_{0} contributes only

−b_{0}(c(X) − 1)⟨c_{1}(𝔰), h⟩^{c(X)−2}⟨Λ, h⟩^{2}. (6.62)

Finally, the term on the right in equation (1.11) with coefficient a_{1} contributes only

a_{1}⟨c_{1}(𝔰), h⟩^{c(X)−2}Q_{X}(h, h). (6.63)

Hence, as a_{0} = 4, b_{0} = 2c(X), and a_{1} = 4\binom{c(X)}{2} by equation (6.59) when m = 0, noting that δ = c(X) and w^{2} ≡ σ (mod 8), and combining observations (6.61), (6.62), and (6.63), we see that equation (1.11) yields

D_{X}^{w}(h^{c(X)}) = 2^{−c(X)} ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+w·c_{1}(𝔰))}SW_{X}(𝔰)

× (4⟨c_{1}(𝔰), h⟩^{c(X)} + 4\binom{c(X)}{2}⟨c_{1}(𝔰), h⟩^{c(X)−2}⟨Λ, h⟩^{2}

− 2c(X)(c(X) − 1)⟨c_{1}(𝔰), h⟩^{c(X)−2}⟨Λ, h⟩^{2}

+4\binom{c(X)}{2}⟨c_{1}(𝔰), h⟩^{c(X)−2}Q_{X}(h, h))

= 2^{2−c(X)} ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+w·c_{1}(𝔰))}SW_{X}(𝔰)

× (⟨c_{1}(𝔰), h⟩^{c(X)} + \binom{c(X)}{2}⟨c_{1}(𝔰), h⟩^{c(X)−2}Q_{X}(h, h)) ,

which proves equation (6.58) for D_{X}^{w}(h^{c(X)}). Thus, by the remark preceding equation (6.57), this proves the second identity in (1.2) and completes the proof of Theorem 1.1.

Remark 6.8. If one attempted to compute D_{X}^{w}, where w is characteristic, instead of D_{X}^{w+Λ}, the requirement that Λ − w also be characteristic would imply that Λ ≡ 0 (mod 2) so Λ = 2Λ_{2} for some Λ_{2} ∈ H^{2}(X; ℤ). If B is non-empty, then c_{1}(𝔰) · Λ = 2(c_{1}(𝔰) · Λ_{2}) = 0 implies Λ_{2}^{2} ≡ 0 (mod 2) so Λ^{2} ≡ 0 (mod 8).

If χ + σ ≡ 4 (mod 8), there exists a class Λ ∈ 2B^{⊥} with Λ^{2} = 4 − (χ + σ) by Lemma 6.6. In this case, Theorem 1.1 would hold for D_{X}^{w} with w characteristic. In the case χ + σ ≡ 0 (mod 8), the identity

c(X) − r(Λ) = Λ^{2} + (χ + σ)

and the requirement that Λ^{2} ≡ 0 (mod 8) would imply that c(X) ≡ r(Λ) (mod 8). Now the Seiberg-Witten moduli spaces appear in level ℓ, where

ℓ = \frac{1}{4}(δ − r(Λ)).

Thus, if we wished to compute degree δ = c(X) Donaldson invariants in the case χ + σ ≡ 0 (mod 8), we would need ℓ ≡ 0 (mod 2). If ℓ = 0, the relations δ ≤ r(Λ) and δ < i(Λ) in [14, Theorem 2.1] imply that δ < c(X). The cases ℓ ≥ 2 have not yet been computed, though work is in progress [19, 20, 21] and some partial results have been reported by Kronheimer and Mrowka [40]. In the meantime, if χ+σ ≡ 0 (mod 8), we cannot yet compute Donaldson invariants D_{X}^{w} of degree c(X) with this methods of this article in full generality.

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Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019

   E-mail address: feehan@math.rutgers.edu

Current address: Department of Mathematics, Trinity College, University of Dublin, Dublin 2, Ireland

   E-mail address: feehan@maths.tcd.ie
   URL: http://www.maths.tcd.ie/∼feehan

Department of Mathematics, Florida International University, Miami, FL 33199

   E-mail address: lenesst@fiu.edu
   URL: http://www.fiu.edu/∼lenesst