PU(2) MONOPOLES. II: TOP-LEVEL SEIBERG-WITTEN MODULI SPACES AND WITTEN’S CONJECTURE IN LOW DEGREES

PAUL M. N. FEEHAN AND THOMAS G. LENESS

Abstract. In this article, a continuation of [10], we complete the proof—for a broad class of four-manifolds—of Witten’s conjecture that the Donaldson and Seiberg-Witten series coincide, at least through terms of degree less than or equal to c−2, where c = − \frac{1}{4} (7χ+11σ) and χ and σ are the Euler characteristic and signature of the four-manifold. We use our computations of Chern classes for the virtual normal bundles for the Seiberg-Witten strata from the companion article [10], a comparison of all the orientations, and the PU(2) monopole cobordism to compute pairings with the links of level-zero Seiberg-Witten moduli subspaces of the moduli space of PU(2) monopoles. These calculations then allow us to compute low-degree Donaldson invariants in terms of Seiberg-Witten invariants and provide a partial verification of Witten’s conjecture.

1. Introduction

1.1. Main results. The purpose of the present article, a continuation of [10], is to complete the proof that Witten’s conjecture [58] relating the Donaldson and Seiberg-Witten invariants holds in “low degrees” for a broad class of four-manifolds, using the PU(2)-monopole cobordism of Pidstrigatch and Tyurin [51]. 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 §4.1) 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

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)

by analogy with the structure of the Donaldson series D_{X}^{w}(h) [35, Theorem 1.7]. 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 [58]. 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). Let c(X) = −\frac{1}{4}(7χ + 11σ). As stated in [10], we have:

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. For any Λ ∈ B^{⊥} and w ∈ H^{2}(X; ℤ) for which Λ^{2} = 2 − (χ + σ) and w − Λ ≡ w_{2}(X) (mod 2), 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} Q_{X}(h,h)}SW_{X}^{w}(h) (mod h^{c(X)}).

(1.2)

Date: July 31, 2000, dg-ga/9712005. First version: December 8, 1997, dg-ga/9712005 (v1), §4–7.

The first author was supported in part by an NSF Mathematical Sciences Postdoctoral Fellowship under grant DMS 9306061 and by NSF grant DMS 9704174.

The order-of-vanishing assertion for the series D_{X}^{w}(h) and SW_{X}^{w}(h) in equation (1.2) was proved in joint work with Kronheimer and Mrowka [8], based on the results in an earlier version [11] of this article and its companion [10].

The background material underlying the statement of Theorem 1.1—including the definition and significance of “abundant” and “effective” four-manifolds—was explained in [10, §1], so we refer the reader to [10] for details and just briefly mention here some aspects of the statement which may be less familiar.

As customary, b_{2}^{+}(X) denotes the dimension of a maximal positive-definite linear subspace H^{2,+}(X; ℝ) for the intersection pairing Q_{X} on H^{2}(X; ℝ). It is implicit in the statement of Theorem 1.1 that we have selected an orientation for H^{1}(X; ℝ) ⊕ H^{2,+}(X; ℝ), and the Donaldson and Seiberg-Witten invariants are computed with respect to this choice.

A four-manifold is abundant if the restriction of Q_{X} to B^{⊥} contains a hyperbolic sublattice [10, Definition 1.2]. This condition ensures that there exist classes Λ ∈ B^{⊥} with prescribed even square, such as Λ^{2} = 2 − (χ + σ). All compact, complex algebraic, simply connected surfaces with b_{2}^{+} ≥ 3 are abundant. There exist simply connected four-manifolds with b_{2}^{+} ≥ 3 which are not abundant, but which nonetheless admit classes Λ ∈ B^{⊥} with prescribed even squares [8, p. 175].

As described in [10, Definition 1.3], a four-manifold is effective if it satisfies Conjecture 3.1 in [8], restated as Conjecture 3.34 in this article. This conjecture asserts that the pairings of Donaldson-type cohomology classes with the link of a Seiberg-Witten moduli subspace of the (compactified) moduli space of PU(2) monopoles are multiples of its Seiberg-Witten invariants, so these pairings are zero when the Seiberg-Witten invariants for that Seiberg-Witten moduli space are trivial.

For any w ∈ H^{2}(X; ℤ), one can define a Donaldson invariant (see §3.4.2 for a detailed description) as a real-linear function [35, p. 595]

D_{X}^{w} : 𝔸(X) → ℝ,

where [35]

(1.3) 𝔸(X) = Sym (H_{even}(X; ℝ)) ⊗ Λ^{•}(H_{odd}(X; ℝ))

is the graded algebra. If z ∈ 𝔸(X) is a monomial then D_{X}^{w}(z) = 0 unless

deg(z) ≡ −2w^{2} − \frac{3}{2}(χ + σ) (mod 8). (1.4)

Recall from [35, Equation (1.5)] that the Donaldson series is defined by

D_{X}^{w}(h) = D_{X}^{w}((1 + \frac{1}{2}x)e^{h}) = ∑_{d≥0} \frac{1}{d!}D_{X}^{w}(h^{d}) + \frac{1}{2d!}D_{X}^{w}(xh^{d}), h ∈ H_{2}(X; ℝ). (1.5)

A four-manifold with b_{1}(X) = 0 and odd b_{2}^{+}(X) ≥ 3 has Kronheimer-Mrowka (KM) simple type [35] if for some w and all z ∈ 𝔸(X),

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

If as in Theorem 1.1, we do not assume that X has KM-simple type, then equation (1.5) only defines D_{X}^{w}(h) as a formal power series and one cannot necessarily recover all invariants of the form D_{X}^{w}(x^{m}h^{d−2m}) from the series (1.5). According to [35, 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} Q_{X}(h,h)} ∑_{r=1}^{s}(−1)^{\frac{1}{2} (w^{2}+K_{r}·w)}a_{r}e^{⟨K_{r},h⟩}, h ∈ H_{2}(X; ℝ).

More generally [33], a four-manifold X has finite type or type τ if

D_{X}^{w}((x^{2} − 4)^{τ} z) = 0,

for some τ ∈ ℕ and all z ∈ 𝔸(X). Kronheimer and Mrowka conjectured that all fourmanifolds X with b_{2}^{+}(X) > 1 have finite type and state an analogous formula for the series D_{X}^{w}(h); proofs of different parts of their conjecture have been reported Frøyshov [23, Corollary 1], Muñoz [49, Corollary 7.2 & Proposition 7.6], and Wieczorek [57, Theorem 1.3].

For a four-manifold X with b_{1}(X) = 0 and odd b_{2}^{+}(X) ≥ 3, Witten’s conjecture [58] asserts that X has KM-simple type if and only if it has SW-simple type; 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). (1.6)

Equation (1.2) therefore tells us that Witten’s formula holds, modulo terms of degree greater than or equal to c(X), at least for four-manifolds satisfying the hypotheses of Theorem 1.1. Equation (1.2) is proved by considering Seiberg-Witten moduli spaces in the top level, ℓ = 0, of the compactified PU(2) monopole moduli space; in order to prove that equation (1.6) holds modulo h^{d} for arbitrary d ≥ c(X) (and the same w, Λ), one needs to compute the contributions of Seiberg-Witten moduli spaces in arbitrary levels ℓ ≥ 0. In [14] we use the case ℓ = 1 to show that equation (1.6) holds mod h^{c(X)+2}.

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 the only Seiberg-Witten moduli spaces with non-trivial invariants lie in the top level of the PU(2)-monopole moduli space. When b_{1}(X) ≥ 0, the Seiberg-Witten invariants for (X, 𝔰), with 𝔰 ∈ Spin^{c}(X), are defined collectively as a real-linear function (see §4.1 for a detailed description),

SW_{X,𝔰} : 𝔹(X) → ℝ, (1.7)

where the graded algebra is given by

𝔹(X) = ℝ[x] ⊗ Λ^{•}(H_{1}(X; ℝ)). (1.8)

Here, Λ^{•}(H_{1}(X; ℝ)) is the exterior algebra on H_{1}(X; ℝ), with γ ∈ H_{1}(X; ℝ) having degree one, and ℝ[x] is the polynomial algebra with generator x of degree two. If z ∈ 𝔹(X) then SW_{X,𝔰}(z) = 0 unless

deg(z) = d_{s}(𝔰),

where d_{s}(𝔰) is the dimension of the Seiberg-Witten moduli space M_{𝔰},

d_{s}(𝔰) = \frac{1}{4}(c_{1}(𝔰)^{2} − 2χ − 3σ). (1.9)

If z = x^{m} ∈ 𝔹(X) and 2m = d_{s}(𝔰), then as customary [34], [46], [58] one has

SW_{X}(𝔰) = SW_{X,𝔰}(z). (1.10)

As in [50, §1], when b_{1}(X) ≥ 0 we call c_{1}(𝔰) ∈ H^{2}(X; ℤ) an SW-basic class if the Seiberg-Witten function (1.7) is non-trivial. If b_{2}^{+}(X) = 1 or b_{1}(X) > 0, there are examples of fourmanifolds whose basic classes have positive-dimensional Seiberg-Witten moduli spaces: ℂℙ^{2} and its blow-ups give examples with b_{2}^{+}(X) = 1 [52], and connected sums of S^{1} × S^{3} and a four-manifold with non-trivial Seiberg-Witten invariants provide examples with b_{1}(X) > 0 [50, §2].

For Λ ∈ H^{2}(X; ℤ), define

i(Λ) = Λ^{2} + c(X) + χ + σ. (1.11)

If S(X) ⊂ Spin^{c}(X) is the subset yielding non-trivial Seiberg-Witten functions (1.7), let

(1.12) 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 for a discussion of the significance of r(Λ, c_{1}(𝔰)) and r(Λ). We then have:

Theorem 1.2. Let X be a four-manifold with b_{2}^{+}(X) ≥ 1. Assume α ⌣ α′ = 0 for every α, α′ ∈ H^{1}(X; ℤ) and that 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. Let z = x^{δ_{0}}ϑh^{δ_{2}}, where h ∈ H_{2}(X; ℝ), ϑ ∈ Λ^{δ_{1}}(H_{1}(X; ℝ)), and x ∈ H_{0}(X; ℤ) is the positive generator, and write deg(z) = 2δ, for δ ∈ \frac{1}{2}ℤ.

(a) If δ < i(Λ) and δ < r(Λ), then

D_{X}^{w}(z) = 0. (1.13)

(b) If δ < i(Λ) and δ = r(Λ), then

D_{X}^{w}(z) = 2^{1− \frac{1}{4} (i(Λ)−δ)−δ_{2}−2δ_{0}}(−1)^{δ_{0}+δ_{1}+ \frac{1}{2} (σ−w^{2})+1}

× ∑_{\{𝔰∈S(X): r(Λ,c_{1}(𝔰))=r(Λ)\}}(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·(w−Λ))}

× H_{χ,σ}(Λ^{2}, deg(z), d_{s}(𝔰), δ_{1})SW_{X,𝔰}(ϑx^{\frac{1}{2} (d_{s}(𝔰)−δ_{1})})⟨c_{1}(𝔰) − Λ, h⟩^{δ_{2}}, (1.14)

with H_{χ,σ} defined in equation (1.15). If \frac{1}{2}(d_{s}(𝔰) − δ_{1}) = 0, then H_{χ,σ}(Λ^{2}, deg(z), d_{s}(𝔰), δ_{1}) = 1. If b_{2}^{+}(X) = 1, then all invariants in equation (1.14) are evaluated with respect to the chambers determined by the same period point in the positive cone of H^{2}(X; ℝ).

(c) If Λ^{2} and δ satisfy

2r(Λ) < 2δ ≤ r(Λ) + \frac{1}{2}(r(Λ) + i(Λ)) − 2,

then equation (1.14) holds with D_{X}^{w}(z) = 0.

Remark 1.3. 1. When δ_{1} > d_{s}(𝔰), the Seiberg-Witten invariant SW_{X,𝔰}(ϑx^{\frac{1}{2} (d_{s}(𝔰)−δ_{1})}) in

equation (1.14) is zero by definition.

2. If z = Y z′ where Y ∈ H_{3}(X; ℤ) and z′ cannot be written as z′ = xz′′ for x ∈ H_{0}(X; ℤ), then equations (1.13) and (1.14) hold but the right-hand-side of (1.14) vanishes.

The hypothesis in Theorems 1.1, 1.2, 1.4, and Corollary 1.5 that X is effective can be eliminated if, in the definition (1.12) of r(Λ), we replace S(X) with the (possibly larger) set of all 𝔰 ∈ Spin^{c}(X) for which the Seiberg-Witten moduli space M_{𝔰} (as defined in [10, §2.3], with perturbations depending on Λ) is non-empty. This additional generality does not seem to be useful in practice, however.

When b_{2}^{+}(X) = 1 and w (mod 2) admits no torsion integral lifts, Lemma 4.1 implies that the walls defining the chamber structure for the Donaldson invariant D_{X}^{w} are given by the walls for the Seiberg-Witten invariants appearing in equation (1.14). Thus, both sides of the equation will change when the period point crosses one of these walls.

The function H_{χ,σ} in equation (1.14) is defined as

H_{χ,σ}(Λ^{2}, deg(z), d_{s}(𝔰), δ_{1}) = 2^{d}P_{d}^{a,b}(0), (1.15) where d is a natural number and a, b are integers given by

d = \frac{1}{2}(d_{s}(𝔰) − δ_{1}),
a = \frac{1}{4} (3r(Λ) + i(Λ)) − \frac{1}{2} deg(z) − d − 2,
b = \frac{1}{2} (deg(z) − 2r(Λ) − d_{s}(𝔰)) − \frac{1}{4}(χ + σ),

and P_{d}^{a,b}(ξ) is a Jacobi polynomial [29, §8.96],

P_{d}^{a,b}(ξ) = ∑_{u=0}^{d}(−1)^{u}\binom{a + d}{d − u}\binom{b + d}{u}(ξ − 1)^{u}(ξ + 1)^{d−u}, ξ ∈ ℂ. (1.16)

The polynomials P_{d}^{a,b}(ξ) may in turn be expressed in terms of hypergeometric functions [29, §9.10], [38], as we explain in §4.4. Since c_{1}(𝔰)^{2} ≡ σ (mod 8), equation (1.9) for d_{s} implies that the expression −\frac{1}{2}d_{s} − \frac{1}{4} (χ + σ) in the definition of b is an integer.

When Λ ∈ B^{⊥} ⊂ H^{2}(X; ℤ) and X has SW-simple type, the expression (1.12) for r(Λ) becomes

r(Λ) = −Λ^{2} + c(X) − (χ + σ), (1.17)

and i(Λ) + r(Λ) = 2c(X), by equation (1.11) for i(Λ). Theorem 1.2 then simplifies to:

Theorem 1.4. Let X be a four-manifold with odd b_{2}^{+}(X) ≥ 3 and b_{1}(X) = 0. Assume that X is effective and has SW-simple type. Suppose that Λ ∈ B^{⊥} and that w ∈ H^{2}(X; ℤ) is a class with w − Λ ≡ w_{2}(X) (mod 2). Let δ ≥ 0 and 0 ≤ m ≤ [δ/2] be integers.

(a) If δ < i(Λ) and δ < r(Λ), then for all h ∈ H_{2}(X; ℝ) we have

D_{X}^{w}(h^{δ−2m}x^{m}) = 0. (1.18)

(b) If δ < i(Λ) and δ = r(Λ) we have

\begin{aligned}D_{X}^{w}(h^{δ−2m}x^{m}) & = 2^{1− \frac{1}{2} (c(X)+δ)}(−1)^{m+1+ \frac{1}{2} (σ−w^{2})} \\ & × ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·w)}SW_{X}(𝔰)⟨c_{1}(𝔰) − Λ, h⟩^{δ−2m}. \end{aligned} (1.19)

(c) If r(Λ) < δ ≤ \frac{1}{2}(r(Λ) + c(X) − 2), then equation (1.19) holds with D_{X}^{w}(h^{δ−2m}x^{m}) = 0.

We recall from [22, §6.1.1] that the symmetric algebra Sym(H_{2}(X; ℝ)) is canonically isomorphic (as a graded algebra) to the algebra P (H_{2}(X; ℝ)) of polynomial functions on H_{2}(X; ℝ). Thus, given the Donaldson invariants D_{X}^{w}(h^{δ−2m}x^{m}), we can recover all invariants of the form D_{X}^{w}(h_{1} ⋯ h_{δ−2m}x^{m}).

Corollary 1.5. [8, Theorem 1.1] Let X be a four-manifold with odd b_{2}^{+}(X) ≥ 3 and b_{1}(X) = 0. Assume that X is effective, abundant, and SW-simple type, with c(X) ≥ 3. Then for any w ∈ H^{2}(X; ℤ) with w ≡ w_{2}(X) (mod 2) we have

SW_{X}^{w}(h) ≡ 0 (mod h^{c(X)−2}).

In §4.6 we give a slightly different and more geometric proof of Corollary 1.5 than provided in [8, §2], using the final case of Theorem 1.4. This result proves a conjecture of Mariño, Moore, and Peradze [40], [41] for four-manifolds of Seiberg-Witten simple type, albeit with the additional hypotheses that the four-manifolds are abundant and effective. The vanishing result for Donaldson invariants in Theorem 1.1 can be sharpened: see Theorem 1.2 in [8].

1.2. Remarks on the hypotheses of Theorems 1.2, 1.4, and 1.1. To prove Theorem 1.2 (and thus Theorems 1.1, 1.4, and Corollary 1.5), we employ the moduli space of PU(2) monopoles, M_{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 Seiberg-Witten monopoles, M_{𝔰}. 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), and

2. Only the top level of the Uhlenbeck compactification M̄_{𝔱} of the moduli space of PU(2) monopoles contains Seiberg-Witten moduli spaces M_{𝔰} with non-trivial invariants (used in Corollary 3.35).

In the proof of Assertions (a) and (b) of Theorem 1.2, one has 2δ = deg(z) = dim M_{κ}^{w} and the hypotheses δ < i(Λ) and δ ≤ r(Λ) ensure that Conditions (1) and (2) hold, respectively. In the proof of Assertion (c), one has dim M_{κ}^{w} = 2r(Λ), which implies that Condition (2) holds while the inequality in the hypothesis of (c) implies that Condition (1) is satisfied.

Assertion (a) follows because the hypothesis δ < r(Λ) implies that there are no Seiberg-Witten moduli spaces contained in M̄_{𝔱} with non-trivial invariants, and so M_{𝔱}/S^{1} is essentially a null-cobordism of the link of M_{κ}^{w}. Assertion (c) follows because the hypothesis on deg(z) implies that deg(z) > M_{κ}^{w}, and only the pairings of Donaldson-type cohomology classes with links of Seiberg-Witten moduli spaces can be non-trivial. In the remaining Assertion (b), the hypotheses imply that the cobordism yields an equality between pairings with the link of M_{κ}^{w} and a sum of pairings with the links of M_{𝔰}. The same remarks apply to the hypotheses in Assertions (a), (b), and (c) in Theorem 1.4.

The assumption that α ⌣ α′ = 0 for all α, α′ ∈ H^{1}(X; ℤ) greatly simplifies the calculation of the Chern classes of the virtual normal bundle of M_{𝔰} in M_{𝔱} (see Corollary 3.30 in [10]) and hence its Segre classes (see Lemma 4.11). It should be possible to remove this condition with more work, but this would take us a little beyond the scope of this article and [10]. We plan to address this point elsewhere.

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 and before Lemma 4.1 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 [8].

1.3. Remarks and conjectures for formulas for Donaldson invariants involving Seiberg-Witten strata in arbitrary levels. The following remarks are intended to convey an outline of the remainder of our work on a proof of Witten’s conjecture in [12], [13], [15], [14]. While some details still remain to be checked, we are confident that the conclusions stated below are correct based on our work thus far, despite their conservatively-stated current status as conjectures rather than firm assertions.

As envisaged in [16], the PU(2)-monopole program proposed by Pidstrigatch and Tyurin [51] for proving Witten’s conjecture [44], [58] uses the oriented cobordism M_{𝔱}/S^{1} between

• The links L_{𝔱,κ}^{w} in M̄_{𝔱}/S^{1} of the anti-self-dual moduli subspace M_{κ}^{w} of M̄_{𝔱}/S^{1}, and • The links L_{𝔱,𝔰} in M̄_{𝔱}/S^{1} of the Seiberg-Witten moduli subspaces, M_{𝔰} × Sym^{ℓ}(X), of

the space of ideal PU(2) monopoles, ∪_{ℓ=0}^{∞}(M_{𝔱_{ℓ}} × Sym^{ℓ}(X)), containing M̄_{𝔱}.

The program therefore has two principal steps, which we now outline. The first step is to define the links L_{𝔱,𝔰} of M_{𝔰} × Sym^{ℓ}(X) for arbitrary ℓ ≥ 0 using the gluing construction of [12], [13], extending the construction in [10] which just treats the case ℓ = 0. The oriented cobordism M_{𝔱}/S^{1} then yields a formula (with deg(z) = 2δ),

D_{X}^{w}(z) = −2^{−δ_{c}} ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{4} (w−Λ+c_{1}(𝔰))^{2}}⟨μ_{p}(z) ⌣ μ_{c}^{δ_{c}}, [L_{𝔱,𝔰}]⟩, (1.20)

where μ_{p}(z) and μ_{c} are Donaldson-type cohomology classes, and δ_{c} = \frac{1}{4}(i(Λ)−δ)−1. Work in progress [15] then strongly indicates that the pairings on the right-hand side of equation (1.20) have the following general form, when b_{1}(X) = 0 and z = x^{m}h^{δ−2m},

⟨μ_{p}(z) ⌣ μ_{c}^{δ_{c}}, [L_{𝔱,𝔰}]⟩ = SW_{X}(𝔰) ∑_{i=0}^{r} δ_{d,i} (⟨c_{1}(𝔰), h⟩, ⟨Λ, h⟩) Q_{X}^{ℓ−i}(h, h), (1.21)

where r = min(ℓ, [δ/2]−m), ℓ = \frac{1}{4}(δ −r(Λ, c_{1}(𝔰))), d = δ −2(m+ℓ−i), and δ_{d,i} is a degreed, homogeneous polynomial (aside from stray powers of (−1) and 2) in two variables with coefficients which are degree-i polynomials in 2χ±3σ, (c_{1}(𝔰)−Λ)^{2}, Λ^{2}, and (c_{1}(𝔰−Λ)·c_{1}(𝔰).

An interesting feature of the formula (1.21) is that one sees a factorization of the pairings ⟨μ_{p}(z) ⌣ μ_{c}^{δ_{c}}, [L_{𝔱,𝔰}]⟩ into a product of SW_{X}(𝔰) and the term δ_{d} = ∑_{i=1}^{r} δ_{d,i}Q_{X}^{ℓ−i}. In particular, the pairing (1.21) vanishes when SW_{X}(𝔰) = 0, implying that X is “effective” in the sense described in §1.1. The factors δ_{d,i} are similar to those appearing in the Kotschick-Morgan conjecture [9], [32] for the form of the wall-crossing formula for the Donaldson invariants of a four-manifold X with b_{2}^{+}(X) = 1. However, δ_{d,i} is a polynomial in two variables while the corresponding term in the conjectured wall-crossing formula for Donaldson invariants is a polynomial in only one variable.

Explicit, direct computations of pairings with the links L_{𝔱,𝔰} of ideal Seiberg-Witten moduli spaces, M_{𝔰}× Sym^{ℓ}(X), are possible when ℓ = 0, 1 or 2: indeed, Theorem 1.1 is proved using the case ℓ = 0 and we prove an ℓ = 1 analogue of Theorem 1.1 in [14], while the case ℓ = 2 would follow by adapting work of Leness in [36]. However, direct computations appear intractable when ℓ is large.

The idea underlying the second step of the program is to use the existence of formulas (1.20) and (1.21) in conjunction with auxiliary arguments to prove Witten’s conjecture, since more direct calculations of the link pairings appear difficult. The work of Göttsche [27] suggests that such indirect strategies should succeed, as he was able to compute the wall-crossing formula for the Donaldson invariants of four-manifolds with b_{2}^{+}(X) = 1 and b_{1}(X) = 0, under the assumption that the Kotschick-Morgan conjecture [32] holds for such four-manifolds. The facts that δ_{d,i} is a function of two variables and both χ and σ may vary independently indicate that this second step in the PU(2)-monopole program is potentially more complicated than that of [27], where the assumption that b_{2}^{+}(X) = 1 implies that σ = 1 − b_{2}^{−}(X) and χ = 3 + b_{2}^{−}(X) (when b_{1}(X) = 0). However, in the case of the PU(2)-monopole program there are more sources of “recursion relations” of the type used by Göttsche, in addition to those arising from the blow-up formulas of Fintushel-Stern [20],

[19] (for Donaldson and Seiberg-Witten invariants). Moreover, there is a rich supply of examples of four-manifolds where Witten’s conjecture is known to hold.

1.4. Remarks on abelian localization. One of the first observations concerning the moduli space of PU(2) monopoles is that the Donaldson stratum, ι(M_{κ}^{w}), and the Seiberg-Witten strata, ι(M_{𝔰}), are fixed-point sets under the circle action given by scalar multiplication on the spinor components of PU(2)-monopole pairs; see [10, §3.1] for a detailed account. This raises the question of whether the technique of abelian localization, as applied to circle actions on compact manifolds [1] or its generalizations to singular algebraic varieties (for example, see [28]), can be usefully applied here to prove Witten’s conjecture. As we indicate below, if the moduli space of PU(2) monopoles were a compact manifold and the Donaldson and Seiberg-Witten strata were smooth submanifolds, then an application of the localization formula would be equivalent to our construction of links and application of the PU(2)-monopole cobordism. There is no saving of labor and the essential point remains, with either equivalent view, to compute the Chern classes of the normal bundles of the fixed-point sets. As the moduli space of PU(2) monopoles is non-compact, equipped with the highly singular Uhlenbeck compactification or somewhat less singular but more complicated bubble-tree compactification, our application of the PU(2)-monopole cobordism can be thought of as an extension of the localization method to those differential-geometric, singular settings.

The technique of abelian localization does not reduce the information about neighborhoods of singularities needed to compute intersection pairings because the localization formula requires a computation of an equivariant Thom or Euler class of the normal bundle of the fixed point set. For example, if F ⊂ M is the fixed point set and N → F is its normal bundle, the equivariant Euler class of N is the Euler class e(N_{S^{1}}) of the bundle

N_{S^{1}} = ES^{1} ×_{S^{1}}N → BS^{1} ×F,

where ES^{1} → BS^{1} is the universal S^{1} bundle over the classifying space BS^{1}. Let

π_{S^{1}} : ES^{1} ×_{S^{1}}M → BS^{1} and ι : ES^{1} ×_{S^{1}}N → ES^{1} ×_{S^{1}}M,

be the projection and embedding maps, respectively. If dim M = m, then because

H_{S^{1}}^{m}(M \ F; ℝ) ≅ H^{m}((M \ F)/S^{1}; ℝ),

any class x ∈ H_{S^{1}}^{m}(M; ℝ) has compact support in ES^{1} ×_{S^{1}}N by dimension counting. The abelian localization formula [1, Equation (3.8)] states that

(π_{S^{1}})_{∗}x = \frac{ι^{∗}x}{e(N_{S^{1}})}/[F]. (1.22)

If h is the pullback of the universal first Chern class from BS^{1} to BS^{1} ×F and π_{F} : BS^{1} ×F → F is the projection, then the splitting principle shows that e(N_{S^{1}}) = ∑_{i=0}^{r} h^{i}π_{F}^{∗} c_{r−i}(N), where r = rank N. A simple algebraic computation (see [14]) shows that computing (π_{S^{1}})_{∗}x using formula (1.22) and finding the inverse of e(N_{S^{1}}) is equivalent to computing the Segre classes of N.

Thus, if M_{𝔱} were a compact manifold and the Seiberg-Witten moduli spaces M_{𝔰} were smooth submanifolds of M_{𝔱}, the abelian localization method would be equivalent to the one used in this article.

If X is a complex algebraic surface, it should be possible to construct the Gieseker compactifiication for the moduli space of PU(2) monopoles over X, by analogy with the construction of Morgan [45] and Li [37] for the moduli space of PU(2) monopoles, and then apply the results of [28] to this compactification. However, one would still need to compute the equivariant Euler classes of the normal sheaves of strata of ideal, reducible PU(2) monopoles, in order to apply [28, Equation (1)]. If X is not algebraic, one would need to solve the non-trivial problem of defining the normal sheaves of these strata in gauge-theoretic compactifications.

1.5. A guide to the article and outline of the proofs of the main results. The present article is a direct continuation of [10] and rather than repeat many of the definitions here, we shall refer to [10]. The reader may wish to consult the notational index in [10], as well as the notational index for the present article which appears just before §2. The first version of this article was distributed in December 1997 as sections 4–7 of the preprint [11].

As in [10], we let 𝔰 = (ρ, W) denote a spin^{c} structure on X, where W is a Hermitian, rank-four bundle over X and use 𝔱 = (ρ, V) to denote a “spin^{u} structure” on X, where V is a Hermitian, rank-eight bundle over X [10, Definition 2.2].

One of the main results (Theorem 3.31) of [10] is a calculation the Chern characters of vector bundles defining tubular neighborhoods of links of Seiberg-Witten moduli spaces in local, “thickened” moduli spaces of PU(2) monopoles M_{𝔱}. In §2 of this article, we compare the orientations of the moduli spaces and anti-self-dual connections, Seiberg-Witten monopoles, and their links in the moduli space of PU(2) monopoles. In §3 we define cohomology classes and dual geometric representatives on the moduli space of PU(2) monopoles and in §4 we prove Theorem 1.2 by counting the intersection of these representatives with links of the moduli spaces of anti-self-dual connections and top-level Seiberg-Witten monopoles.

Compatible choices of orientations for all the moduli spaces appearing in the stratification [10, Equation 3.17] of M_{𝔱} and of the links L_{𝔱,κ}^{w} and L_{𝔱,𝔰} are a basic requirement of the cobordism method and a discussion of our orientation conventions is taken up in §2. As in [10, Equation (2.40)], we let M_{𝔱}^{∗,0} ⊂ M_{𝔱} denote subspace represented by PU(2) monopoles which are neither zero-section pairs (corresponding to anti-self-dual connections) or reducible pairs (corresponding to Seiberg-Witten monopoles). In §2.1 we show that M_{𝔱}^{∗,0}/S^{1} (the smooth locus or top stratum of M_{𝔱}/S^{1}) is orientable, with an orientation determined by a choice of an orientation for the moduli space of anti-self-dual connections, M_{κ}^{w}, as explained further in §2.2: this allows us to compute the oriented intersection of one-manifolds with the link L_{𝔱,κ}^{w}, where the one-manifolds arise as the intersection of geometric representatives of the cohomology classes on M_{𝔱}^{∗,0}/S^{1}. We also define an orientation for M_{𝔱}^{∗,0}/S^{1} induced by an orientation of M_{𝔰}, as in §2.3, and this allows us to compute the oriented intersection of one-manifolds with the link L_{𝔱,𝔰}. In §2.4 we compare the two orientations of M_{𝔱}^{∗,0}/S^{1} naturally induced by those of M_{κ}^{w} and M_{𝔰}. Finally, in §2.5 we compare the natural orientations of these links with the orientations obtained by considering them as boundaries of the complement in M_{𝔱}/S^{1} of small open neighborhoods of the anti-self-dual and Seiberg-Witten strata.

In §3.1 and 3.2 we describe the cohomology classes μ_{p}(β) on the moduli space M_{𝔱}^{∗}/S^{1} and μ_{c} on M_{𝔱}^{∗,0}/S^{1} and their dual geometric representatives V(β) and W, following the methods of [4], [6], [7], [35] for the classes μ_{p}(β). A technical complication not present when dealing solely with M_{κ}^{w} is that the lower strata of M̄_{𝔱} have smaller codimension than those of M̄_{κ}^{w}. The unique continuation property for reducible PU(2) monopoles [17, Theorem 5.11] plays a role here analogous to that of the unique continuation property for reducible anti-self-dual SO(3) connections in [7], [35]. In §3.3, we show that the closures V̄(β) and W̄ of these geometric representatives in M̄_{𝔱}/S^{1} meet the lower strata of M̄_{𝔱}^{∗,0}/S^{1} transversely, that is, in a subspace of the expected codimension away from the reducible and zero section solutions appearing in lower levels. Thus, the closure of the one-manifolds will have boundaries in M̄_{κ}^{w} or in some stratum M_{𝔰} × Σ of reducible PU(2) monopoles, where Σ ⊂ Sym^{ℓ}(X). The hypotheses of Theorem 1.2 exclude consideration of the more difficult case ℓ > 0. In §3.4 we show that the number of points, counted with sign, in the boundaries of the one-manifolds defined by an appropriate choice of geometric representatives in the link L_{𝔱,κ}^{w} of the stratum M_{κ}^{w} is given by a multiple of the Donaldson invariant, thus completing the proof of Theorem 3.33. In the course of proving this result we also show that M_{𝔱}^{∗,0} is nonempty for sufficiently negative p_{1}(𝔱) — see Proposition 3.30 in §3.4.

In §4 we calculate the intersection of these geometric representatives with the link L_{𝔱,𝔰} of the stratum M_{𝔰} and show that it is given by a multiple of the Seiberg-Witten invariant associated to the spin^{c} structure 𝔰 (see Theorem 4.13). The geometric representatives, in general, intersect the strata M_{𝔰} in sets of higher than expected dimension, so our calculation of the link pairings here may be viewed as a differential-geometric analogue of the “excess intersection theory” calculations discussed in [24]. Combining the link pairing calculations of §3 and §4 and applying the cobordism M_{𝔱}^{∗,0}/S^{1} then yields the formulas for Donaldson invariants in terms of Seiberg-Witten invariants in Theorem 1.2, from which Theorems 1.1 and 1.4 and Corollary 1.5 are derived in §4.6.

Acknowledgments. The authors thank Ron Fintushel, Tom Mrowka, Peter Ozsváth, and Zoltán Szabó for helpful conversations. We are especially grateful to Tom Mrowka for his many helpful comments during the course of this work, for bringing a correction to some examples in [11] to our attention, as well pointing out that our Theorem 1.4 in [11] could be elegantly rephrased and specialized to give the version stated as Theorem 1.4 here (and as Theorem 2.1 in [8]). We thank Dietmar Salamon for generously providing us with a pre-publication version of his book [52]. We thank Victor Pidstrigatch for inviting us to present our work at the Warwick Symposium on Geometry and Topology (2000). We thank the Columbia University Mathematics Department, the Institute for Advanced Study, Princeton, and the Max Planck Institute für Mathematik, Bonn for their generous support and hospitality during a series of visits while this article and its companion [10] were being prepared. Finally, we thank the anonymous referee and Simon Donaldson for their editorial suggestions and comments on the previous versions of this article.

Index of Notation

𝔸(X) Equation (1.3) 𝔹(X) Equation (1.8) B_{κ}^{w} [10, §2.1.6] C_{χ,σ}(·) Theorem 4.13 C_{𝔱}, C̃_{𝔱}, C_{𝔱}^{∗,0}, C_{𝔱}^{∗}, C_{𝔱}^{0} [10, §2.1.5] D_{A}, D_{A,ϑ}, D_{B}, D_{B,ϑ} Equation (2.4) D_{𝔱} §2.1 D^{n}, D^{t} [10, Equation (3.35)] D_{X}^{w} Equation (3.31) D_{X}^{w} Equation (1.5) 𝔽_{κ}^{w} Equation (3.4) 𝔽_{𝔱} Equation (3.2) G_{𝔰} [10, §2.3] G_{𝔱} [10, Definition 2.6] H_{χ,σ}(·) Equation (1.15) K_{A,δ} Equation (3.46) 𝕃_{𝔰} Equation (4.1) 𝕃_{𝔱} Equation (3.12) L_{𝔱,κ}^{w}, L_{𝔱,κ}^{w,ε} Equation (3.43) L_{𝔱,𝔰} [10, Definition 3.22] M_{κ}^{w}, M̄_{κ}^{w} [10, §2.1.6 & 2.2] M_{𝔰}, M̃_{𝔰} [10, §2.3] M_{𝔱}, M_{𝔱}^{∗,0} [10, Equations (2.33) & (2.40)] M̄_{𝔱} [10, Equation (2.46)] M̄_{𝔱}^{asd} Equation (3.28) M̄_{𝔱}^{∗}, M̄_{𝔱}^{0}, M̄_{𝔱}^{∗,0} Equation (3.18) M̄_{𝔱}^{≥ε}, M̄_{𝔱}^{∗,≥ε} Equation (3.19) M_{𝔱}(Ξ, 𝔰) [10, Definition 3.20] N_{𝔱}(Ξ, 𝔰), Ñ_{𝔱}(Ξ, 𝔰), [10, §3.5] N_{𝔱}^{ε}(Ξ, 𝔰), N_{𝔱}^{≤ε}(Ξ, 𝔰) §2.5 O^{asd}(Ω, w) Definition 2.3 O^{red}(Ω, 𝔱, 𝔰) Definition 2.5 O_{A}, O′_{A} Lemma 3.22 P_{d}^{a,b} Equation (1.16) SW_{X} Equation (1.10) SW_{X,𝔰} Equation (4.5) SW_{X}^{w} Equation (1.1) V(z), W §3.2.3 & §3.2.4 V̄(z), W̄ Definition 3.14 X̃ §3.4.2 Z_{A} Equation (3.40) (a)_{n} Equation (4.38) c(X) §1.1 c_{1}(𝔱), p_{1}(𝔱), w_{2}(𝔱) [10, Equation (2.19)] d_{a}(𝔱) Equation (3.21) d_{s}(𝔰) Equation (1.9) deg(z) Equation (3.15) det D_{A,Φ}, det D_{𝔱} Equation (2.1) det δÂ, det δ_{E} Equation (2.2) f_{A} Lemma 3.22 g_{A} Equation (3.53) i(Λ) Equation (1.11) ℓ Equation (3.42) n_{a}(𝔱) Equation (3.21) n′_{s}(𝔱, 𝔰), n′′_{s}(𝔱, 𝔰) Equation (4.25) o(Ω, w) §2.2 o_{𝔱}(w, 𝔰) Equation (3.66) r(Λ), r(Λ, 𝔰) Equations (1.12) & (1.17) 𝔰 [10, Definition 2.1] 𝔰^{±} §4.1 s_{i}, s_{i}(N) Lemma 4.10 𝔱 [10, Definition 2.2] 𝔱̃ Lemma 4.19 𝔱_{ℓ} [10, Equation (2.44)] Ξ [10, Definition 3.20] γ_{A} Equation (3.36) δ_{E} §2.1 δ_{c} Equation (3.26) δ_{i} Equation (3.15) δ_{p} Equation (4.33) ι [10, Equations (3.4) & (3.9)] κ §3.1 μ_{c} Equation (3.12) μ_{p} Equation (3.3) μ_{𝔰} Equations (4.3) & (4.4) ν Definition 4.3 π_{B} Equation (2.6) ϱ Equation (4.10) ϱ_{L} Equation (4.11) σ, χ, §1.1 φ_{A} Equation (3.37) ω(K, o) Equation (3.48) ω(L, ∂O) Equation (3.47) ω(Z ∩ K, ∂O) Equation (3.50) ω(Z), ω(Z) Convention 3.23

2. Orientations of moduli spaces

Following the pattern in [5] and [7, §5.4 & §7.1.6], we first show that M_{𝔱}^{∗,0} is an orientable manifold and then show that its orientation is canonically determined by a choice of homology orientation of our four-manifold X and an integral lift w of w_{2}(𝔱). The orientation for M_{𝔱}^{∗,0} will be invariant under the circle action and thus give an orientation for M_{𝔱}^{∗,0}/S^{1}. We also obtain relations between the orientations of the smooth, top stratum M_{𝔱}^{∗,0}/S^{1}, the stratum M_{κ}^{w} ↪ M_{𝔱}/S^{1} defined by the anti-self-dual moduli space, and the strata M_{𝔰} ↪ M_{𝔱}/S^{1} defined by the Seiberg-Witten moduli spaces.

2.1. Orientability of moduli spaces of PU(2) monopoles. In this section we show that M_{𝔱}^{∗,0} is orientable.

As in [10, §2.1.5], we let C̃_{𝔱} denote the pre-configuration space of pairs (A, Φ), where A is a spin connection on V = V^{+}⊕V^{−} with fixed determinant connection A^{det} = 2A_{Λ} on det(V^{+}) and Φ is a section of V^{+}. We defined C_{𝔱} = C̃_{𝔱}/G_{𝔱}, where G_{𝔱} is the group spin^{u} automorphisms of V [10, Definition 2.6].

Recall that D_{A,Φ} = d_{A,Φ}^{∗,0}+d_{A,Φ}^{1} is the “deformation operator” corresponding to the elliptic deformation complex [10, Equation (2.47] for the moduli spaces M_{𝔱}. Let det D_{𝔱} be the real determinant line bundle over the pre-configuration space C̃_{𝔱}, with fiber over (A, Φ) ∈ C̃_{𝔱} given by

(2.1) det D_{A,Φ} = Λ^{max}(Ker D_{A,Φ}) ⊗ Λ^{max}(Coker D_{A,Φ})^{∗}.

(See [7, §5.2.1] for the construction of determinant line bundles for families of elliptic operators.) The kernel and cokernel of D_{A,Φ} are equivariant with respect to the actions of group G_{𝔱} ×_{{±1}} S^{1}. The stabilizer, in G_{𝔱} ×_{{±1}} S^{1}, of the pair (A, Φ) acts trivially on the fibers of det D_{𝔱} because this stabilizer is connected and the structure group of the fiber of det D_{𝔱} (which is a real line) is {±1} and thus disconnected. Hence, the bundle det D_{𝔱} descends to C_{𝔱}/S^{1} and so to C_{𝔱} as well. We will show that the bundle det D_{𝔱} → C_{𝔱}/S^{1} is trivial.

Motivated by the remarks of [22, p. 330], we say that an orientation for M_{𝔱} is a choice of orientation for the real line bundle det D_{𝔱} (restricted to M_{𝔱}): an orientation of the fibers of det D_{𝔱} gives orientations for the real lines

det D_{A,Φ} = Λ^{max}(H_{A,Φ}^{1}) ⊗ Λ^{max}(H_{A,Φ}^{0} ⊕ H_{A,Φ}^{2})^{∗}, [A, Φ] ∈ M_{𝔱}.

If [A, Φ] is a smooth point of M_{𝔱}, so Coker D_{A,Φ} ≅ H_{A,Φ}^{0} ⊕ H_{A,Φ}^{2} = 0, then Ker D_{A,Φ} = H_{A,Φ}^{1} = T_{A,Φ}M_{𝔱} and

det D_{A,Φ} = Λ^{max}(H_{A,Φ}^{1}) = Λ^{max}(T_{[A,Φ]}M_{𝔱}),

so an orientation for det D_{𝔱} defines an orientation for the open manifold M_{𝔱}^{∗,0} of smooth points of M_{𝔱}. Therefore, det D_{𝔱} is an orientation bundle for M_{𝔱} and M_{𝔱} is orientable if det D_{𝔱} is trivial. As in [5], we show that M_{𝔱} is orientable because the bundle det D_{𝔱} → C_{𝔱}/S^{1} has a nowhere vanishing section, that is, its first Stiefel-Whitney class vanishes.

Suppose E → X is a rank-two, complex Hermitian bundle with c_{1}(E) = w and p_{1}(𝔰𝔲(E)) = −4κ. Denote the group of determinant-one, unitary automorphisms of E by G_{κ}^{w} and the space of SO(3) connections on 𝔰𝔲(E) by A_{κ}^{w}. Over the quotient space of connections B_{κ}^{w} = A_{κ}^{w}/G_{κ}^{w} there is an orientation bundle det δ_{E} (see [7, Equation (5.4.2)]) with fiber over [Â] ∈ B_{κ}^{w} given by

(2.2) det δÂ = Λ^{max}(Ker δÂ) ⊗ Λ^{max}(Coker δÂ), coming from the rolled-up deformation complex for the anti-self-dual moduli space M_{κ}^{w},

δÂ = d∗ ; Â + d+ ; Â : C^{∞}(Λ^{1} ⊗ 𝔰𝔲(E)) → C^{∞}((Λ^{0} ⊕ Λ^{+}) ⊗ 𝔰𝔲(E)). (2.3)

Thus, an orientation for det δ_{E} defines an orientation for manifold M_{κ}^{w}, since

det(d∗ ; Â + d+ ; Â) = Λ^{max}(H1 ; Â) ⊗ Λ^{max}(H0 ; Â ⊕ H2 ; Â)^{∗},

We recall the following result of Donaldson:

Proposition 2.1. [5, Corollary 3.27] The bundle det δ_{E} → B_{κ}^{w} is topologically trivial.

We now show that det D_{𝔱} is trivial using the fact that det δ_{E} is trivial, where 𝔱 = (ρ, V), V = W ⊗ E, we identify 𝔤_{𝔱} ≅ 𝔰𝔲(E) and so w_{2}(𝔱) ≡ w (mod 2) and p_{1}(𝔱) = −4κ. We shall denote

D_{A,ϑ} = D_{A} + ρ(ϑ) and D_{B,ϑ} = D_{B} + ρ(ϑ), (2.4)

where D_{A} : C^{∞}(V^{+}) → C^{∞}(V^{−}) and D_{B} : C^{∞}(W^{+}) → C^{∞}(W^{−}) are the Dirac operators defined by spin connections A on V and B on W, respectively [10, §2.2 & §2.3].

Lemma 2.2. The bundle det D_{𝔱} → C_{𝔱}/S^{1} is topologically trivial.

Proof. We recall that the K-theory isomorphism class of an index bundle over a compact topological space depends only on the homotopy class of its defining family of Fredholm operators (see, for example, [2, p. 69]). Moreover, the isomorphism class of the determinant line bundle (over a possibly non-compact topological space) depends only on the homotopy class of the family of Fredholm operators [46, Lemma 6.6.1]. In particular, the defining family of Fredholm operators D_{A,Φ}, parameterized by [A, Φ] ∈ C_{𝔱}/S^{1}, is homotopic through D_{A,tΦ}, t ∈ [0, 1], to the family of Fredholm operators D_{A,0} = (d∗ ; Â+d+ ; Â)⊕D_{A,ϑ} parameterized by [A, Φ] ∈ C_{𝔱}/S^{1}. Thus,

(2.5) det D_{A,Φ} ≅ det(d∗ ; Â + d+ ; Â) ⊗ det D_{A,ϑ}.

Let det D_{V} be the real determinant line bundle over C_{𝔱}/S^{1} associated to the family of perturbed Dirac operators, D_{A,ϑ}, where [A, Φ] ∈ C_{𝔱}. Let

π_{B} : C_{𝔱}/S^{1} → B_{κ}^{w}, [A, Φ] ↦ [Â] (2.6)

be the projection. Equation (2.5) implies there is an isomorphism det D_{𝔱} ≅ π_{B}^{∗} det δ_{E} ⊗ det D_{V} of real determinant line bundles, so

w_{1}(det D_{𝔱}) = π_{B}^{∗}w_{1}(det δ_{E}) + w_{1}(det D_{V}).

Because the Dirac operators D_{A,ϑ} have complex kernels and cokernels, the real line bundle det D_{V} → C_{𝔱}/S^{1} is topologically trivial and hence w_{1}(det D_{V}) = 0. By Proposition 2.1 we have w_{1}(det δ_{E}) = 0. Combining these observations yields w_{1}(det D_{𝔱}) = 0.

2.2. Orientations of moduli spaces of PU(2) monopoles and anti-self-dual connections. We introduce an orientation for the PU(2)-monopole moduli space M_{𝔱} determined by an orientation for the moduli space M_{κ}^{w} ↪ M_{𝔱} of anti-self-dual connections.

An orientation for the line bundle det D_{𝔱} determines an orientation for M_{𝔱}. The space C̃_{𝔱} is connected, so the quotients C_{𝔱} and C_{𝔱}/S^{1} are connected and a choice of orientation for det D_{𝔱} is equivalent to a choice of orientation for a fiber det D_{A,Φ} over a point [A, Φ]. The proof of Lemma 2.2 provides a method of orienting det D_{𝔱} from an orientation for det δ_{E}, and thus from a homology orientation and integral lift w of w_{2}(𝔱), using the isomorphism (2.5) of real determinant lines. Indeed, it suffices to choose an orientation for the line det(d∗ ; Â + d+ ; Â) and thus an orientation for det δ_{E} and choose the orientation of det D_{V} induced from the complex orientations of the complex kernel and cokernel of D_{A,ϑ}.

To fix our conventions and notation, we outline Donaldson’s method for orienting det(d∗ ; Â+ d+ ; Â), and thus det δ_{E}, given a homology orientation Ω and an integral lift w of w_{2}(𝔱): the detailed construction is described in [5, §3]. Suppose E ≅ \underline{ℂ} ⊕ L is a Hermitian, rank-two vector bundle over X, where \underline{ℂ} = X × ℂ and L is a complex line bundle with c_{1}(L) = w. Then 𝔰𝔲(E) ≅ i\underline{ℝ} ⊕ L has w_{2}(𝔰𝔲(E)) ≡ c_{1}(L) (mod 2), where \underline{ℝ} = X × ℝ. If d_{ℂ} ⊕ A_{L} is a reducible connection with respect to the splitting of E, where d_{ℂ} is the trivial connection on \underline{ℂ}, and Â = d_{ℝ}⊕A_{L} is the corresponding reducible connection on 𝔰𝔲(E), where d_{ℝ} the trivial connection on \underline{ℝ}. Then the induced rolled-up deformation complex for the anti-self-dual equation (2.3) splits as

d∗ ; Â + d+ ; Â = (d^{∗} + d^{+}) ⊕ (d_{A_{L}}^{∗} + d_{A_{L}}^{+}), (2.7)

where,

d^{∗} + d^{+} : C^{∞}(iΛ^{1}) → C^{∞}(iΛ^{0} ⊕ iΛ^{+}), (2.8)

d_{A_{L}}^{∗} + d_{A_{L}}^{+} : C^{∞}(Λ^{1} ⊗_{ℝ} L) → C^{∞}((Λ^{0} ⊕ Λ^{+}) ⊗_{ℝ} L). (2.9)

The real determinant line,

det(d^{∗} + d^{+}) ≅ Λ^{max}(H^{1}(X; ℝ)) ⊗ Λ^{max}(H^{0}(X; ℝ) ⊕ H^{+}(X; ℝ))^{∗},

is oriented by a choice of “homology orientation” Ω [5, §3], that is, an orientation for H^{1}(X; ℝ) ⊕ H^{2,+}(X; ℝ), while H^{0}(X; ℝ) is oriented by the choice of orientation for X [46, §6.6]. The operator d_{A_{L}}^{∗} + d_{A_{L}}^{+} is complex linear, and hence the complex orientations of its complex kernel and cokernel determine an orientation for the real line det(d_{A_{L}}^{∗} + d_{A_{L}}^{+}). Thus, an orientation for det(d∗ ; Â + d+ ; Â) is defined by the class w and homology orientation Ω.

An isomorphism between any two pairs of Hermitian, rank-two complex vector bundles E, E′ over X with first Chern class w can be constructed by splicing in |c_{2}(E) − c_{2}(E′)| copies of SU(2) bundles over S^{4} with second Chern class one. Given a U(2) connection on the bundle over X with smaller second Chern class, we obtain a U(2) connection on the other U(2) bundle by splicing in copies of the one-instanton on S^{4}. The excision principle [5, §3], [7, §7.1] implies that an orientation for one of the pair det δ_{E}, det δ_{E′} determines an orientation for the other.

For the moduli space M_{κ}^{w} of anti-self-dual SO(3) connections, we let o(Ω, w) denote the orientation determined by the class w ∈ H^{2}(X; ℤ) and corresponding split U(2) bundle, \underline{ℂ} ⊕ L, with c_{1}(L) = w, together with a homology orientation Ω.

Definition 2.3. Let w ∈ H^{2}(X; ℤ) be an integral lift of w_{2}(𝔱). The orientation O^{asd}(Ω, w) for the line bundle det D_{𝔱} over C_{𝔱}/S^{1}, and so for the moduli space M_{𝔱}, is defined by:

• The orientation of a fiber det D_{A,Φ} over a point [A, Φ] ∈ C_{𝔱}, via isomorphism (2.5),

• The complex orientation for det D_{A,ϑ}, and

• The orientation o(Ω, w) for det(d∗ ; Â + d+ ; Â).

For the moduli space of anti-self-dual connections on an SO(3) bundle, we shall need to compare orientations defined by different integral lifts of its second Stiefel-Whitney class:

Lemma 2.4. [7, p. 283] Let X be a closed, oriented, Riemannian four-manifold and let Ω be a homology orientation. If w, w′ ∈ H^{2}(X; ℤ) obey w ≡ w′ (mod 2), then

o(Ω, w′) = (−1)^{\frac{1}{4} (w−w′)^{2}}o(Ω, w).

2.3. Orientations of moduli spaces of PU(2) and Seiberg-Witten monopoles. We introduce an orientation for the PU(2)-monopole moduli space M_{𝔱} determined by an orientation for a Seiberg-Witten moduli space M_{𝔰} ↪ M_{𝔱}.

Let (A, Φ) = ι(B, Ψ) = (B⊕B⊗A_{L}, Ψ) be a reducible pair in C̃_{𝔱}, with respect to a splitting V = W ⊕W ⊗L, where 𝔰 = (ρ, W) and 𝔱 = (ρ, V) and ι : C̃_{𝔰} ↪ C̃_{𝔱} denotes the embedding (see Lemma 3.11 in [10]). Recall from [10, §3.4] that the deformation operator D_{ι(B,Ψ)} admits a splitting D_{ι(B,Ψ)} = D_{ι(B,Ψ)}^{t} ⊕ D_{ι(B,Ψ)}^{n} into tangential and normal components given by [10, Equations (3.21) & (3.22)]; the splitting is G_{𝔰}-equivariant with respect to the inclusion G_{𝔰} ↪ G_{𝔱} of automorphism groups in [10, Equation (3.10)]. Hence, we have an isomorphism of real determinant lines,

(2.10) det D_{ι(B,Ψ)} ≅ det D_{ι(B,Ψ)}^{t} ⊗ det D_{ι(B,Ψ)}^{n}.

Furthermore, by comparing equations (2.12), (2.17), and (2.19) with [10, Equations (3.26) & (3.32], we see that the rolled-up Seiberg-Witten elliptic deformation complex is identified with the rolled-up tangential deformation complex (3.34) in [10]. This identifies an orientation for the line det D_{B,Ψ} with an orientation for det D_{ι(B,Ψ)}^{t}. Combined with the isomorphism (2.10), this yields

(2.11) det D_{ι(B,Ψ)} ≅ det D_{B,Ψ} ⊗ det D_{ι(B,Ψ)}^{n}.

The pair ι(B, Ψ) ∈ C̃_{𝔱} is a fixed point of the S^{1} action on C̃_{𝔱} induced by the S^{1} action on V = W ⊕ W ⊗ L given by the trivial action on the factor W and the action by scalar multiplication on L (see [10, Equation (3.2)]). The operator

D_{ι(B,Ψ)}^{n} : C^{∞}(Λ^{1} ⊗ L) ⊕ C^{∞}(W^{+} ⊗ L) → C^{∞}(L) ⊕ C^{∞}(Λ^{+} ⊗ L) ⊕ C^{∞}(W^{−} ⊗ L)

is gauge equivariant and thus, because ι(B, Ψ) is a fixed point of this S^{1} action, is complex linear. Hence, D_{ι(B,Ψ)}^{n} is complex linear and and the complex orientations on its complex kernel and cokernel induce an orientation for det D_{ι(B,Ψ)}^{n}.

We recall that a homology orientation Ω defines an orientation for M_{𝔰} [46, §6.6]. As in [10, §2.3], we let C̃_{𝔰} denote the pre-configuration space of pairs (B, Ψ), where 𝔰 = (ρ, W), B is a spin connection on W, and Φ is a section of W^{+}; then C_{𝔰} = C̃_{𝔰}/G_{𝔰} is the configuration space, where G_{𝔰} ≅ Map(X, S^{1}) is the group of spin^{c} automorphisms of W. If (B, 0) is a point in C̃_{𝔰} then from [10, Equations (2.61) & (2.62)], the rolled-up Seiberg-Witten elliptic deformation complex is given by

D_{B,0} : C^{∞}(iΛ^{1}) ⊕ C^{∞}(W^{+}) → C^{∞}(iΛ^{0} ⊕ iΛ^{+}) ⊕ C^{∞}(W^{−}).

According to [10, Equations (2.59), (2.60), & (2.62)], we have

D_{B,0} = (d^{∗} + d^{+}) ⊕ D_{B,ϑ},

where d^{∗} + d^{+} is the operator in (2.8) and D_{B,ϑ} is the Dirac operator in (2.4). Thus,

(2.12) det D_{B,0} ≅ det(d^{∗} + d^{+}) ⊗ det D_{B,ϑ}.

The determinant line bundle det D_{𝔰} with fibers det D_{B,Ψ} is topologically trivial over C_{𝔰}, so M_{𝔰} is orientable and, as C_{𝔰} is connected, an orientation for the real line det D_{B,0} defines an orientation for det D_{𝔰}. A homology orientation Ω determines an orientation for det(d^{∗}+d^{+}). Since the Dirac operator D_{B,ϑ} is complex linear, the complex orientation for its complex kernel and cokernel defines an orientation for the real line det D_{B,ϑ}. The product of these orientations then defines an orientation for det D_{B,0} and hence for det D_{𝔰} and M_{𝔰}.

Definition 2.5. The orientation O^{red}(Ω, 𝔱, 𝔰) for the real line det D_{A,Φ}, and so for the line bundle det D_{𝔱} and the moduli space M_{𝔱}, is defined, through the isomorphism (2.11), by:

• The orientation for det D_{B,Ψ}, and thus det D_{𝔰}, given by the homology orientation Ω,

• The complex orientation for det D_{ι(B,Ψ)}^{n}.

2.4. Comparison of orientations of moduli spaces of PU(2) monopoles. We now compare the different possible orientations for M_{𝔱} which we have defined in the preceding sections.

Lemma 2.6. Let 𝔱 be a spin^{u} structure on an oriented four-manifold X and let Ω be a homology orientation. Suppose that w is an integral lift of w_{2}(𝔱) and that 𝔱 admits a splitting 𝔱 = 𝔰 ⊕ 𝔰 ⊗ L, for some complex line bundle L. Then,

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

Proof. By Definition 2.3, the difference between O^{asd}(Ω, w) and O^{asd}(Ω, c_{1}(L)) is equal to the difference between the orientations o(Ω, w) and o(Ω, c_{1}(L)) for the moduli spaces of antiself-dual connections on SO(3) bundles with second Stiefel-Whitney classes w (mod 2) and c_{1}(L) (mod 2), respectively. Since 𝔤_{𝔱} ≅ i\underline{ℝ} ⊕ L and w_{2}(𝔱) ≡ w (mod 2) by hypothesis, we have c_{1}(L) ≡ w (mod 2) and so Lemma 2.4 applies to compute the difference in orientations.

To see the second equality, write 𝔱 = (ρ, V) and 𝔰 = (ρ, W) and let

(A, Φ) = ι(B, 0) = (B ⊕ B ⊗ A_{L}, 0)

be a pair in C̃_{𝔱} which is reducible with respect to the splitting V = W ⊕ W ⊗ L and which has a vanishing spinor component, with A_{L} = A_{Λ} ⊗ (B^{det})^{∗}. Recall from [10, Lemma 2.9] that Â is then reducible with respect to the splitting 𝔤_{𝔱} = i\underline{ℝ} ⊕ L and can be written as Â = d_{ℝ} ⊕ A_{L}. The Dirac operator D_{A,ϑ} also splits,

D_{A,ϑ} = D_{B,ϑ} ⊕ D_{B⊗A_{L},ϑ}, (2.13)

where

D_{B,ϑ} : C^{∞}(W^{+}) → C^{∞}(W^{−}) and D_{B⊗A_{L},ϑ} : C^{∞}(W^{+} ⊗ L) → C^{∞}(W^{−} ⊗ L).

The isomorphism (2.5) of determinant lines giving the orientation O^{asd}(Ω, c_{1}(L)) to the line det D_{A,0} and the decompositions (2.7) of d∗ ; Â + d+ ; Â and (2.13) of D_{A,ϑ} at a reducible connection A yield the isomorphisms

det D_{A,0} ≅ det(d∗ ; Â + d+ ; Â) ⊗ det D_{A,ϑ}

≅ det(d^{∗} + d^{+}) ⊗ det(d_{A_{L}}^{∗} + d_{A_{L}}^{+}) ⊗ det D_{B,ϑ} ⊗ det D_{B⊗A_{L},ϑ}. (2.14)

The operators d_{A_{L}}^{∗} + d_{A_{L}}^{+}, D_{B,ϑ}, and D_{B⊗A_{L},ϑ} are complex linear and thus have complex kernels and cokernels. By Definition 2.3, the orientation O^{asd}(Ω, c_{1}(L)) is defined by choosing a homology orientation Ω for det(d^{∗}+d^{+}), and the complex orientation on the remaining factors on the right-hand-side of (2.14).

On the other hand, the isomorphisms (2.11) and (2.12) of determinant lines giving the orientation O^{red}(Ω, 𝔱, 𝔰) to the line det D_{A,0} yield the isomorphisms

det D_{A,0} ≅ det D_{B,0} ⊗ det D_{ι(B,0)}^{n}

≅ det(d^{∗} + d^{+}) ⊗ det D_{B,ϑ} ⊗ det(d_{A_{L}}^{∗} + d_{A_{L}}^{+}) ⊗ det D_{B⊗A_{L},ϑ}. (2.15)

By Definition 2.5, the orientation O^{red}(Ω, 𝔱, 𝔰) for det D_{A,0} is induced by the isomorphism (2.15), a choice of homology orientation Ω for det(d^{∗} + d^{+}), and the complex orientation on the remaining factors on the right-hand side of (2.15).

The isomorphisms (2.14) and (2.15) thus yield the same orientation of det D_{A,0}, and therefore O^{asd}(Ω, c_{1}(L)) = O^{red}(Ω, 𝔱, 𝔰).

2.5. Orientations of links of strata of reducible PU(2) monopoles. We shall need to compute the oriented intersections of codimension-one submanifolds of M_{𝔱}^{∗,0}/S^{1} with links L_{𝔱,𝔰} in M_{𝔱}^{∗,0}/S^{1} of the strata ι(M_{𝔰}). These computations (see §4) are performed most naturally with a “complex orientation” of the link L_{𝔱,𝔰} induced from the complex structure on the fibers of the “virtual normal bundle” of M_{𝔰} . We then compare this orientation with the “boundary orientation” of L_{𝔱,𝔰} induced from an orientation of M_{𝔱}^{∗,0}/S^{1} when the link is oriented as a boundary of an open subspace of M_{𝔱}^{∗,0}/S^{1}. Our orientation conventions for the link L_{𝔱,κ}^{w} of the stratum ι(M_{κ}^{w}) are explained in §3.4.3. We assume throughout this subsection that there are no zero-section pairs in M_{𝔰}.

Suppose that Y is a connected, finite-dimensional, orientable manifold with a free circle action. We give S^{1} ⊂ ℂ its usual orientation. If λ_{S^{1}} is a vector in T_{y}Y which is tangent to an S^{1} orbit through y ∈ Y, then an orientation λ_{Y} for det(T_{y}Y) and an orientation λ_{Y/S^{1}} for det(T_{y}(Y/S^{1})) determine one another through the convention

λ_{Y} = λ_{S^{1}} ∧ λ̃_{Y/S^{1}}, (2.16)

where λ̃_{Y/S^{1}} ∈ Λ^{dim Y −1}(T_{y}Y) satisfies π_{∗}(λ̃_{Y/S^{1}}) = λ_{Y/S^{1}} and π : Y → Y/S^{1} is the projection. In particular, orientations for M_{𝔱} and M_{𝔱}/S^{1} determine one another via convention (2.16).

Recall from [10, §3.5.3] that the “thickened moduli space” M_{𝔱}(Ξ, 𝔰) ⊂ C_{𝔱}^{∗,0} is a finitedimensional S^{1}-invariant manifold, defined by a choice of finite-rank, S^{1}-equivariant, trivial “stabilizing” or “obstruction” bundle Ξ over an open neighborhood of ι(M_{𝔰}) in C_{𝔱}^{0}. Then M_{𝔰} ↪ M_{𝔱}(Ξ, 𝔰) is a smooth submanifold with S^{1}-equivariant normal bundle N_{𝔱}(Ξ, 𝔰) → M_{𝔰} and tubular neighborhood defined by the image of the S^{1}-equivariant smooth embedding,

γ : N_{𝔱}(Ξ, 𝔰) ↪ M_{𝔱}(Ξ, 𝔰).

An open neighborhood of ι(M_{𝔰}) in the moduli space M_{𝔱} is recovered as the zero locus of an S^{1}-equivariant section φ of the S^{1}-equivariant vector bundle γ^{∗}Ξ → N_{𝔱}(Ξ, 𝔰):

γ (φ^{−1}(0) ∩ N_{𝔱}(Ξ, 𝔰)) ⊂ M_{𝔱}.

The section φ vanishes transversely on N_{𝔱}(Ξ, 𝔰) − M_{𝔰}. As in [10, Definition 3.22] we define the link of the stratum ι(M_{𝔰}) to be

L_{𝔱,𝔰} = γ (φ^{−1}(0) ∩ ℙN_{𝔱}(Ξ, 𝔰)) ⊂ M_{𝔱}/S^{1},

where ℙN_{𝔱}(Ξ, 𝔰) is the projectivization of the complex vector bundle N_{𝔱}(Ξ, 𝔰). Via the diffeomorphism,

L_{𝔱,𝔰} ≅ φ^{−1}(0) ∩ ℙN_{𝔱}(Ξ, 𝔰),

we can take the right-hand side as our model for the link, where φ is a section of the complex vector bundle γ^{∗}Ξ → ℙN_{𝔱}(Ξ, 𝔰).

We now define the complex orientation for the link L_{𝔱,𝔰}. The tangent space of ℙN_{𝔱}(Ξ, 𝔰) is oriented by an orientation on M_{𝔰} and the complex structure on the fibers. To be precise, at a point [B, Ψ, η] in the fiber of ℙN_{𝔱}(Ξ, 𝔰) over [B, Ψ] ∈ M_{𝔰}, the inclusion of the fiber gives an exact sequence of tangent spaces,

0 → T_{[η]}(ℙN_{𝔱}(Ξ, 𝔰)|_{[B,Ψ]}) → T_{[B,Ψ,η]}ℙN_{𝔱}(Ξ, 𝔰) → T_{[B,Ψ]}M_{𝔰} → 0,

and thus an isomorphism of determinant lines,

(2.17) Λ^{max}(T_{[η]}(ℙN_{𝔱}(Ξ, 𝔰)|_{[B,Ψ]})) ⊗ Λ^{max}(T_{[B,Ψ]}M_{𝔰}) ≅ Λ^{max}(T_{[B,Ψ,η]}ℙN_{𝔱}(Ξ, 𝔰)).

According to [10, Lemma 3.23], the section φ vanishes transversely at any point [B, Ψ, η] in an open neighborhood of the zero section M_{𝔰} of N_{𝔱}(Ξ, 𝔰), provided [B, Ψ, η] ∉ M_{𝔰}. Thus, at [B, Ψ, η] ∈ γ^{−1}(L_{𝔱,𝔰}) = φ^{−1}(0) ∩ ℙN_{𝔱}(Ξ, 𝔰), for η ≠ 0, the differential of φ and the diffeomorphism γ induce an exact sequence,

0 → T_{γ[B,Ψ,η]}L_{𝔱,𝔰} → T_{[B,Ψ,η]}ℙN_{𝔱}(Ξ, 𝔰) → (γ^{∗}Ξ)_{[B,Ψ,η]} → 0,

since T_{γ[B,Ψ,η]}L_{𝔱,𝔰} ≅ Ker(Dφ)_{[B,Ψ,η]} and Ran(Dφ)_{[B,Ψ,η]} = (γ^{∗}Ξ)_{[B,Ψ,η]}. This exact sequence and the isomorphism (2.17) induce an isomorphism

Λ^{max}(T_{γ[B,Ψ,η]}L_{𝔱,𝔰}) ≅ Λ^{max}(T_{[B,Ψ,η]}ℙN_{𝔱}(Ξ, 𝔰)) ⊗ (Λ^{max}(Ξ_{γ[B,Ψ,η]}))^{∗}
≅ Λ^{max}(T_{[B,Ψ]}M_{𝔰}) ⊗ Λ^{max}(T_{[η]}(ℙN_{𝔱}(Ξ, 𝔰)|_{[B,Ψ,η]}))
⊗ (Λ^{max}(Ξ_{γ[B,Ψ,η]}))^{∗}.

(2.18)

The fibers of the bundle Ξ → M_{𝔱}(Ξ, 𝔰) are preserved under the S^{1} action. The complex structure defined by this S^{1} action gives an orientation for Λ^{max}(Ξ_{γ[B,Ψ,η]}).

Definition 2.7. The complex orientation of the link L_{𝔱,𝔰} is defined through the isomorphism (2.18) with the orientations of the terms on the right-hand-side of (2.18) given by:

• The orientation of M_{𝔰} defined by a choice of homology orientation Ω,

• The complex orientation of the bundle γ^{∗}Ξ → ℙN_{𝔱}(Ξ, 𝔰),

• The complex orientation of the tangent space of a fiber of ℙN_{𝔱}(Ξ, 𝔰).

Although the complex orientation given by Definition 2.7 is the natural orientation to use when computing intersection numbers with L_{𝔱,𝔰}, we shall need to orient L_{𝔱,𝔰} as a boundary when using M_{𝔱}^{∗,0}/S^{1} as a cobordism. We describe this procedure next.

Suppose Z ⊂ Y is a compact submanifold of an oriented, Riemannian manifold Y, with normal bundle N → Z. If r⃗ is the outward-pointing radial vector at a point y on the boundary ∂N of the tubular neighborhood, also denoted N, then an orientation λ_{Y} for det(T_{y}Y) and an orientation λ_{∂N} for det(T_{y}(∂N)) determine one another through the convention

λ_{Y} = −r⃗ ∧ λ_{∂N}, (2.19)

choosing the sign in equation (2.19) so that the link ∂N is the boundary of Y − N.

For [A, Φ] ∈ L_{𝔱,𝔰}, choose an outward-pointing radial vector with respect to the thickened tubular neighborhood N_{𝔱}^{<ε}(Ξ, 𝔰)/S^{1},

(2.20)  r⃗ ∈ T_{[A,Φ]}(M_{𝔱}(Ξ, 𝔰)/S^{1}) ≅ T_{[A,Φ]}(N_{𝔱}(Ξ, 𝔰)/S^{1})

≅ ℝ · r⃗ ⊕ T_{[A,Φ]}(N_{𝔱}^{ε}(Ξ, 𝔰)/S^{1}). (2.21)

Because the section φ of γ^{∗}Ξ vanishes transversely on both N_{𝔱}(Ξ, 𝔰)/S^{1} and its ε-sphere bundle, for generic ε, we have isomorphisms

(2.22)  T_{[A,Φ]}(N_{𝔱}(Ξ, 𝔰)/S^{1}) ≅ T_{[A,Φ]}(M_{𝔱}/S^{1}) ⊕ Ξ_{[A,Φ]},
(2.23)  T_{[A,Φ]}(N_{𝔱}^{ε}(Ξ, 𝔰)/S^{1}) ≅ T_{[A,Φ]}L_{𝔱,𝔰} ⊕ Ξ_{[A,Φ]}.

Through the isomorphism (2.20), let π_{M/S^{1}}r⃗ be the orthogonal projection of r⃗ onto the subspace T_{[A,Φ]}(M_{𝔱}/S^{1}) in equation (2.22). If π_{M/S^{1}}r⃗ = 0, we would have r⃗ ∈ Ξ_{[A,Φ]} and thus tangent to N_{𝔱}^{ε}(Ξ, 𝔰)/S^{1} at [A, Φ] by equation (2.23), contradicting our choice of r⃗. Since π_{M/S^{1}}r⃗ ≠ 0, a comparison of the isomorphisms (2.21), (2.22), and (2.23) yields

T_{[A,Φ]}(M_{𝔱}/S^{1}) ≅ (π_{M/S^{1}}r⃗) · ℝ ⊕ T_{[A,Φ]}L_{𝔱,𝔰}. (2.24)

Hence, we make the

Definition 2.8. Given an orientation λ_{M/S^{1}} of M_{𝔱}/S^{1} and an outward-pointing radial vector r⃗ with respect to the tubular neighborhood N_{𝔱}^{<ε}(Ξ, 𝔰)/S^{1}, we define the boundary orientation λ_{∂M/S^{1}} of L_{𝔱,𝔰} by

λ_{M/S^{1}} = −π_{M/S^{1}}r⃗ ∧ λ_{∂M/S^{1}}. (2.25)

Lemma 2.9. The complex orientation (Definition 2.7) of the link L_{𝔱,𝔰} agrees with the boundary orientation (Definition 2.8) of L_{𝔱,𝔰} determined by the orientation O^{red}(Ω, 𝔱, 𝔰) for M_{𝔱}/S^{1}.

Proof. The orientation O^{red}(Ω, 𝔱, 𝔰) of M_{𝔱} is defined through the isomorphism (2.10), using the orientation for det D_{ι(B,Ψ)}^{t} (and thus the tangent space for M_{𝔰}) given by the homology orientation Ω, and the complex orientation for det D_{ι(B,Ψ)}^{n}. From [10, Equation (3.55)] we have an isomorphism det(D^{n}) ≅ det([N_{𝔱}(Ξ, 𝔰)] − [Ξ]) and thus, for any [B, Ψ] ∈ M_{𝔰} an isomorphism,

(2.26) det D_{ι(B,Ψ)}^{n} ≅ Λ^{max}(N_{𝔱}(Ξ, 𝔰)|_{[B,Ψ]}) ⊗ (Λ^{max}(Ξ_{[B,Ψ]}))^{∗},

which preserves the orientations defined by the complex structures of both sides. The orientation O^{red}(Ω, 𝔱, 𝔰) of M_{𝔱} determines one for M_{𝔱}/S^{1} through convention (2.16) and a boundary orientation for L_{𝔱,𝔰} through convention (2.25).

On the other hand, the complex orientation for L_{𝔱,𝔰} uses, through equation (2.18), the complex orientation for the complex projective space given by the fiber of ℙN_{𝔱}(Ξ, 𝔰). Comparing equation (2.26) with equation (2.18) shows that the difference between the two orientations lies in how the fibers of the projections ℙN_{𝔱}(Ξ, 𝔰) → M_{𝔰} and N_{𝔱}(Ξ, 𝔰) → M_{𝔰} are oriented. The boundary orientation λ_{∂M/S^{1}} for L_{𝔱,𝔰} induced by O^{red}(Ω, 𝔱, 𝔰) on M_{𝔱} begins with the complex orientation for the fiber of the projection N_{𝔰}(Ξ, 𝔰) → M_{𝔰}, uses convention (2.16) to define an orientation for the fiber of N_{𝔰}(Ξ, 𝔰)/S^{1} → M_{𝔰}, and then uses convention (2.19) to define an orientation for the boundary N_{𝔱}^{ε}(Ξ, 𝔰)/S^{1} = ℙN_{𝔱}(Ξ, 𝔰) of the bundle N_{𝔱}^{≤ε}(Ξ, 𝔰)/S^{1}. Hence, it is enough to compare these two methods of orienting the fibers of ℙN_{𝔱}(Ξ, 𝔰).

We denote the fibers of N_{𝔱}(Ξ, 𝔰), N_{𝔱}^{ε}(Ξ, 𝔰), and ℙN_{𝔱}(Ξ, 𝔰) by ℂ^{k}, (ℂ^{k})^{ε}, and ℙ^{k−1} = (ℂ^{k})^{ε}/S^{1}, respectively. If ℂ^{k} has a complex basis {r⃗, v_{1}, …, v_{k−1}}, then the complex orientation of ℂ^{k} is defined by

\begin{aligned}λ_{ℂ^{k}} & = r⃗ ∧ ir⃗ ∧ v_{1} ∧ iv_{1} ∧ ⋯ ∧ v_{k−1} ∧ iv_{k−1} \\ & = −ir⃗ ∧ r⃗ ∧ v_{1} ∧ iv_{1} ∧ ⋯ ∧ v_{k−1} ∧ iv_{k−1}. \end{aligned} (2.27)

If λ_{ℙ^{k−1}} is the complex orientation for ℙ^{k−1} and π : ℂ^{k} \ {0} → ℙ^{k−1} is the projection, then

π_{∗}(v_{1} ∧ iv_{1} ∧ ⋯ ∧ v_{k−1} ∧ iv_{k−1}) = λ_{ℙ^{k−1}},

because {v_{1}, …, v_{k−1}} is a complex basis for the tangent space to ℙ^{k−1} at π(r⃗), so equation (2.27) yields the following relation between the complex orientations of ℂ^{k} and ℙ^{k−1}:

λ_{ℂ^{k}} = −ir⃗ ∧ r⃗ ∧ λ_{ℙ^{k−1}}. (2.28)

On the other hand, if ℂ^{k}/S^{1} is oriented through convention (2.19) by −r⃗∧λ_{ℙ^{k−1}}, the boundary orientation of the link ℙ^{k−1} is equal to its complex orientation, λ_{ℙ^{k−1}}. By convention (2.16), the orientation −r⃗∧λ_{ℙ^{k−1}} for ℂ^{k}/S^{1} is induced by the orientation −ir⃗∧r⃗∧π^{∗}λ_{ℙ^{k−1}} for ℂ^{k}, which is equal to the complex orientation λ_{ℂ^{k}} by equation (2.28). Hence, the complex and boundary orientations of ℙ^{k−1} agree.

Therefore, the complex orientation agrees with the boundary orientation for L_{𝔱,𝔰}, induced by the orientation O^{red}(Ω, 𝔱, 𝔰) through the conventions (2.25) and (2.16).

3. Cohomology classes on moduli spaces

In this section we introduce cohomology classes on the moduli space M_{𝔱}^{∗,0} (see §3.1) and construct geometric representatives for these cohomology classes (see §3.2). The PU(2) monopole program uses the moduli space M_{𝔱}^{∗,0}/S^{1} as a cobordism between the link L_{𝔱,κ}^{w} of the anti-self-dual moduli space stratum, ι(M_{κ}^{w}) ⊂ M_{𝔱}, and the links L_{𝔱,𝔰} of the Seiberg-Witten strata, ι(M_{𝔰}) ⊂ M_{𝔱}, giving an equality between the pairings of the cohomology classes with these links. The following geometric description should help motivate the constructions of this section.

The intersection of the geometric representatives with M_{𝔱}^{∗,0} is a family of oriented onemanifolds. The links L_{𝔱,κ}^{w} and L_{𝔱,𝔰} of the strata of zero-section and reducible monopoles described in [10, Definitions 3.7 & 3.22] are oriented hypersurfaces in M_{𝔱}^{∗,0}/S^{1}. The intersection of these hypersurfaces with the one-dimensional manifolds given by the intersection of the geometric representatives is thus an oriented collection of points. We would like to use the family of oriented one-manifolds to show that the total signed count of the points in the intersection of the geometric representatives with the links is zero (being an oriented boundary). This would give an equality between the oriented count of points in the link of the stratum of zero-section monopoles with the oriented count of points in the links of the strata of the reducible monopoles. In §3.4 we show that the intersection of the geometric representatives with the link L_{𝔱,κ}^{w} is a multiple of the Donaldson invariant. In §4 we show that the intersection of the geometric representatives with the links L_{𝔱,𝔰} can be expressed in terms of Seiberg-Witten invariants. Hence, the cobordism gives a relation between these two invariants.

In practice, the above argument does not work in the simple manner just described because M_{𝔱}^{∗,0}/S^{1} is non-compact: the non-compactness phenomenon responsible for the difficulty is due to Uhlenbeck bubbling. Geometrically, this means there can be one-manifolds in the intersection of the geometric representatives with one boundary on a link and the other end approaching a reducible in a lower level of M̄_{𝔱}/S^{1}. Let

M̄_{𝔱}^{∗,0} ⊂ M̄_{𝔱} (3.1)

be the subspace consisting of points [A, Φ, x] where (A, Φ) is neither a zero-section nor a reducible pair. In §3.3 we describe the intersection of the closure of the geometric representatives with the lower Uhlenbeck levels of M̄_{𝔱}^{∗,0}/S^{1} and show that for appropriate choices of geometric representatives these intersections are empty. Therefore, the ends of the one-manifolds in M_{𝔱}^{∗,0}/S^{1} do not approach the lower levels of M̄_{𝔱}^{∗,0}/S^{1}.

However, there may still be one-manifolds in the intersection of the geometric representatives with ends approaching reducible monopoles in lower Uhlenbeck levels of M̄_{𝔱}/S^{1}. Theorem 3.33 gives a relationship between the Donaldson and Seiberg-Witten invariants when there are no reducible monopoles in the lower levels of M̄_{𝔱} and thus the ends of the one-manifolds do not approach the lower levels of M̄_{𝔱}/S^{1}. To extend this argument to the case where there are reducible monopoles in the lower levels of the compactification requires a description of neighborhoods of the lower strata precise enough to allow the definition of links of these strata of lower-level reducibles. As we show in §4, the geometric representatives intersect the strata of reducible monopoles in sets of larger than expected codimension. Thus, in the case of the reducibles, we cannot cut down by geometric representatives as we do with the zero-section monopoles and restrict our attention to a generic point. Rather, we are forced to describe the entire link. When the reducible monopoles lie in a lower level, these links can be extremely complicated. A description of neighborhoods the strata of lower-level reducibles, sufficient to define links, will be given in [12], [13].

We work with geometric representatives rather than cohomology classes for two reasons. First, describing the closure of a geometric representative in a compactification appears to be simpler than calculating the extension of a cohomology class. Second, the topology near points in the lower levels of M̄_{𝔱} need not be locally finite (for example, there may be infinitely many path-connected components). Hence, it is not known if L_{𝔱,κ}^{w} is triangulable and thus it may not have a fundamental class to pair with the cohomology classes described in §3.1. This problem also leads us to work in the category of smoothly stratified spaces rather than that of piecewise-linear spaces.

3.1. The cohomology classes. In this subsection, we define the cohomology classes on the moduli spaces, following the prescriptions of [4], [6], [7], [35]. These classes arise from a universal SO(3) bundle, just as in Donaldson theory, and a universal line bundle.

Recall that C̃_{𝔱}^{∗} denotes the subspace of pairs which are not reducible, C̃_{𝔱}^{0} denotes the subspace of pairs which are not zero-section pairs, and C̃_{𝔱}^{∗,0} = C̃_{𝔱}^{∗} ∩ C̃_{𝔱}^{0} (see [10, §2.1.5]). We define a universal SO(3) bundle:

𝔽_{𝔱} = C̃_{𝔱}^{∗}/S^{1} ×_{G_{𝔱}} 𝔤_{𝔱} → C_{𝔱}^{∗}/S^{1} × X. (3.2)

The action of G_{𝔱} in (3.2) is diagonal so, for u ∈ G_{𝔱} and (A, Φ, ξ) ∈ C̃_{𝔱}^{∗,0} × 𝔤_{𝔱}, one has

((A, Φ), ξ) ↦ (u(A, Φ), uξ).

We now define

μ_{p} : H_{•}(X; ℝ) → H^{4−•}(C_{𝔱}^{∗}/S^{1}; ℝ), β ↦ −\frac{1}{4}p_{1}(𝔽_{𝔱})/β. (3.3)

Following [7, Definition 5.1.11] we can also define a universal SO(3) bundle over the quotient space of SO(3) connections,

𝔽_{κ}^{w} = A_{κ}^{w,∗} ×_{G_{κ}^{w}} F → B_{κ}^{w,∗} × X, (3.4)

where F is an SO(3) bundle over X with κ = −\frac{1}{4}p_{1}(F) and w is an integral lift of w_{2}(F), and G_{κ}^{w} is the group of special unitary automorphisms of the U(2) bundle E with 𝔰𝔲(E) = F, so p_{1}(F) = p_{1}(𝔰𝔲(E)), and c_{1}(E) = w. As in [7, Definition 5.1.11], we define cohomology classes on B_{κ}^{w,∗} via

ν_{p} : H_{•}(X; ℝ) → H^{4−•}(B_{κ}^{w,∗}; ℝ), β ↦ −\frac{1}{4}p_{1}(𝔽_{κ}^{w})/β. (3.5)

Comparing (3.3) and (3.5), we see that there must be a simple relation, which we now describe, between the cohomology classes defined by these two SO(3) bundles.

Recall from [10, §2.1.3] that if F = 𝔰𝔲(E) and V = W ⊗E, then we have an identification of automorphism groups, G_{κ}^{w} ≅ G_{𝔱}, and isomorphisms

A_{κ}^{w}(X) ≅ A_{𝔱}, Â ↦ A, and B_{κ}^{w}(X) ≅ B_{𝔱}, [Â] ↦ [A]. (3.6)

Hence, denoting 𝔤_{𝔱} = 𝔰𝔲(E), we have an isomorphism of SO(3) bundles

𝔽_{κ}^{w} ≅ A_{𝔱}^{∗} ×_{G_{𝔱}} 𝔤_{𝔱} → B_{𝔱}^{∗} × X. (3.7)

Furthermore, there are natural embeddings

ι : A_{𝔱} ↪ C̃_{𝔱}, A ↦ (A, 0), and ι : B_{𝔱} ↪ C_{𝔱}, [A] ↦ [A, 0]. (3.8)

Using (3.8) together with the isomorphism (3.7) and the definition (3.2) of 𝔽_{𝔱}, we see that

(π_{B} × id_{X})^{∗}𝔽_{κ}^{w} = 𝔽_{𝔱} and (ι × id_{X})^{∗}𝔽_{𝔱} = 𝔽_{κ}^{w}, (3.9)

where π_{B} : C_{𝔱}^{∗}/S^{1} → B_{κ}^{w,∗} is the restriction of the map (2.6) to C_{𝔱}^{∗}/S^{1}. Since π_{B} is a deformation retract, we obtain the following relation between the cohomology classes on C_{𝔱}^{∗}/S^{1} and B_{κ}^{w,∗}.

Lemma 3.1. If β ∈ H_{•}(X; ℝ), then π_{B}^{∗}ν_{p}(β) = μ_{p}(β) or, equivalently, ι^{∗}μ_{p}(β) = ν_{p}(β).

Because (ι × id_{X})^{∗}μ_{p} = ν_{p}, we shall henceforth write μ_{p} for both μ_{p} and ν_{p}.

Lastly, we define a universal complex line bundle,

𝕃_{𝔱} = C_{𝔱}^{∗,0} ×_{(S^{1},×−2)} ℂ → C_{𝔱}^{∗,0}/S^{1}, (3.10)

where the S^{1} action defining 𝕃_{𝔱} is given, for [A, Φ] ∈ C_{𝔱}^{∗,0}, e^{iθ} ∈ S^{1}, and z ∈ ℂ, by

([A, Φ], z) ↦ ([A, e^{iθ}Φ], e^{2iθ}z) . (3.11)

The factor of 2 is necessary in the action (3.11) because −1 ∈ S^{1} acts on C̃_{𝔱} as −id_{V} ∈ G_{t} and thus −1 ∈ S^{1} acts trivially on C_{𝔱}^{∗,0}. The negative sign in the quotient (3.10) indicates that the S^{1} action is diagonal, and is chosen to give a more convenient sign in Lemma 3.28. We then define an additional cohomology class in H^{2}(C_{𝔱}^{∗,0}/S^{1}; ℝ),

μ_{c} = c_{1}(𝕃_{𝔱}). (3.12)

In Lemma 3.28 we will show that the class μ_{c} is non-trivial on the link of the subspace ι(M_{κ}^{w}) ⊂ M_{𝔱}/S^{1}. Thus, μ_{c} does not extend over ι(B_{𝔱}^{∗}) ⊂ C_{𝔱}^{∗}/S^{1} as the restriction of an extension to a contractible neighborhood of point [A, 0] ∈ ι(B_{𝔱}^{∗}) would have to be trivial, contradicting Lemma 3.28.

3.2. Geometric representatives. To avoid having to define the link of M_{κ}^{w} in M_{𝔱} as a homology class, we work with geometric representatives of these cohomology classes. We define geometric representatives V(β) and W to represent μ_{p}(β) and μ_{c} respectively. To facilitate the description of the intersection of the closures of V(β) and W in M̄_{𝔱} with the lower strata, we construct geometric representatives with certain localization properties— they are pulled back from configuration spaces over proper open subsets U ⊂ X. We let B_{κ}^{w}(U) and C_{𝔱}(U) denote the quotient spaces of connections and pairs respectively on U ⊆ X, where κ = −\frac{1}{4}p_{1}(𝔱) and w is an integral lift of w_{2}(𝔱).

3.2.1. Stratified spaces. We begin by recalling a definition of a stratified space (see [48, Definition 11.0.1]) that will be sufficient for the purposes of defining our intersection pairings.

Definition 3.2. [26], [42], [48] A smoothly stratified space Z is a topological space with a smooth stratification given by a disjoint union, Z = Z_{0} ∪ Z_{1} ∪ ⋯ ∪ Z_{n}, where the strata Z_{i} are smooth manifolds. There is a partial ordering among the strata, given by Z_{i} < Z_{j} if Z_{i} ⊂ Z̄_{j}. There is a unique stratum of highest dimension, Z_{0}, such that Z̄_{0} = Z, called the top stratum. If Y, Z are smoothly stratified spaces, a map f : Y → Z is smoothly stratified if f is a continuous map, there are smooth stratifications of Z and Y such that f preserves strata, and restricted to each stratum f is a smooth map. A subspace Y ⊂ Z is smoothly stratified if the inclusion is a smoothly stratified map.

Remark 3.3. If Z is a smoothly stratified space and f : Z → ℝ is a smoothly stratified map, that is, f is a continuous map which is smooth on each stratum, then for generic values of ε, the preimage f^{−1}(ε) is a smoothly stratified subspace of Z.

We shall use the following definition of a geometric representative:

Definition 3.4. [35, p 588]. Let Z be a smoothly stratified space. A geometric representative for a rational cohomology class μ of dimension c on Z is a closed, smoothly stratified subspace V of Z together with a rational coefficient q, the multiplicity, satisfying

1. The intersection Z_{0} ∩ V of V with the top stratum Z_{0} of Z has codimension c 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 c is obtained by choosing a smooth singular cycle representing h whose intersection with all strata of V has the codimension dim Z_{0} − c 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.

Definition 3.5. Let V_{1}, …, V_{n} be geometric representatives on a compact, smoothly stratified space Z with multiplicities q_{1}, …, q_{n}. Assume

1. The sum of the codimensions of the V_{i} is equal to the dimension of the top stratum Z_{0} of Z.

2. For every smooth stratum Z_{s} of Z, the smooth submanifolds V_{i} ∩ Z_{s} intersect trans-

versely. Then dimension-counting and the definition of a geometric representative imply that the intersection ∩_{i}V_{i} is a finite collection of points in the top stratum Z_{0}:

V_{1} ∩ ⋯ ∩ V_{n} = {v_{1}, …, v_{N}} ⊂ Z_{0}.

Let ε_{j} = ±1 be the sign of this intersection at v_{j}. Then we define the intersection number of the V_{i} in Z by setting

# (V_{1} ∩ ⋯ ∩ V_{n} ∩ Z) = (∏_{i=1}^{n} q_{i}) ∑_{j=1}^{N} ε_{j}.

A cobordism between two geometric representatives V and V′ in Z with the same multiplicity is a geometric representative W ⊂ Z × [0, 1] which is transverse to the boundary and with W ∩ Z × {0} = V and W ∩ Z × {1} = V′, with the obvious orientations of normal bundles.

The definition of intersection number does not change if V_{i} is replaced by V′_{i} and there is a cobordism between V_{i} and V′_{i} whose intersection with the other geometric representatives is transverse in each stratum. One can see this by observing that the intersection of the cobordism W with the other geometric representatives will be a collection of one-manifolds contained in Z_{0} because the lower strata of Z have codimension two. The boundaries of these one manifolds are the points in the two intersections

V_{1} ∩ ⋯ ∩ V_{n} and V_{1} ∩ … V_{i−1} ∩ V′_{i} ∩ ⋯ ∩ V_{n},

giving the equality of oriented intersection numbers.

Remark 3.6. In Definition 3.5, it is necessary to assume that the geometric representatives have transverse intersection in each stratum because we cannot assume there are perturbations of the geometric representatives which do intersect transversely. The definition of a smoothly-stratified space in Definition 3.2 does not control the topology of one strata near another. If there is “control data” on a neighborhood of one strata in another (see [26, p. 42], as is true for Whitney-stratified spaces, then such perturbations are constructed in [26, §1.3]. Instead, we will construct our geometric representatives pulling them back from a smooth manifold where one can assume that generic choices of the geometric representatives intersect transversely.

3.2.2. Preliminaries for localization. By construction, our geometric representatives will be “determined by restriction to submanifolds” of X, in the sense that they have the following localization property:

Definition 3.7. Let U ⊂ X be a submanifold. A geometric representative V in B_{κ}^{w,∗} or C_{𝔱}^{∗,0}/S^{1} is determined by restriction to U ⊂ X if there is a geometric representative V_{U} in B_{κ}^{w,∗}(U) or C_{𝔱}^{∗,0}(U)/S^{1} such that V = r_{U}^{−1}(V_{U}), where r_{U} is the map given by restricting connections or pairs to the submanifold U.

This localization property will allow a partial description of the intersection of the closures of the geometric representatives in the subspace M̄_{𝔱}^{∗,0}/S^{1} with the lower strata in this compactification. The technical issue which has complicated this localization technique since its introduction in [6], [7] (see [7, p. 192]) is that there can be pairs (connections) which are irreducible on X but are reducible when restricted to a submanifold Y ⊂ X. The bundles over B_{κ}^{w,∗}(Y), whose sections define the geometric representative, do not extend over B_{κ}^{w}(Y). Therefore, the pullback of these sections do not have good properties (transversality, for example) over the subspace of connections in B_{κ}^{w,∗}(X) which are reducible when restricted to Y. When working with the moduli space M_{κ}^{w} of anti-self-dual SO(3) connections, this problem can be overcome, if X is simply connected, by working with a tubular neighborhood ν(Y) of Y ⊂ X. The local-to-global reducibility result of [7, Lemma 4.3.21] implies that any anti-self-dual connection which is irreducible on X must be irreducible on ν(Y) if X is simply connected. If X is not simply connected, there can be “twisted reducible” connections, (see [10, Lemma 3.5] or [35, Lemma 2.4]) which are irreducible on X but reducible when restricted to a tubular neighborhood. The notion of a “suitable open neighborhood” of Y (see Definition 3.8) was introduced in [35] to deal with the problem of twisted reducibles. Any anti-self-dual connection which is irreducible on X must be irreducible on a suitable open neighborhood. We can then define geometric representatives by pulling back sections of bundles over B_{κ}^{w,∗}(U(Y)), where U(Y) is a suitable open neighborhood of Y. When points [Â_{α}] in this geometric representative approach a point [Â_{0}, x] in a lower Uhlenbeck level where the support of x is disjoint from U(Y), the point [Â_{0}] will also be in this geometric representative. Thus, either [Â_{0}] is in the geometric representative or the support of x meets U(Y). Both of these conditions have high enough codimension in the lower Uhlenbeck levels of the compactification to ensure, via dimension-counting arguments (see [35, pp. 592–593]), that the intersection of the geometric representatives is compactly supported in the top level M_{κ}^{w} of the Uhlenbeck compactification M̄_{κ}^{w}.

We begin by recalling the following definition of Kronheimer and Mrowka:

Definition 3.8. [35, p. 589] A smooth submanifold-with-boundary or open set U ⫋ X is suitable if the induced map H_{1}(U; ℤ/2ℤ) → H_{1}(X; ℤ/2ℤ) is surjective.

Let Y ⊂ X be a submanifold with tubular neighborhood ν(Y). If D is a set of embedded loops generating H_{1}(X; ℤ/2ℤ), which are mutually disjoint and transversal to Y, then a tubular neighborhood ν(Y ∪ D) of Y ∪ D ⊂ X is a suitable neighborhood Y. By tubular neighborhood of the possibly singular space Y ∪ D, we mean a smoothing of the union of the tubular neighborhoods of Y and of D.

Remark 3.9. If [Â] ∈ M_{κ}^{w,∗} then the restriction of Â to any suitable open neighborhood is irreducible by unique continuation [7, Lemma 4.3.21]. Hence, the only irreducible anti-self-dual connections which could be reducible when restricted to an open set are the twisted reducibles (see [10, §3.2] or [35, p. 586]). However, the homology condition in Definition 3.8 excludes this possibility.

The corresponding local-to-global reducibility result for PU(2) monopoles which are not zero-section pairs [17, Theorem 5.11] is stronger than that for anti-self-dual connections:

Theorem 3.10. [17, Theorem 5.10] Suppose (A, Φ) is a solution to the perturbed PU(2) monopole equations [10, Equation (2.32)] over a connected, oriented, smooth four-manifold X with smooth Riemannian metric such that (A, Φ) is reducible on a non-empty open subset U ⊂ X. Then (A, Φ) is reducible on X if

• Φ ≢ 0 on X, or

• Φ ≡ 0, and M_{κ}^{w} contains no twisted reducibles or U is suitable.

Both suitable and tubular neighborhoods of submanifolds are open subsets of X and thus have codimension zero. However, a tubular neighborhood admits a retraction onto the submanifold while a suitable neighborhood admits a retraction onto the union of the submanifold and a collection of loops. Hence, these neighborhoods may, for the purposes of counting intersection points, be thought of as having codimension equal to that of the submanifold or to that of the union of the submanifold and some loops in X. In this sense, the suitable neighborhood of a point has smaller codimension than a tubular neighborhood of a point. Because the lower strata of M̄_{𝔱} do not have codimension as large as those of M̄_{κ}^{w} and because the suitable neighborhoods do not have codimension as large as tubular neighborhoods, we will require an additional technical condition on the elements of H_{•}(X; ℝ) to ensure the intersection of the geometric representatives does not intersect the lower levels away from the reducible pairs.

Let Y ⊂ X be a smooth submanifold. If Y is a manifold with boundary, the manifold structure of the configuration space B_{κ}^{w}(Y) is described in, say, [6, p. 262], [7, p. 192], [56, §2(a)]; the corresponding slice result for C_{𝔱}(Y)/S^{1} can be obtained from the slice result for manifolds without boundary [17, Proposition 2.12] by taking into account the Neumann boundary conditions as in [56]. Let r_{Y} : C_{𝔱}/S^{1} → C_{𝔱}(Y)/S^{1} and r_{Y} : B_{κ}^{w} → B_{κ}^{w}(Y) denote the restriction maps defined by [A, Φ] ↦ [A|_{Y} , Φ|_{Y}] and [Â] ↦ [Â|_{Y}], respectively. We will use the same notation for the restriction map on any domain.

We define C_{𝔱}^{∗}(X, U) to be the quotient space of pairs on X which are irreducible when restricted to U, let C_{𝔱}^{0}(X, U) denote the quotient space of pairs on X which are not zerosection pairs when restricted to U, and let C_{𝔱}^{∗,0}(X, U) = C_{𝔱}^{∗}(X, U) ∩ C_{𝔱}^{0}(X, U). The space B_{κ}^{w,∗}(X, U) is similarly defined.

If [A, Φ] ∈ M_{𝔱}^{∗}, then Theorem 3.10 implies that the restriction of the SO(3) connection Â to the suitable neighborhood ν(Y ∪ D) cannot be reducible; if [A, Φ] ∈ M_{𝔱}^{∗,0}, so we further assume Φ ≢ 0, then Theorem 3.10 implies that the restriction of the connection Â to ν(Y) cannot be reducible. There is a disjoint decomposition

M_{𝔱}^{∗} = M_{𝔱}^{∗,0} ∪ ι(M_{κ}^{w,∗}).

The unique continuation result for reducible anti-self-dual SO(3) connections [7, Lemma 4.3.21] and PU(2) monopoles (Theorem 3.10), and the preceding decomposition and remarks yield Lemma 3.11. Let U ⊂ X be an open subset and let Y ⊂ X be a submanifold. Then the following inclusions hold:

π_{B}(M_{𝔱}^{∗,0}) ⊂ B_{κ}^{w,∗}(X, U) and M_{κ}^{w,∗} ⊂ B_{κ}^{w,∗}(X, ν(Y ∪ D)),

where, as in equation (2.6), π_{B} : C_{𝔱} → B_{κ}^{w} is the projection [A, Φ] ↦ [Â].

We can now proceed to construct geometric representatives for the classes μ_{p}(β) ∈ H^{•}(M_{𝔱}^{∗}/S^{1}; ℝ) and μ_{c} ∈ H^{•}(M_{𝔱}^{∗,0}/S^{1}; ℝ).

3.2.3. The geometric representatives for μ_{p}. Let Y ⊂ X be a smooth submanifold and let β = [Y] ∈ H_{•}(X; ℝ). Let ν(Y ∪ D) be a suitable open neighborhood of Y. In [35, p. 588–595], geometric representatives for the classes μ_{p}(β) ∈ H^{•}(M_{κ}^{w,∗}; ℝ),

r_{ν(Y ∪D)}^{−1}(V(β)) ⊂ B_{κ}^{w,∗}(X, ν(Y ∪ D)),

are defined which have the property that they are determined by

V(β) ⊂ B_{κ}^{w,∗}(ν(Y ∪ D)).

Let (r_{ν(Y ∪D)}π_{B})^{−1}(V(β)) be the preimage of this geometric representative with respect to the map

π_{B} : C_{𝔱}^{∗}(X, ν(Y ∪ D))/S^{1} → B_{κ}^{w,∗}(X, ν(Y ∪ D)).

The following result is a clear consequence of the definitions and Lemma 3.11.

Lemma 3.12. If Y ⊂ X is a smooth submanifold, representing a class β ∈ H_{•}(X; ℝ), then (r_{ν(Y ∪D)}π_{B})^{−1}(V(β)) ⊂ M_{𝔱}^{∗}/S^{1} is a geometric representative for μ_{p}(β) ∈ H^{4−•}(M_{𝔱}^{∗}/S^{1}; ℝ) and is determined by restriction to ν(Y ∪ D) ⊂ X.

Henceforth, we shall abuse notation slightly and write V(β) for V(β) ⊂ B_{κ}^{w,∗}(ν(Y ∪ D)), for its preimage r_{ν(Y ∪D)}^{−1}(V(β)) ⊂ B_{κ}^{w,∗}(X, ν(Y ∪ D)), and for (r_{ν(Y ∪D)}π_{B})^{−1}(V(β)) ⊂ C_{𝔱}^{∗}(X, ν(Y ∪ D))/S^{1}.

3.2.4. A representative for the determinant class. Recall that C_{𝔱}^{∗,0} is an S^{1} bundle over C_{𝔱}^{∗,0}/S^{1}. Let ν(x) be a tubular neighborhood of x and let s be a generic, smooth, C^{0} bounded section of the line bundle

𝕃_{𝔱}(ν(x)) = C_{𝔱}^{∗,0}(ν(x)) ×_{(S^{1},×−2)} ℂ → C_{𝔱}^{∗,0}(ν(x)). (3.13)

The action of S^{1} in the above is the same as that in (3.11), for the definition (3.10) of the universal line bundle 𝕃_{𝔱} → C_{𝔱}^{∗,0}. The pullback r_{ν(x)}^{∗}𝕃_{𝔱}(ν(x)) is thus isomorphic to the

restriction of 𝕃_{𝔱} to C_{𝔱}^{∗,0}(X, ν(x)). We define a geometric representative by

W = (r_{ν(x)}^{∗}s)^{−1}(0) ⊂ C_{𝔱}^{∗,0}(X, ν(x)). (3.14)

Since M_{𝔱}^{∗,0} ⊂ C_{𝔱}^{∗,0}(X, ν(x)) by Theorem 3.10 and the unique continuation theorem for the Dirac operator, [17, Lemma 5.12], the proof of the next lemma is then clear.

Lemma 3.13. For a generic choice of section s, the zero locus W of the section r_{ν(x)}^{∗}s is a

geometric representative for μ_{c} ∈ H^{2}(M_{𝔱}^{∗,0}/S^{1}; ℝ) and is determined by restriction to ν(x).

Let 1 ∈ 𝔸(X) be the element of degree zero. If z = β_{1} ⋯ β_{r} ∈ 𝔸(X), we write

(3.15)  δ_{i} = ∑_{{p|β_{p}∈H_{i}(X;ℝ)}}1 and deg(z) = ∑_{i=0}^{4}(4 − i) deg_{i}(z).

For monomials z = β_{1} ⋯ β_{r}, we set

\begin{aligned}μ_{p}(z) & = μ_{p}(β_{1}) ⌣ ⋯ ⌣ μ_{p}(β_{r}), \\ V(z) & = V(β_{1}) ∩ ⋯ ∩ V(β_{r}), \end{aligned} (3.16)

and define μ_{p}(z) and V(z) for arbitrary elements z ∈ 𝔸(X) by ℝ-linearity [35, p. 595]. We write

(3.17) μ_{c}^{m} = \underbrace{μ_{c} ⌣ ⋯ ⌣ μ_{c}}_{m times} and W^{m} = \underbrace{W ∩ ⋯ ∩ W}_{m times},

for products of the class μ_{c} and its dual W (with the understanding that the copies of W in the above representative are defined with different points x and different transversely intersecting sections s in Lemma 3.13).

3.3. Extension of the geometric representatives. The Uhlenbeck closure M̄_{𝔱} of the PU(2) monopole moduli space M_{𝔱} is described in [10, §2.2]. The space M̄_{𝔱} is compact [10, Theorem 2.12], [17, Theorem 1.2]. We shall need to consider the following subspaces of M̄_{𝔱}:

M̄_{𝔱}^{∗} = {[A, Φ, x] ∈ M̄_{𝔱} : A is irreducible},
M̄_{𝔱}^{0} = {[A, Φ, x] ∈ M̄_{𝔱} : Φ ≢ 0}, (3.18)

and so, as defined in (3.1), M̄_{𝔱}^{∗,0} = M̄_{𝔱}^{∗} ∩ M̄_{𝔱}^{0}. We also define

M_{𝔱}^{≥ε} = {[A, Φ] ∈ M_{𝔱} : ∥Φ∥_{L^{2}}^{2} ≥ ε} ⊂ M_{𝔱}^{0}, (3.19)

and the subspace M_{𝔱}^{∗,≥ε} ⊂ M_{𝔱}^{∗,0} is defined analogously. By [10, Theorem 2.13], the dimension of the highest stratum M_{𝔱}^{∗,0} of M̄_{𝔱} is given by

dim M_{𝔱}^{∗,0} = d_{a}(𝔱) + 2n_{a}(𝔱), (3.20)

where

\begin{aligned}d_{a}(𝔱) & = −2p_{1}(𝔱) − \frac{3}{2}(χ + 2σ), \\ n_{a}(𝔱) & = \frac{1}{4}(p_{1}(𝔱) + c_{1}(𝔱)^{2} − σ). \end{aligned} (3.21)

Recall that the spin^{u} structure 𝔱_{ℓ} defined in [10, Equation (2.44)] has p_{1}(𝔱_{ℓ}) = p_{1}(𝔱) + 4ℓ and so equation (3.20) implies dim M_{𝔱_{ℓ}} = dim M_{𝔱} − 6ℓ. The strata of M_{𝔱_{ℓ}}/S^{1} × Sym^{ℓ}(X) then have codimension at least 2ℓ relative to the top stratum. Thus we can calculate the intersection of geometric representatives whose intersections with the lower strata have the expected codimensions because this ensures that (by the usual dimension-counting argument) the intersection of the geometric representatives will be in the top stratum.

Definition 3.14. The closures of the geometric representatives, V(β), W, in M̄_{𝔱} are denoted by V̄(β), W̄, respectively. For z = β_{1} ⋯ β_{r} ∈ 𝔸(X), the positive generator x ∈ H_{0}(X; ℤ), and an integer m ≥ 0, we denote

(3.22)  V̄(z) = V̄(β_{1}) ∩ ⋯ ∩ V̄(β_{r}) and W̄^{m} = \underbrace{W̄ ∩ ⋯ ∩ W̄}_{m times} .

We shall see in Lemma 3.15 that these closures intersect the lower strata of M̄_{𝔱}^{∗}/S^{1} in sets of the appropriate codimension, except for V̄(x) where x ∈ H_{0}(X; ℤ) (see the remarks following the proof of Lemma 3.15), and thus are geometric representatives on the compactification, away from the zero-section and reducible monopoles. The description of the intersection of V̄(β) and W̄ with the lower strata given in this section is incomplete, as it does not give the multiplicities of all components of these intersections. A more complete description will be given in [15] using information about neighborhoods of the lower strata in M̄_{𝔱}^{∗} obtained from gluing maps.

Recall that Sym^{ℓ}(X) is a smoothly stratified space, the strata being enumerated by partitions of ℓ ∈ ℕ. For i = 1, …, ℓ, let π_{i} : X × ⋯ × X → X be projection onto the ith factor. For any subset Y ⊂ X, let S^{ℓ}(Y) be the image of π_{1}^{−1}(Y) ∪ ⋯ ∪ π_{ℓ}^{−1}(Y) in Sym^{ℓ}(X) under the projection X^{ℓ} → Sym^{ℓ}(Y). If Σ ⊂ Sym^{ℓ}(X) is a smooth stratum, we define S_{Σ}(Y) = S^{ℓ}(Y) ∩ Σ. Let π_{Σ} : M_{𝔱_{ℓ}} × Σ → Σ be the projection.

On each space M_{𝔱_{ℓ}}^{∗}/S^{1} and M_{𝔱_{ℓ}}^{∗,0}/S^{1} there are geometric representatives V_{ℓ}(β) and W_{ℓ} defined by the same construction as V(β) and W, except using the bundles 𝔤_{𝔱_{ℓ}} instead of 𝔤_{𝔱}. We write V_{ℓ}(β) and W_{ℓ} for the pullbacks of these geometric representatives to M_{𝔱_{ℓ}} × Sym^{ℓ}(X).

Lemma 3.15. Let ℓ ≥ 0 be an integer, let Σ ⊂ Sym^{ℓ}(X) be a smooth stratum, and let β ∈ H_{•}(X; ℝ).

1. If β has a smoothly embedded representative Y ⊂ X with a suitable neighborhood ν(D ∪ Y) and x ∈ X has a tubular neighborhood ν(x), then

(a) V̄(β) ∩ (M_{𝔱_{ℓ}}^{∗}/S^{1} × Σ) ⊂ V_{ℓ}(β) ∪ π_{Σ}^{−1}(S_{Σ}(ν(Y ∪ D))),

(b) W̄ ∩ (M_{𝔱_{ℓ}}^{∗,0}/S^{1} × Σ) ⊂ W_{ℓ} ∪ π_{Σ}^{−1}(S_{Σ}(ν(x))).

2. If ι(M_{𝔰}) ⊂ M_{𝔱} and β ∈ H_{2}(X; ℝ) is a two-dimensional class with ⟨c_{1}(𝔱)−c_{1}(𝔰), β⟩ ≠ 0, and γ ∈ H_{1}(X; ℝ), then we have the following reverse inclusions:

(a) ι(M_{𝔰}) ⊂ V̄(β),

(b) ι(M_{𝔰}) ⊂ V̄(γ),

(c) ι(M_{𝔰}) ⊂ V̄(x),

(d) ι(M̄_{κ}^{w}) ∪ ι(M_{𝔰}) ⊂ W̄.

Remark 3.16. From the expression for μ_{p}(β) in Corollary 4.7, when β ∈ H_{3}(X; ℝ), one can see that μ_{p}(β) extends across ι(M_{𝔰}). Thus, V̄(β) should be transverse to ι(M_{𝔰}) and so V̄(β) ∩ ι(M_{𝔰}) would be a codimension-one subset of ι(M_{𝔰}) in this case.

Proof. Here we only prove Assertion (1) . Assertions (2a), (2b), and (2c) will be shown in Corollary 4.7, while Assertion (2d) will follow from Lemma 3.28.

We prove Assertion (1a) about V̄(β) by restricting pairs to the complement of the set π_{Σ}^{−1}(S_{Σ}(ν(Y ∪ D))). We assume that [A_{α}, Φ_{α}] ∈ V(β) is a sequence of points in M_{𝔱}^{∗}/S^{1} converging to the point

[A_{∞}, Φ_{∞}, y] ∈ (M_{𝔱_{ℓ}}^{∗}/S^{1} × Σ) − π_{Σ}^{−1}(S_{Σ}(ν(Y ∪ D))).

Given a suitable neighborhood U = ν(Y ∪ D) of Y ⊂ X − y, we may choose a positive constant r such that

U ⊂ X − ∪_{y∈y}B(y, r).

By the definition of Uhlenbeck convergence, (A_{α}, Φ_{α}) converges in the C^{∞} topology to (A_{∞}, Φ_{∞}) on X − ∪_{y∈y}B(y, r), modulo gauge transformations, and thus

lim_{α→∞}[A_{α}|_{U}, Φ_{α}|_{U}] = [A_{∞}|_{U}, Φ_{∞}|_{U}]. (3.23)

Let V_{Y} (β) ⊂ B_{κ}^{w,∗}(ν(Y ∪ D)) be the geometric representative whose pullback defines V(β). By Lemma 3.12, if [A_{α}, Φ_{α}] ∈ V(β), then [(A_{α}, Φ_{α})|_{U}] ∈ V_{Y} (β). Because V_{Y} (β) is closed (see the definition in [35, pp. 588–592]), equation (3.23) implies that [(A_{∞}, Φ_{∞})|_{U}] ∈ V_{Y} (β) and thus [A_{∞}, Φ_{∞}] ∈ V_{ℓ}(β).

The same argument proves Assertion (1b) concerning W̄, except one observes that one can replace a suitable neighborhood U of x with a tubular neighborhood ν(x).

Lemma 3.15 shows that the intersection of V̄(β) with the lower levels of M̄_{𝔱}^{∗}/S^{1} has the same codimension as that of V(β) in M_{𝔱}^{∗}/S^{1}, unless β ∈ H_{0}(X; ℤ). This is only significant if z = β_{1} ⋯ β_{r} contains both a three-dimensional and a four-dimensional homology class. Then the loops in the suitable neighborhood ν(D ∪ x) may intersect the three-manifold Y. In general, if U_{i} is a suitable neighborhood of a smooth representative of β_{i} , then for any x ∈ X, the inequality

∑_{{i: x∈U_{i}}}(4 − dim β_{i}) ≤ 5, (3.24)

holds [35, Equation (2.7)]. If there are either no three-dimensional classes among the β_{i} or no four-dimensional classes among the β_{i}, the inequality (3.24) can be improved to

∑_{{i: x∈U_{i}}}(4 − dim β_{i}) ≤ 4. (3.25)

If z = β_{1} ⋯ β_{r} ∈ 𝔸(X) and there is a collection of suitable neighborhoods U_{i} of smooth representatives of β_{i} satisfying (3.25), then we call z intersection-suitable.

Lemma 3.17. If z = β_{1} ⋯ β_{r} ∈ 𝔸(X) and either β_{i} ∉ H_{0}(X; ℤ) for i = 1, …, r or β_{i} ∉ H_{3}(X; ℝ) for i = 1, …, r then z is intersection-suitable.

Let z ∈ 𝔸(X) and δ_{c} be a non-negative integer which satisfy

deg(z) + 2δ_{c} = d_{a} + 2n_{a} − 2, (3.26)

so for generic choices of the geometric representatives, V(z) ∩ W^{δ_{c}} ∩ M_{𝔱}^{∗,0}/S^{1} is a collection of one-manifolds. If

[A_{0}, Φ_{0}, y] ∈ V̄(z) ∩ W̄^{δ_{c}} ∩ M̄_{𝔱}^{∗,0}/S^{1},

where y ∈ Sym^{ℓ}(X), then by equation (3.24) and Lemma 3.15 we have

[A_{0}, Φ_{0}] ∈ V_{ℓ}(z′) ∩ W_{ℓ}^{δ_{c}−j},

for some z′ = β_{i_{1}} ⋯ β_{i_{q}} and 1 ≤ i_{1} < ⋯ < i_{q} ≤ r, where 0 ≤ j ≤ δ_{c}. The preceding intersection has codimension greater than or equal to

deg(z) + 2δ_{c} − 5ℓ = d_{a} + 2n_{a} − 2 − 5ℓ.

Then, because dim(M_{𝔱_{ℓ}}/S^{1}) = d_{a} + 2n_{a} − 1 − 6ℓ, the intersection

V_{ℓ}(z′) ∩ W_{ℓ}^{δ_{c}−j} ∩ M_{𝔱_{ℓ}}/S^{1} (3.27)

has dimension less than or equal to

dim(M_{𝔱_{ℓ}}/S^{1}) − (deg(z) + 2δ_{c} − 5ℓ) = 1 − ℓ.

Hence, there could be a point in V̄(z) ∩ W̄^{δ_{c}} ∩ M̄_{𝔱}^{∗,0}/S^{1} contained in the level X × M_{𝔱_{1}}^{∗,0}/S^{1}, where ℓ = 1. However, if z is intersection-suitable, the dimension of the intersection (3.27) is 1 − 2ℓ, using equation (3.25), so intersection will be empty if ℓ > 0. Thus, we have the following corollary to Lemma 3.15.

Corollary 3.18. Let z ∈ 𝔸(X) be intersection-suitable and let δ_{c} be a non-negative integer satisfying

deg(z) + 2δ_{c} = dim(M_{𝔱}^{∗,0}/S^{1}) − 1 = d_{a} + 2n_{a} − 2.

Then for generic choices of geometric representatives, the intersection

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

is a collection of one-dimensional manifolds, disjoint from the lower strata of M̄_{𝔱}^{∗}/S^{1}.

Remark 3.19. The restriction that z ∈ 𝔸(X) be intersection-suitable is a technical one and it should be possible to remove it; we plan to address this point in a subsequent paper.

3.4. Geometric representatives and zero-section monopoles. Our goal in this subsection is to show that the signed count of the points in the intersection of the geometric representatives with the link of Donaldson moduli space, #(V̄(z) ∩ W̄^{n_{a}−1} ∩ L_{𝔱,κ}^{w}), can be expressed in terms of the intersection number #(V̄(z) ∩ M̄_{κ}^{w}), which defines a Donaldson invariant (at least when w is chosen so that no SO(3) bundle with second Stiefel-Whitney class w (mod 2) admits a flat connection); the conclusion is stated as Proposition 3.29.

3.4.1. Geometric representatives on the stratum of zero-section monopoles. From the construction of the Uhlenbeck compactifications for M_{κ}^{w} and M_{𝔱}, the smoothly stratified embedding [10, Equation (3.5)],

ι : M_{κ}^{w} ↪ M_{𝔱}, [Â] ↦ [A, 0],

extends (see [10, §2.2], [17, §4], [7, §4.4]) to smoothly stratified embedding

ι : M̄_{κ}^{w} ↪ M̄_{𝔱}, [Â, x] ↦ [A, 0, x],

where κ = −\frac{1}{4}p_{1}(𝔱) and w is any integral lift of w_{2}(𝔱). Define

M̄_{𝔱}^{asd} = {[A, Φ, x] ∈ M̄_{𝔱} : Φ = 0} ⊂ M̄_{𝔱}. (3.28)

The space L_{𝔱,κ}^{w} defined in [10, Definition 3.7] serves as a link for M̄_{𝔱}^{asd}. Although the definitions imply that

ι(M̄_{κ}^{w}) ⊂ M̄_{𝔱}^{asd},

the reverse inclusion might not be true. For example, suppose [Â_{0}] is the gauge-equivalence class of a flat connection on an SO(3) bundle F over X with w_{2}(F) = w (mod 2) and −\frac{1}{4}p_{1}(F) = 0. The Uhlenbeck compactification M̄_{κ}^{w} might not contain all points [Â_{0}, x] ∈ M_{0}^{w} × Sym^{ℓ}(X) because there can be obstructions to gluing [54] onto flat connections, as the Freed-Uhlenbeck generic metrics theorem does not guarantee they are smooth points of their moduli spaces [7], [21]. However, there could be a sequence of points [A_{α}, Φ_{α}] ∈ M_{𝔱} converging to [A_{0}, 0, x] in the Uhlenbeck topology but no sequence [Â′_{α}] ∈ M_{κ}^{w} also converging to [Â_{0}, x].

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

Observe that a class v ∈ H^{2}(X; ℤ/2ℤ) is good if and only if no line bundle L over X with first Chern class c_{1}(L) ≡ v (mod 2) admits a flat connection or if and only if no SO(3) bundle over X with second Stiefel-Whitney class v admits a flat connection. Thus, Lemma 3.2 in [10] gives a criterion for v to be good.

Hence, if w_{2}(𝔱) is good, there are no flat SO(3) connections in M̄_{𝔱} and, when there no obstructions to gluing (for example, when the metric on X is generic in the sense of [7, 21]), it follows from Taubes’ gluing theorem for anti-self-dual SO(3) connections [53], [55] that

M̄_{𝔱}^{asd} ⊂ ι(M̄_{κ}^{w}).

The preceding discussion yields

Lemma 3.21. Let 𝔱 be a spin^{u} structure on a closed, oriented four-manifold X with generic metric, b_{2}^{+}(X) > 0 and w_{2}(𝔱) ≡ w (mod 2), for w ∈ H^{2}(X; ℤ). If w (mod 2) is good, then

M̄_{𝔱}^{asd} = ι(M̄_{κ}^{w}). (3.29)

The constraint that w (mod 2) is good is also used to separate the strata of zero-section PU(2) monopoles from the strata of reducible monopoles, so that the moduli space of PU(2) monopoles gives a smooth cobordism between their links. Therefore, when w (mod 2) is good, equation (3.29) holds, we have a disjoint union

M̄_{𝔱} = M̄_{𝔱}^{0} ∪ ι(M̄_{κ}^{w}),

and L_{𝔱,κ}^{w} is a link of ι(M̄_{κ}^{w}) ⊂ M̄_{𝔱}/S^{1}.

3.4.2. A definition of the Donaldson invariants. Fix w ∈ H^{2}(X; ℤ) and let z ∈ 𝔸(X) be a monomial whose degree deg(z) satisfies equation (1.4). Then let κ ∈ \frac{1}{4}ℤ be defined by

deg(z) = 8κ − \frac{3}{2} (χ + σ) . (3.30)

Let X̃ = X#\overline{ℂℙ}^{2} denote the blow-up of X and let e ∈ H_{2}(X̃; ℤ) be the exceptional class and let PD[e] be its Poincaré dual. Since −4κ = w^{2} (mod 4) by equations (1.4) and (3.30) and thus −4(κ + 1/4) = (w + PD[e])^{2} (mod 4), we can find an SO(3) bundle F̃ → X̃ with p_{1}(F̃) = −4(κ + 1/4) and w_{2}(F̃) = (w + PD[e]) (mod 2) [25, Theorem 1.4.20]. We can therefore define M_{κ+1/4}^{w+PD[e]}(X̃), the moduli space of anti-self-dual SO(3) connections on F̃. Then the Donaldson invariant is defined by [35, p. 594]

D_{X}^{w}(z) = # (V̄(ze) ∩ M̄_{κ+1/4}^{w+PD[e]}(X̃)) , (3.31)

where the moduli space M_{κ+1/4}^{w+PD[e]}(X̃) is given the orientation o(Ω, w + PD[e]) and on the right-hand side, we consider z to be a monomial in 𝔸(X̃) via the inclusion

H_{2}(X; ℝ) ⊂ H_{2}(X̃; ℝ) = H_{2}(X; ℝ) ⊕ ℝ[e] and 𝔸(X) ⊂ 𝔸(X̃).

The Donaldson invariant is independent of the choice of generic geometric representatives and, when b_{2}^{+}(X) > 1, independent of the metric.

If M_{κ}^{w}(X) is given the orientation o(Ω, w), a well-known special case of the blow-up formula [18, Lemma 3.13], [31, Theorem 6.9] implies that

# (V̄(z) ∩ M̄_{κ}^{w}(X)) = # (V̄(ze) ∩ M̄_{κ+1/4}^{w+PD[e]}(X̃)) , (3.32)

when the intersection number on the left is well-defined, for example, when w is good in the sense of Definition 3.20. However, the blow-up trick [47] ensures that the intersection number on the right is well-defined for arbitrary w ∈ H^{2}(X; ℤ), since w +PD[e] ∈ H^{2}(X̃; ℤ) is good.

In [35, p. 585], Kronheimer and Mrowka require that b_{2}^{+}(X) − b_{1}(X) be odd. However, for the purposes of defining the Donaldson invariants for a closed four-manifold, with b_{1}(X) possibly non-zero, one can have non-trivial Donaldson invariants when b_{2}^{+}(X) − b_{1}(X) is even as they point out in [35, p. 595]. The reason for the constraint is that their structure theorem is only stated for the case b_{1}(X) = 0; the invariants become more difficult to compute when b_{1}(X) > 0. If b_{1}(X) = 0, then the Donaldson invariants are necessarily trivial unless b_{2}^{+}(X) is odd (and the moduli spaces of anti-self-dual connections are even dimensional).

When b_{2}^{+}(X) = 1, the Donaldson invariant (3.31) depends on the “chamber” in H^{2}(X̃; ℝ) defined by metric g̃ on X̃. Specifically, if ω(g̃) is the unique unit-length harmonic two-form which is self-dual with respect to g̃—the period point for g̃—and lies in the positive cone (determined by the homology orientation) of H^{2}(X̃; ℝ), then the intersection number on the right-hand-side of equation (3.31) changes whenever the sign of ω(g̃) ⌣ α changes for some α ∈ H^{2}(X̃; ℤ) satisfying

(3.33) α ≡ w + PD[e] (mod 2) and α^{2} = −4(κ + 1/4) + 4ℓ, for some ℓ ∈ ℕ.

(The classes α correspond to split SO(3) bundles over X̃, namely \underline{ℝ} ⊕ L with c_{1}(L) = α, so they have first Pontrjagin number α^{2} and second Stiefel-Whitney class α (mod 2); see [31],

[32] for further explanation.) For any α ∈ H^{2}(X̃; ℝ) which is not torsion, the subset

{h ∈ H^{2}(X̃; ℝ) : h ⌣ h > 0 and h ⌣ α = 0}

of the positive cone of H^{2}(X̃; ℝ) is an α-wall. If α obeys condition (3.33), then α is nontorsion since w+PD[e] (mod 2) is good and the resulting α-wall is called a (w+PD[e], −4κ− 1)-wall. The connected components of the complement in the positive cone of H^{2}(X̃; ℝ) of the union of (w + PD[e], −4κ − 1)-walls are called (w + PD[e], −4κ − 1)-chambers. Hence, the intersection pairing in definition (3.31) changes if the period point ω(g̃) moves from one (w + PD[e], −4κ − 1)-chamber to another.

We now discuss how a choice of a metric g on X determines a chamber in the positive cone of H^{2}(X̃; ℝ) ≅ ℝ[e] ⊕ H^{2}(X; ℝ). Assume first that w (mod 2) is good in the sense of Definition 3.20. Therefore, if β ∈ H^{2}(X; ℤ) satisfies

(3.34) β ≡ w (mod 2) and β^{2} = −4κ + 4ℓ, for some ℓ ∈ ℕ,

then β is non-torsion and thus defines a (w, −4κ)-wall in H^{2}(X; ℝ). Moreover, α = β + PD[e] satisfies condition (3.33) and defines a (w + PD[e], −4κ − 1)-wall in H^{2}(X̃; ℝ). This establishes an inclusion of (w, −4κ)-chambers in H^{2}(X; ℝ) into related (w+PD[e], −4κ−1)- chambers in H^{2}(X̃; ℝ).

If g is a generic metric on X, then the period point ω(g) ∈ H^{2}(X; ℝ) does not lie on any wall and there is a unique chamber in the positive cone of H^{2}(X; ℝ) which contains ω(g) [7, Corollary 4.3.15].

If the metric g̃ on X̃ is constructed by splicing together a generic metric g on X and the Fubini-Study metric on \overline{ℂℙ}^{2} along a “long neck”, then ω(g̃) converges to ω(g) in L^{2} as the length of the neck tends to infinity, viewing both ω(g̃) and ω(g) as elements of H^{2}(X̃; ℝ). Thus, ω(g̃) will lie in the chamber in the positive cone of H^{2}(X̃; ℝ) related to the chamber in the positive cone of H^{2}(X; ℝ) containing ω(g). Thus, when w (mod 2) is good, the intersection pairing (3.31) defining the invariant D_{X}^{w}(z) is defined with respect to the chamber in H^{2}(X̃; ℝ) related to the chamber in H^{2}(X; ℝ) determined by the period point ω(g).

If w (mod 2) is not good, one can have torsion classes β ∈ H^{2}(X; ℤ) satisfying condition (3.34) and thus α = β + PD[e] ∈ H^{2}(X̃; ℤ) satisfying (3.33). The corresponding (w + PD[e], −4κ − 1)-wall is given by

{h ∈ H^{2}(X̃; ℝ) : h ⌣ h > 0 and h ⌣ PD[e] = 0}.

Hence, the period point ω(g) ∈ H^{2}(X; ℝ) for any metric g on X lies in this PD[e]-wall, since H^{2}(X̃; ℝ) ≅ ℝ[e] ⊕ H^{2}(X; ℝ), and the fact that ω(g̃) converges to ω(g) as the length of the neck tends to infinity does not determine the chamber of ω(g̃) without a delicate analysis of the sign of ω(g̃) ⌣ PD[e] (see [59]). We plan to address the case b_{2}^{+}(X) = 1 elsewhere and so in the present article, if b_{2}^{+}(X) = 1, we only consider the dependence of the invariants D_{X}^{w}(z) on the chamber in H^{2}(X; ℝ) when w is good.

3.4.3. Geometric representatives on the link of the stratum of zero-section monopoles. We now turn to the arguments leading to a proof of Proposition 3.29, which expresses the intersection number

# (V̄(z) ∩ W̄^{n_{a}−1} ∩ L_{𝔱,κ}^{w}) (3.35)

in terms of

# (V̄(z) ∩ ι(M̄_{κ}^{w})) = #(V̄(z) ∩ M̄_{κ}^{w}),

which is equal to the Donaldson invariant D_{X}^{w}(z) when w is good.

Note that by the construction of the geometric representatives and definition of the Donaldson invariants [7], [35, §2], one has

V̄(z) ∩ M̄_{κ}^{w} = V(z) ∩ M_{κ}^{w}.

That is, the intersection is contained in the top stratum M_{κ}^{w} of the compactification M̄_{κ}^{w}. Therefore, to calculate the intersection number (3.35) it will be enough to examine small neighborhoods of the points in the intersection V(z) ∩ ι(M_{κ}^{w}). Such neighborhoods are described by Kuranishi models, which we now describe.

Suppose [A, 0] ∈ V(z) ∩ ι(M_{κ}^{w}). Applying the Kuranishi method to describe the zero locus of the PU(2) monopole equations using [10, Corollary 3.6], we obtain a smooth S^{1}- equivariant embedding

γ_{A} : O_{A} ⊂ T_{[Â]}M_{κ}^{w} ⊕ Ker D_{A,ϑ} → (A, 0) + Ker d_{A,0}^{∗} ⊂ C̃_{𝔱}, (3.36)

of a precompact, open S^{1}-invariant neighborhood O_{A} of the origin with image γ_{A}(O_{A}/S^{1}) ⊂ C_{𝔱}/S^{1}, where S^{1} acts on the domain by scalar multiplication on Ker D_{A,ϑ}, and a smooth S^{1}-equivariant map

(3.37) φ_{A} : O_{A} ⊂ T_{[Â]}M_{κ}^{w} ⊕ Ker D_{A,ϑ} → Coker D_{A,ϑ},

such that

(3.38)  γ_{A} ((φ_{A}^{−1}(0) ∩ O_{A})/S^{1}) = M_{𝔱}/S^{1} ∩ γ_{A}(O_{A}/S^{1}),
(3.39)  γ_{A} ((T_{[Â]}M_{κ}^{w} ⊕ {0}) ∩ O_{A}) = ι(M_{κ}^{w}) ∩ γ_{A}(O_{A}),

are open neighborhoods of the point [A, 0] in M_{𝔱}/S^{1} and ι(M_{κ}^{w}), respectively. Because points in M_{𝔱}^{∗,0} are regular, the map φ_{A} vanishes transversely on

O_{A} − (T_{[Â]}M_{κ}^{w} ⊕ {0}).

Compare the proof of Assertion (4) in [10, Theorem 3.21]. For convenience, we set

Z_{A} = φ_{A}^{−1}(0) ∩ O_{A}. (3.40)

Equation (3.38) implies that the S^{1}-equivariant embedding γ_{A} descends to a homeomorphism from a neighborhood of the origin onto a neighborhood of [A, 0],

γ_{A} : (φ_{A}^{−1}(0) ∩ O_{A})/S^{1} ≅ M_{𝔱}/S^{1} ∩ γ_{A}(O_{A}/S^{1}), (3.41)

which restricts to a diffeomorphism on a smooth stratum.

In [10, §3.2], we constructed the link L_{𝔱,κ}^{w} using the S^{1}-invariant “distance function”,

ℓ : C_{𝔱} → [0, ∞), [A, Φ] ↦ ∥Φ∥_{L^{2}}^{2}. (3.42)

The function ℓ extends continuously over M̄_{𝔱} if we set ℓ([A, Φ, x]) = ∥Φ∥_{L^{2}}^{2}. For generic, positive, small ε we have [10, Definition 3.7]

L_{𝔱,κ}^{w,ε} = ℓ^{−1}(ε) ∩ M̄_{𝔱}/S^{1}, (3.43) and denote L_{𝔱,κ}^{w,ε} by L_{𝔱,κ}^{w} when the value of ε is not relevant. To compute the pairing (3.35) with L_{𝔱,κ}^{w,ε}, we must first describe V̄(z) in a neighborhood of ι(M_{κ}^{w}) in M_{𝔱}.

Lemma 3.22. Let 𝔱 be a spin^{u} structure over a four-manifold X with w_{2}(𝔱) ≡ w (mod 2), where w ∈ H^{2}(X; ℤ) and w (mod 2) is good. Suppose that deg(z) ≥ dim M_{κ}^{w} and z is intersection-suitable, as defined before Lemma 3.17. Denote V̄(z) ∩ M̄_{κ}^{w} = {[Â_{i}]_{i=1}^{N}}. Then for each [Â] ∈ V̄(z)∩M̄_{κ}^{w}, there is an open neighborhood O′_{A} ⊂ O_{A} of the origin in T_{[Â]}M_{κ}^{w} ⊕ Ker D_{A,ϑ}, where O_{A} is the open neighborhood defining the Kuranishi model (3.36), such that:

1. There is a smooth, S^{1}-invariant map,

f_{A} : O′_{A} ∩ ({0} ⊕ Ker D_{A,ϑ}) → T_{[Â]}M_{κ}^{w},

with f_{A}(0) = 0 and (Df_{A})_{0} = 0 such that

γ_{A}^{−1}(V(z)) ∩ O′_{A} = {(f_{A}(ϕ), ϕ) : ϕ ∈ O′_{A} ∩ ({0} ⊕ Ker D_{A,ϑ})}.

2. There is a positive constant ε_{0} such that for all ε < ε_{0},

V̄(z) ∩ L_{𝔱,κ}^{w,ε} ⊂ ⋃_{i=1}^{N} γ_{A_{i}} (O′_{A_{i}}) /S^{1},

where the union on the right above is disjoint.

3. For each ε ∈ (0, ε_{0}) with ε_{0} as in (2), there is a positive constant δ such that all (a, ϕ) ∈ γ_{A}^{−1} (V(z)) ∩ O′_{A} satisfy ∥ϕ∥_{L^{2}}^{2} > δ.

Proof. Consider V(z) as a smooth submanifold of B_{κ}^{w,∗}. If π_{B} : C_{𝔱}^{∗} → B_{κ}^{w,∗} is the projection, then the composition π_{B} ∘ γ_{A} is a smooth map from O_{A} to B_{κ}^{w,∗}. The manifolds M_{κ}^{w} and V(z) intersect transversely in B_{κ}^{w,∗} at [Â] = π_{B} ∘ γ_{A}(0, 0). The restriction of γ_{A} to

O_{A} ∩ (T_{[Â]}M_{κ}^{w} ⊕ {0})

is an embedding onto an open neighborhood of [Â] in M_{κ}^{w}, so the composition π_{B} ∘ γ_{A} is transverse to V(z) at the origin. Thus, restricted to a sufficiently small open neighborhood O′′_{A} ⊂ O_{A} of the origin (0, 0) in T_{[Â]}M_{κ}^{w} ⊕ Ker D_{A,ϑ}. the map π_{B} ∘ γ_{A} is transverse to V(z) in B_{κ}^{w,∗}. Hence γ_{A}^{−1}(V(z)) ∩ O′′_{A} is a smooth manifold. By shrinking the neighborhood O′′_{A} we can assume that γ_{A}^{−1} (V(z)) ∩ O′′_{A} and T_{[Â]}M_{κ}^{w} ⊕ {0} intersect only at the origin, since

V(z) ∩ M_{κ}^{w} = [Â] = π_{B} ∘ γ_{A}(0, 0). We now prove that

T_{(0,0)} (γ_{A}^{−1}(V(z)) ∩ O′′_{A}) = {0} ⊕ Ker D_{A,ϑ}. (3.44)

First, note that because the derivative of γ_{A} at the origin is the inclusion of T_{[Â]}M_{κ}^{w} ⊕

Ker D_{A,ϑ} into Ker d_{A,0}^{0,∗} by construction of the Kuranishi model, we have the inclusion:

{0} ⊕ Ker D_{A,ϑ} ⊂ Ker(D(π_{B} ∘ γ_{A}))_{(0,0)} ⊂ T_{(0,0)} (γ_{A}^{−1}(V(z)) ∩ O′′_{A}) .

Because (D(π_{B} ∘ γ_{A}))_{(0,0)} maps T_{[Â]}M_{κ}^{w,∗} ⊕ {0} onto the normal bundle of V(z) in B_{κ}^{w,∗}, the above inclusion is an equality. Equation (3.44) implies that if

π_{K,A} : T_{[Â]}M_{κ}^{w} ⊕ Ker D_{A,ϑ} → Ker D_{A,ϑ}

is the projection onto Ker D_{A,ϑ} then the derivative of the restriction of π_{K,A},

π_{K,A} : γ_{A}^{−1}(V(z)) ∩ O′′_{A} → Ker D_{A,ϑ} (3.45)

at the origin is an isomorphism. Therefore, for small enough O′′_{A} the map (3.45) is a diffeomorphism onto a neighborhood O_{K,A} of the origin in Ker D_{A,ϑ}. If O′_{A} = O′′_{A} ∩ π_{K,A}^{−1}(O_{K,A}), then O′_{A} ∩ {0} ⊕ Ker D_{A,ϑ} ⊂ O_{K,A} and π_{K,A} restricted to γ_{A}^{−1}(V(z)) ∩ O′_{A} is still a diffeomorphism onto O_{K,A} with inverse as described.

By shrinking the sets O′_{A}, we can assume the images γ_{A}(O′_{A}) are disjoint, proving the final statement in Assertion (2). Suppose ε_{α} is a sequence of positive numbers converging to zero. If Assertion (2) were not true, there would be a sequence

{[A_{α}, Φ_{α}, x_{α}]}_{α=1}^{∞} ⊂ V̄(z) ∩ M̄_{𝔱},

satisfying ℓ([A_{α}, Φ_{α}, x_{α}]) = ε_{α} and [A_{α}, Φ_{α}, x_{α}] not in any γ_{A_{i}}(O′_{A_{i}}). Since V̄(z) ∩ M̄_{𝔱} is compact, there would be a convergent subsequence, also denoted {[A_{α}, Φ_{α}, x_{α}]}_{α=1}^{∞}, converging to [A_{∞}, Φ_{∞}, x_{∞}]. Because ℓ is continuous on M̄_{𝔱}, we would have ℓ([A_{∞}, Φ_{∞}, x_{∞}]) = 0, so [A_{∞}, Φ_{∞}, x_{∞}] ∈ ι(M̄_{κ}^{w}) by Lemma 3.21, with Φ_{∞} = 0. Then,

[A_{∞}, 0, x_{∞}] ∈ V̄(z) ∩ ι(M̄_{κ}^{w}) = V(z) ∩ ι(M_{κ}^{w}),

and thus x = ∅ and [Â_{∞}] ∈ {[Â_{1}, …, Â_{N}]}. The images γ_{A_{i}}(O′_{A_{i}}) contain open neighborhoods of the points {[Â_{1}], …, [Â_{N}]} in M̄_{𝔱}, so for large enough α the sequence must lie in the union of these images, contradicting the assumption that [A_{α}, Φ_{α}, x_{α}] is not in any γ_{A_{i}}(O′_{A_{i}}). This proves Assertion (2).

We use contradiction to prove Assertion (3): if it were not true, then there would be a sequence {(a_{α}, ϕ_{α})}_{α=1}^{∞} in γ_{A}^{−1}(V(z)) ∩ O′_{A} with ℓ(γ_{A}(a_{α}, ϕ_{α})) = ε and ϕ_{α} ∈ O_{K,A} with lim_{α→∞} ∥ϕ_{α}∥_{L^{2}}^{2} = 0. By Assertion (1) this sequence could be written (a_{α}, ϕ_{α}) = (f_{A}(ϕ_{α}), ϕ_{α}). Because the sequence {ϕ_{α}} ⊂ Ker D_{A,ϑ} converged to zero in L^{2} and D_{A,ϑ} is elliptic, it would converge to zero in L_{ℓ}^{2} (for ℓ ≥ 2). Since f_{A} is continuous on O_{K,A}, we would have lim_{α→∞} a_{α} = lim_{α→∞} f_{A}(ϕ_{α}) = 0 and, as ℓ ∘ γ_{A} is continuous on O_{K,A},

lim_{α→∞} ℓ(γ_{A}(a_{α}, ϕ_{α})) = ℓ(γ_{A}(0, 0)) = ℓ(0) = 0,

contradicting our assumption that for all α we have ℓ(γ_{A}(a_{α}, ϕ_{α})) = ε. This proves Assertion (3) and completes the proof of the lemma.

Because the spaces γ_{A}(O′_{A}) suffice to cover the intersection V̄(z) ∩ L_{𝔱,κ}^{w,ε} for ε sufficiently small by Lemma 3.22, we shall henceforth restrict the domain of γ_{A} to O′_{A}.

We define a link of the submanifold T_{[Â]}M_{κ}^{w} ⊕ {0} ⊂ T_{[Â]}M_{κ}^{w} ⊕ Ker D_{A,ϑ} by setting

K_{A,δ} = {(a, ϕ) ∈ T_{[Â]}M_{κ}^{w} ⊕ Ker D_{A,ϑ} : ∥ϕ∥_{L^{2}}^{2} = δ}. (3.46)

The link K_{A,δ}/S^{1} is more convenient to work with than the level sets of ℓ ∘ γ_{A} defining γ_{A}^{−1} (L_{𝔱,κ}^{w,ε}). We will in Lemma 3.26 that the two links are related by an oriented cobordism. Therefore, prior to showing this equivalence, we first discuss the orientation of the spaces.

An orientation O for M_{𝔱}^{∗,0} determines an orientation

ω(L, ∂O) (3.47)

for L_{𝔱,κ}^{w,ε} by considering L_{𝔱,κ}^{w,ε} as the boundary of the subspace ℓ^{−1}([ε, ∞)) ⊂ M_{𝔱}/S^{1} and using the convention (2.19).

An orientation o of T_{[Â]}M_{κ}^{w} determines an orientation of K_{A,δ}/S^{1} by identifying K_{A,δ}/S^{1} with T_{[Â]}M_{κ}^{w} × ℂℙ^{k−1}, where k = dim_{ℂ} Ker D_{A,ϑ} and taking the complex orientation of ℂℙ^{k−1}, denoted by

ω(K, o). (3.48)

We now describe a convention for orienting smooth submanifolds.

Convention 3.23. Suppose Z and M are manifolds which intersect transversely. Then an orientation O for M and an orientation ω(Z) for the normal bundle of Z determine an orientation for M ∩ Z, which we write as O/ω(Z).

From equation (3.37) and the fact that φ_{A} vanishes transversely on (O′_{A} − (T_{[Â]}M_{κ}^{w} ⊕ {0})/S^{1}, the fibers of the normal bundle of Z_{A}/S^{1} in (O′_{A}−(T_{[Â]}M_{κ}^{w} ⊕{0})/S^{1} are naturally identified with Coker D_{A,ϑ}. Let ω(Z) be the orientation of this normal bundle of Z_{A}/S^{1} obtained by giving Coker D_{A,ϑ} the complex orientation.

By Convention 3.23, the orientation ω(K, o) of K_{A,δ}/S^{1} and orientation ω(Z) of the normal bundle of Z_{A}/S^{1} determine a “complex orientation” of Z_{A} ∩ K_{A,δ}/S^{1},

ω(K, o)/ω(Z), (3.49)

where o is the orientation for T_{[Â]}M_{κ}^{w}, by analogy with Definition 2.7 which this construction matches.

However, to compare the orientations of L_{𝔱,κ}^{w,ε} and Z_{A} ∩ K_{A,δ}/S^{1} with those of other links in a cobordism formula such as equation (3.70), it is natural to orient Z_{A} ∩ K_{A,δ}/S^{1} as a boundary of the cobordism. If O is an orientation of M_{𝔱}^{∗,0}/S^{1}, we obtain an orientation

ω(Z ∩ K, ∂O) (3.50)

for Z_{A} ∩ K_{A,δ}/S^{1} by identifying this manifold, via the map γ_{A}, with the boundary of

M_{𝔱} \ γ_{A} (T_{[Â]}M_{κ}^{w} × B_{A}(0, δ) ∩ O′_{A}) ,

where B_{A}(0, δ) ⊂ Ker D_{A,ϑ} is the ball of radius δ, and using convention (2.19). The proof of the following lemma is the same as that of Lemma 2.9.

Lemma 3.24. Let w be an integral lift of w_{2}(𝔱). Fix an orientation o = o(Ω, w) of M_{κ}^{w}. Let O = O^{asd}(Ω, w) be the orientation for M^{∗,0}/S^{1} in Definition 2.3. Then, the orientations (3.49) and (3.50) for Z_{A} ∩ K_{A,δ}/S^{1} agree, that is

ω(K, o)/ω(Z) = ω(Z ∩ K, ∂O).

By the definition of a geometric representative (Definition 3.4), the normal bundle of V(z) in B_{κ}^{w,∗} has an orientation, which we denote by ω(V). Because V(z) intersects

π_{B} ∘ γ_{A}(K_{A,δ}/S^{1}), π_{B} ∘ γ_{A}(Z_{A} ∩ K_{A,δ}/S^{1}), and π_{B}(L_{𝔱,κ}^{w,ε})

transversely, the orientations ω(V), ω(K, o), ω(Z), ω(Z ∩ K, ∂O) and ω(L, ∂O) determine orientations

ω(K, o)/ω(V) for γ_{A}^{−1}(V(z)) ∩ K_{A,δ}/S^{1},
\begin{matrix}(ω(K, o)/ω(Z)) /ω(V) \\ ω(Z ∩ K, ∂O)/ω(V) \end{matrix}\} for γ_{A}^{−1}(V(z)) ∩ Z_{A} ∩ K_{A,δ}/S^{1},
ω(L, ∂O)/ω(V) for V(z) ∩ L_{𝔱,ε}^{w,ε}.

(3.51)

Observe that Lemma 3.24 implies that

(ω(K, o)/ω(Z)) /ω(V) = ω(Z ∩ K, ∂O)/ω(V), (3.52)

if o = o(Ω, w) and O = Ω^{asd}(Ω, w).

Lemma 3.25. Let w be an integral lift of w_{2}(𝔱). Let ε(A) = ±1 be the signed intersection number of V(z) and M_{κ}^{w} at [Â], where M_{κ}^{w} is given the orientation o(Ω, w). Then the following map is a diffeomorphism,

g_{A} = f_{A} × id_{Ker D_{A,ϑ}} : ℂℙ^{k−1} → γ_{A}^{−1} (V(z)) ∩ K_{A,δ}/S^{1}, (3.53)

where ℂℙ^{k−1} = ℙ(Ker D_{A,ϑ}) and f_{A} is defined in Assertion (1) of Lemma 3.22. If ℂℙ^{k−1} has the complex orientation and γ_{A}^{−1} (V(z)) ∩ K_{A,δ}/S^{1} has the orientation ω(K, o)/ω(V) of (3.51) for o = o(Ω, w), then g_{A} preserves orientation if and only if ε(A) = 1.

Proof. By Assertion (1) of Lemma 3.22, the intersection γ_{A}^{−1} (V(z)) ∩ K_{A,δ}/S^{1} is given by the S^{1} quotient of the graph of f_{A} restricted to S^{2k−1} ⊂ Ker D_{A,ϑ}. Thus, g_{A} gives the desired diffeomorphism. Because V(z) and M_{κ}^{w} intersect transversely in B_{κ}^{w,∗} at [Â], the normal bundle of V(z) in B_{κ}^{w,∗} is identified with T_{[Â]}M_{κ}^{w} but their orientations agree if and only if ε(A) = 1. The result then follows from the definition (3.48) of the orientation ω(K, o) of K_{A,δ}/S^{1} determined by the orientation o(Ω, w) of T_{[Â]}M_{κ}^{w}.

Lemma 3.26. Continue the assumptions and notation of Lemma 3.22. For ε sufficiently small and δ as in Assertion (3) of Lemma 3.22 and generic, there is a smooth, compact, and oriented cobordism between

γ_{A}^{−1} (V(z) ∩ L_{𝔱,κ}^{w,ε}) , (3.54)

with the orientation ω(L, ∂O)/ω(V) of (3.51) for O = O^{asd}(Ω, w), and the manifold

Z_{A} ∩ γ_{A}^{−1} (V(z)) ∩ K_{A,δ}/S^{1}, (3.55)

with the orientation (ω(K, o)/ω(Z)) /ω(V) of (3.51), where o = o(Ω, w).

Proof. As before, we let B_{A}(0, δ) ⊂ Ker D_{A,ϑ} be the open ball of radius δ. Assertion (3) of Lemma 3.22 yields the inclusion

γ_{A} ((T_{[Â]}M_{κ}^{w} × B_{A}(0, δ)) ∩ O′_{A}/S^{1}) ∩ V̄(z) ∩ M̄_{𝔱}/S^{1} ⊂ ℓ^{−1}([0, ε)).

Then for generic values of ε and δ,

M̄_{𝔱}/S^{1} ∩ V̄(z) ∩ ℓ^{−1}([0, ε]) ∩ γ_{A}(O′_{A}) − γ_{A} (T_{[Â]}M_{κ}^{w} × B_{A}(0, δ)/S^{1}) (3.56)

is a smooth manifold with boundaries given by the manifold (3.54) and by

M̄_{𝔱}^{0} ∩ V̄(z) ∩ γ_{A} ((T_{[Â]}M_{κ}^{w} × ∂B̄_{A}(0, δ)) ∩ O′_{A}/S^{1}) , (3.57)

which is diffeomorphic, via γ_{A}, to the manifold (3.55).

By Assertion (2) of Lemma 3.22, the compact set V̄(z) ∩ M̄_{𝔱} ∩ ℓ^{−1}([0, ε]) is contained in a finite, disjoint union ∪_{i=1}^{N}γ_{A_{i}} (O′_{A_{i}}/S^{1}), where V̄(z) ∩ M̄_{κ}^{w} = {[Â_{i}]}_{i=1}^{N}. Thus, each component

V̄(z) ∩ M̄_{𝔱} ∩ ℓ^{−1}([0, ε]) ∩ γ_{A} (O′_{A}/S^{1}) ,

of this disjoint union is compact. Therefore, the space (3.56) is a compact and smooth cobordism between the manifold (3.54) and the manifold (3.57) which, as previously noted, is diffeomorphic to the manifold (3.55).

Let M_{𝔱}^{∗,0}/S^{1} ∩ V(z) have the orientation determined by the orientation O^{asd}(Ω, w) of M_{𝔱}/S^{1} and the orientation ω(V) of the normal bundle of V(z). The manifold (3.56) has codimension zero in M_{𝔱}^{∗,0}/S^{1} ∩ V(z) and thus inherits an orientation from M_{𝔱}^{∗,0}/S^{1} ∩ V(z). Hence, the manifold (3.56) defines an oriented cobordism. The orientation of the manifold (3.54) given by viewing it as a component of the boundary of the oriented manifold (3.56) is then equal to −ω(L, ∂O)/ω(V) (as defined in (3.51)) for O = O^{asd}(Ω, w). The negative sign arises because the orientation ω(L, ∂O) of (3.47) is defined by viewing L_{𝔱,κ}^{w,ε} as the boundary of ℓ^{−1}([ε, ∞)).

The orientation of the manifold (3.55) given by considering it as a component of the boundary of the oriented manifold (3.56) is then equal to ω(Z ∩ K, ∂O)/ω(V) (as defined in (3.51)) for O = O^{asd}(Ω, w) which, by equation (3.52) is equal to (ω(K, o)/ω(Z)) /ω(V) for o = o(Ω, w).

Recall that two oriented manifolds (M_{i}, ω_{i}) for i = 0, 1 are cobordant if there is an oriented manifold W whose oriented boundary is (M_{0}, −ω_{0}) ∪ (M_{1}, ω_{1}) [30, p. 170]. Therefore, the manifold (3.56) gives the desired cobordism.

We now give a cohomological description of the zero locus Z_{A} of the obstruction map:

Lemma 3.27. Continue the hypotheses and notation of Lemmas 3.22 and 3.25. Assume that n_{a}(𝔱) = Index_{ℂ} D_{A,ϑ} is positive and let c = dim_{ℂ} Coker D_{A,ϑ}. If ℂℙ^{k−1} ≅ ℙ(Ker D_{A,ϑ}) has the complex orientation and h ∈ H^{2}(ℂℙ^{k−1}; ℤ) is the positive generator, then for generic δ > 0, there is a smooth submanifold T of ℂℙ^{k−1} which is Poincaré dual to h^{c} such that the restriction of the map g_{A} of definition (3.53) to T gives a diffeomorphism,

g_{A} : T ≃ Z_{A} ∩ γ_{A}^{−1} (V(z)) ∩ K_{A,δ}/S^{1}.

If T is oriented as the Poincaré dual of h^{c} and Z_{A} ∩ γ_{A}^{−1} (V(z)) ∩ K_{A,δ}/S^{1} is oriented by ω(Z ∩ K, o)/ω(V) from (3.51) where o = o(Ω, w), then the restriction of g_{A} to T is orientation preserving if and only if ε(A) = 1, where ε(A) is defined in Lemma 3.25.

Proof. As noted before Lemma 3.22, the Kuranishi map φ_{A} in (3.37) vanishes transversely on O′_{A} − (T_{[Â]}M_{κ}^{w} ⊕ {0}). For generic δ, the map φ_{A} vanishes transversely on γ_{A}^{−1} (V(z)) ∩

K_{A,δ}/S^{1} because V(z) is transverse to M_{𝔱}^{∗,0}/S^{1} by construction. This implies that the zero locus of φ_{A} is Poincaré dual to the Euler class of the vector bundle (3.58) of which φ_{A} is a section. We define a smooth submanifold,

T = g_{A}^{−1}(φ_{A}^{−1}(0)) ⊂ ℂℙ^{k−1},

and observe that the diffeomorphism g_{A} in equation (3.53) restricts to the desired diffeomorphism of T.

From the definition of φ_{A} in equation (3.37) and of g_{A} in equation (3.53), we see that the composition φ_{A} ∘ g_{A} can be viewed as an S^{1}-equivariant map

φ_{A} ∘ g_{A} : S^{2k−1} → ℂ^{c},

where ℂ^{c} ≅ Coker D_{A,ϑ}, and thus a section of the vector bundle

S^{2k−1} ×_{S^{1}} ℂ^{c} → ℂℙ^{k−1}, (3.58)

where the S^{1} action is diagonal since (φ_{A} ∘ g_{A})(e^{iθ}z) = e^{iθ}(φ_{A} ∘ g_{A})(z), for z ∈ S^{2k−1} and e^{iθ} ∈ S^{1}. Because the action is diagonal, the Euler class of this bundle is h^{c} (see [10, Lemma 3.27] for a further explanation of the sign).

By Lemma 3.25, the diffeomorphism g_{A} defines an orientation-preserving diffeomorphism from ℂℙ^{k−1} to γ_{A}^{−1} (V(z)) ∩ K_{A,δ}/S^{1} (with the orientation ω(K, o)/ω(V) of (3.51)) if and only if ε(A) = 1. Recall that the orientation (ω(K, o)/ω(Z)) /ω(V) from (3.51) of Z_{A} ∩ γ_{A}^{−1} (V(z)) ∩ K_{A,δ}/S^{1} is given by the orientation ω(K, o)/ω(V) of γ_{A}^{−1} (V(z)) ∩ K_{A,δ}/S^{1} and the orientation ω(Z) of the normal bundle of Z_{A}/S^{1}. Thus, if T is oriented as the Poincaré dual of h^{c} and thus has the orientation determined by the complex orientation of ℂℙ^{k−1} and the complex orientation of the normal bundle of T, then the restriction of g_{A} to T is orientation preserving if and only if ε(A) = 1.

The final tool needed to compute intersection numbers with L_{𝔱,κ}^{w} in equation (3.60) is the following description of γ_{A}^{−1}(W):

Lemma 3.28. Continue the notation and assumptions of Lemmas 3.22, 3.25, and 3.27. Then (γ_{A} ∘ g_{A})^{∗}μ_{c} = 2h, where μ_{c} is the cohomology class (3.12).

Proof. Recall that μ_{c} is the first Chern class of the line bundle 𝕃_{𝔱} in definition (3.10). The embedding γ_{A} and the map g_{A} are S^{1}-equivariant so, noting that 𝕃_{𝔱} is defined by the S^{1} action in equation (3.11), we have

(γ_{A} ∘ g_{A})^{∗}𝕃_{𝔱} ≅ S^{2k−1} ×_{(S^{1},×−2)} ℂ → S^{2k−1}/S^{1}, (3.59)

The bundle (3.59) has first Chern class 2h, the sign being positive because the S^{1} action is diagonal (see [10, Lemma 3.27]).

Using Lemma 3.28 we can prove the assertion of Lemma 3.15 that ι(M̄_{κ}^{w}) ⊂ W̄:

Proof of Assertion (2d) in Lemma 3.15. Lemma 3.28 shows that W will have non-trivial intersection with the normal cone of any point in ι(M_{κ}^{w}) ⊂ M_{𝔱}, where by “normal cone” we mean γ_{A}(Z_{A} ∩ {0} ⊕ Ker D_{A,ϑ})/S^{1}. Therefore, the closure of W will contain all points in ι(M_{κ}^{w}) and thus ι(M̄_{κ}^{w}) ⊂ W̄.

We can now compute the intersection with the link.

Proposition 3.29. Let 𝔱 be a spin^{u} structure on a four-manifold X, with w an integral lift of w_{2}(𝔱) and w (mod 2) is good. We further assume that d_{a}(𝔱) = dim M_{κ}^{w} ≥ 0 and that n_{a}(𝔱) = Index_{ℂ} D_{A} > 0. Let δ_{c} be a non-negative integer such that

deg(z) + 2δ_{c} = d_{a} + 2n_{a} − 2 = dim(M_{𝔱}^{∗,0}/S^{1}) − 1.

Suppose z ∈ 𝔸(X) has degree deg(z) ≥ d_{a} and is intersection-suitable. If L_{𝔱,κ}^{w,ε} ∩ ι(M_{κ}^{w}) is oriented as the boundary of M_{𝔱}^{∗,≥ε}/S^{1}, where M_{𝔱}^{∗,0}/S^{1} is given the orientation O^{asd}(Ω, w), then there is a positive constant ε_{0} such that for generic ε ∈ (0, ε_{0}),

(3.60) # (V̄(z) ∩ W̄^{δ_{c}} ∩ L_{𝔱,κ}^{w,ε}) = \begin{cases}2^{n_{a}−1}#(V̄(z) ∩ M̄_{κ}^{w}), & if deg(z) = d_{a}, \\ 0, & if deg(z) > d_{a}. \end{cases}

Moreover, these intersection numbers are independent of the choice of generic ε < ε_{0}.

Proof. By Assertion (2) of Lemma 3.22, the pairing (3.60) is a sum of local terms,

\begin{matrix}# (V̄(z) ∩ W̄^{δ_{c}} ∩ L_{𝔱,κ}^{w,ε}) \\ = ∑_{[A]∈V̄(z)∩M̄_{κ}^{w}} # (V̄(z) ∩ W̄^{δ_{c}} ∩ L_{𝔱,κ}^{w,ε} ∩ γ_{A} (O′_{A})) , \end{matrix} (3.61)

where O′_{A} is the neighborhood defined in Lemma 3.22. If deg(z) > dim M_{κ}^{w}, the intersection V̄(z) ∩ M̄_{κ}^{w} is empty and so the sum is trivial. Hence, we can assume deg(z) = dim M_{κ}^{w}, so δ_{c} = n_{a}(𝔱) − 1. Let c = dim_{ℂ} Coker D_{A,ϑ}, so k = dim Ker D_{A,ϑ} = n_{a} + c. If V̄(z) has multiplicity q (in the sense of Definition 3.4), then we can evaluate the terms in the sum in equation (3.61) as

# (V̄(z) ∩ W̄^{n_{a}−1} ∩ L_{𝔱,κ}^{w,ε} ∩ γ_{A} (O′_{A}))

= # (γ_{A}^{−1} (V(z) ∩ W̄^{n_{a}−1} ∩ ℓ^{−1}(ε)) ∩ Z_{A}/S^{1}) (Equations (3.41) & (3.43))

= # (γ_{A}^{−1} \binom{}{¯W^{n_{a}−1}} ∩ γ_{A}^{−1} (sV (z)) ∩ K_{A,δ} ∩ Z_{A}/S^{1}) (Lemma 3.26)

= q# ((γ_{A} ∘ g_{A})^{−1} \binom{}{¯W^{n_{a}−1}} ∩ g_{A}^{−1}(Z_{A}/S^{1}) ∩ ℂℙ^{n_{a}+c−1}) (Lemma 3.25)

= qε(A) ⟨(2h)^{n_{a}−1} ⌣ h^{c}, [ℂℙ^{n_{a}+c−1}]⟩ (Lemmas 3.27 and 3.28)

= qε(A)2^{n_{a}−1}.

Hence, equation (3.61) simplifies to give

# (V̄(z) ∩ W̄^{δ_{c}} ∩ L_{𝔱,κ}^{w,ε}) = q2^{n_{a}−1} ∑_{[A]∈V̄(z)∩M̄_{κ}^{w}} ε(A) = 2^{n_{a}−1}# (V̄(z) ∩ M̄_{κ}^{w}) ,

completing the proof of the proposition.

As an application of Lemma 3.27, we explain why the moduli space M_{𝔱} contains solutions to the PU(2) monopole equations [10, Equation (2.32)] which are distinct from the anti-self-dual or reducible solutions. Lemma 3.27 yields the following analogue of Taubes’ existence theorem for solutions to the anti-self-dual equation for SO(3) connections:

Proposition 3.30. Let 𝔱 be a spin^{u} structure on a four-manifold X, where we allow b_{2}^{+}(X) ≥ 0, and suppose w_{2}(𝔱) ≡ w (mod 2), for w ∈ H^{2}(X; ℤ). Assume that w (mod 2) is good. If n_{a}(𝔱) > 0, then for a generic, C^{∞} pair (g, ρ), consisting of a Riemannian metric and Clifford map, and generic, C^{∞} parameters (τ, ϑ), the moduli space M_{𝔱}^{∗,0}(g, ρ, τ, ϑ) of irreducible, non-zero-section PU(2) monopoles is non-empty if the moduli space M_{κ}^{w,∗} of irreducible, anti-self-dual SO(3) connections on 𝔤_{𝔱} is non-empty.

Proof. This follows from the fact that the Euler class of the obstruction bundle in the Kuranishi model of [A, 0] ∈ M_{𝔱} is non-trivial by Lemma 3.27.

3.5. The cobordism formula. If z ∈ 𝔸(X) is intersection-suitable and deg(z) + 2δ_{c} = dim(M_{𝔱}^{∗,0}/S^{1}) − 1 = d_{a} + 2n_{a} − 2, then Corollary 3.18 tells us that the intersection

V̄(z) ∩ W̄^{δ_{c}} ∩ M̄_{𝔱}^{∗,≥ε}/S^{1} (3.62)

is a union of smooth one-dimensional manifolds in M_{𝔱}^{∗,≥ε}/S^{1}. The boundaries of these onemanifolds will lie either on L_{𝔱,κ}^{w} or in a neighborhood of some reducible monopole, possibly in a lower level. Proposition 3.29 describes the intersection of this family of one-manifolds with the component L_{𝔱,κ}^{w} of the boundary of M̄_{𝔱}^{∗,≥ε}/S^{1}. In particular, we see there are finitely many points in this boundary.

If w_{2}(𝔱) is good then (noting that we always assume b_{2}^{+}(X) > 0), for any splitting 𝔱 = 𝔰 ⊕ 𝔰 ⊗ L, the space M_{𝔰} contains no zero-section monopoles, [10, Corollary 3.3]. In [10, Definition 3.22] we constructed a homology class [L_{𝔱,𝔰}] of the link of the family of reducibles M_{𝔰} contained in the top level M_{𝔱}^{0}/S^{1}. By Lemma 2.6, the orientations O^{asd}(Ω, w) and O^{red}(Ω, 𝔱, 𝔰) differ by \frac{1}{4}(w − c_{1}(L))^{2}. From the definition of the geometric representatives and Lemma 2.9, we have:

Lemma 3.31. Let 𝔱 be a spin^{u} structure over a four-manifold X, with w an integral lift of w_{2}(𝔱). Assume that w (mod 2) is good. If z ∈ 𝔸(X) is intersection-suitable and deg(z) + 2δ_{c} = d_{a} + 2n_{a} − 2, then

# (V̄(z) ∩ W̄^{δ_{c}} ∩ L_{𝔱,𝔰}) = (−1)\frac{1}{4}(w − c_{1}(L))^{2}⟨μ_{p}(z) ⌣ μ_{c}^{δ_{c}}, [L_{𝔱,𝔰}]⟩,

where L_{𝔱,𝔰} is given the boundary orientation determined by O^{asd}(Ω, w) on the left hand side and the complex orientation of Definition 2.7 on the right hand side of the above identity.

We now characterize the spin^{c} structures, 𝔰, for which 𝔱 = 𝔰 ⊕ 𝔰′.

Lemma 3.32. A spin^{u} structure 𝔱 on X admits a splitting, 𝔱 = 𝔰 ⊕ 𝔰′, if and only if

(c_{1}(𝔱) − c_{1}(𝔰))^{2} = p_{1}(𝔱). (3.63)

Proof. Assume 𝔱 = 𝔰⊕𝔰′. We may write 𝔰′ = 𝔰⊗L for some line bundle L, with 𝔱 = (ρ, V) and 𝔰 = (ρ, W). Then V = W ⊗E, where E = \underline{ℂ}⊕L, and 𝔤_{𝔱} ≅ i\underline{ℝ}⊕L. Thus, c_{1}(L) = c_{1}(𝔱)−c_{1}(𝔰) obeys c_{1}(L)^{2} = p_{1}(𝔤_{𝔱}) = p_{1}(𝔱), as desired.

Conversely, suppose c_{1}(𝔱) − c_{1}(𝔰) obeys condition (3.63). Let L be a complex line bundle with c_{1}(L) = c_{1}(𝔱)−c_{1}(𝔰). From Lemma 2.3 in [10] we know that V ≅ W ⊗E for a complex rank-two bundle E determined up to isomorphism by 𝔰 and 𝔱. Then

c_{1}(E) = \frac{1}{2}c_{1}(V^{+}) − c_{1}(W^{+}) = c_{1}(𝔱) − c_{1}(𝔰) = c_{1}(L),

while

c_{2}(E) = −\frac{1}{4}(p_{1}(𝔰𝔲(E)) − c_{1}(E)^{2}) = −\frac{1}{4}(p_{1}(𝔱) − c_{1}(L)^{2}) = 0,

where the final equality follows from the fact that p_{1}(𝔱) = c_{1}(L)^{2} by hypothesis. Hence, E ≅ \underline{ℂ} ⊕ L and so V ≅ W ⊕ W ⊗ L, as desired.

If 𝔰 is a spin^{c} structure with c_{1}(𝔰) obeying condition (3.63), there is a topological embedding M_{𝔰} ↪ M_{𝔱} of M_{𝔰} into the top level of M̄_{𝔱} [10, Lemma 3.13]. More generally, if c_{1}(𝔰) obeys

(c_{1}(𝔱) − c_{1}(𝔰))^{2} = p_{1}(𝔱) + 4ℓ, (3.64)

for some non-negative integer ℓ, then there is a topological embedding of M_{𝔰} into the lowerlevel PU(2)-monopole moduli space M_{𝔱_{ℓ}} × Sym^{ℓ}(X), where 𝔱_{ℓ} is a spin^{u} structure with p_{1}(𝔱_{ℓ}) = p_{1}(𝔱) + 4ℓ [10, Equation (2.44)].

If the reducibles in M̄_{𝔱} appear only in the top Uhlenbeck level M_{𝔱}, then M̄_{𝔱}^{∗,≥ε}/S^{1} is a cobordism between the link L_{𝔱,κ}^{w,ε} of the stratum defined by the anti-self-dual moduli space, ι(M_{κ}^{w}), and the links L_{𝔱,𝔰} of the strata of reducibles, ι(M_{𝔰}). Counting the points in the boundary of the oriented, one-dimensional manifold (3.62) then gives the identity:

# (V̄(z) ∩ W^{δ_{c}} ∩ L_{𝔱,κ}^{w}) = − ∑_{{𝔰: 𝔰⊕𝔰′=𝔱}} # (V̄(z) ∩ W^{δ_{c}} ∩ L_{𝔱,𝔰}) . (3.65)

Let w be an integral lift of w_{2}(𝔱) defining the orientation O^{asd}(Ω, w) of M_{𝔱}^{∗,0}/S^{1}. Lemmas 2.6 and 2.9 imply that the orientation O^{asd}(Ω, w) and the complex orientation for the link L_{𝔱,𝔰} differ by

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

(3.66)

Equation (3.65), Proposition 3.29 and Lemma 3.31 then yield the following result.

Theorem 3.33. Let 𝔱 be a spin^{u} structure on an oriented, smooth four-manifold X with b_{2}^{+}(X) > 0 and w_{2}(𝔱) ≡ w (mod 2), for w ∈ H^{2}(X; ℤ). Assume that w (mod 2) is good. Suppose z ∈ 𝔸(X) has degree

d_{a}(𝔱) ≤ deg(z) ≤ d_{a}(𝔱) + 2n_{a}(𝔱) − 2, (3.67)

and is intersection-suitable. Assume that the set of isomorphism classes of spin^{c} structures, 𝔰 ∈ Spin^{c}(X), defining reducible PU(2) monopoles in M̄_{𝔱} all obey condition (3.63), and so non-empty Seiberg-Witten moduli strata ι(M_{𝔰}) appear only in the top level, M_{𝔱}.

(a) If for all 𝔰 ∈ Spin^{c}(X) with M_{𝔰} non-empty we have

(c_{1}(𝔱) − c_{1}(𝔰))^{2} < p_{1}(𝔱), (3.68)

so M̄_{𝔱} contains no reducible monopoles, then

# (V̄(z) ∩ M̄_{κ}^{w}(X)) = 0. (3.69)

(b) If deg(z) = d_{a} and o_{𝔱}(w, 𝔰) is as defined in equation (3.66), then

# (V̄(z) ∩ M̄_{κ}^{w}(X)) = −2^{1−n_{a}} ∑_{{𝔰: 𝔰⊕𝔰′=𝔱}}(−1)^{o_{𝔱}(w,𝔰)}⟨μ_{p}(z) ⌣ μ_{c}^{n_{a}−1}, [L_{𝔱,𝔰}]⟩, (3.70)

where the class [L_{𝔱,𝔰}] is defined by the complex orientation of Definition 2.7 on L_{𝔱,𝔰}.

(c) If d_{a} < deg(z) ≤ d_{a} + 2n_{a} − 2 and δ_{c} ∈ ℕ is defined by deg(z) + δ_{c} = d_{a} + 2n_{a} − 2, then

∑_{{𝔰: 𝔰⊕𝔰′=𝔱}}(−1)^{o_{𝔱}(w,𝔰)}⟨μ_{p}(z) ⌣ μ_{c}^{δ_{c}}, [L_{𝔱,𝔰}]⟩ = 0. (3.71)

The sums in equations (3.70) and (3.71) are necessarily finite, because there are only finitely many spin^{c} structures 𝔰 with M_{𝔰} non-empty and thus L_{𝔱,𝔰} non-empty. Theorem 3.33 can be strengthened to a more useful form if we assume

Conjecture 3.34. [8, Conjecture 3.1] Continue the notation of Theorem 3.33. Suppose that 𝔱_{ℓ} = 𝔰⊕𝔰′, where p_{1}(𝔱_{ℓ}) = p_{1}(𝔱)+4ℓ and ι(M_{𝔰}) is contained in the level M_{𝔱_{ℓ}}×Sym^{ℓ}(X), for some natural number ℓ ≥ 0. Then the pairing #(V̄(z) ∩ W̄^{δ_{c}} ∩ L_{𝔱,𝔰}) is a multiple of SW_{X}(𝔰) and thus vanishes if the Seiberg-Witten function SW_{X,𝔰} is trivial.

See §4.1 for a definition of the Seiberg-Witten invariants. The motivation for this conjecture is discussed in [16] and almost certainly does hold, one of our current goals being to provide a complete proof in the near future. The difficulty lies in the construction of the link L_{𝔱,𝔰} of a Seiberg-Witten moduli space when ℓ = ℓ(𝔱, 𝔰) > 0. We show that Conjecture 3.34 holds when ℓ = 0 in Theorem 4.13 and when ℓ = 1 in [14]. By adapting Leness’s proof of the wall-crossing formula in [36], we can also see that the conjecture holds when ℓ = 2.

Corollary 3.35. Given Conjecture 3.34, we can relax the hypothesis of Theorem 3.33 — that non-empty Seiberg-Witten moduli spaces M_{𝔰} appear only in the top level M_{𝔱}—to the weaker requirement that Seiberg-Witten moduli spaces M_{𝔰} with non-trivial Seiberg-Witten functions SW_{X,𝔰} appear only in the top level M_{𝔱}. Then the conclusions of Theorem 3.33 hold without change.

Remark 3.36. The Seiberg-Witten stratum ι(M_{𝔰}) corresponding to a splitting 𝔱_{ℓ} = 𝔰 ⊕ 𝔰′ lies in level

ℓ(𝔱, 𝔰) = \frac{1}{8}(d_{a} − 2r(Λ, 𝔰))

of the space of ideal PU(2) monopoles containing M̄_{𝔱}, from the definition (1.12) of r(Λ, 𝔰), where d_{a} is the dimension of the anti-self-dual moduli space M_{κ}^{w} ↪ M_{𝔱}. Then, by definition (1.12) of r(Λ), the Seiberg-Witten strata with non-trivial invariants are contained only in levels ℓ of this space of ideal PU(2) monopoles, where

0 ≤ ℓ ≤ \frac{1}{8}(d_{a} − 2r(Λ)).

See the proof of Theorem 1.2 in §4.6.

4. Intersection with the link of a stratum of top-level reducibles

In this section we calculate the pairings appearing on the right-hand-side of equation (3.70) in Theorem 3.33 under the additional assumption that for all α, α′ ∈ H^{1}(X; ℤ) one has α ∪ α′ = 0. We begin by giving a definition of the Seiberg-Witten invariants which is appropriate for the perturbations we use in our version of the Seiberg-Witten equations [10, §2.3]. Recall that the link L_{𝔱,𝔰} constructed in [10, §3.5] is diffeomorphic via a map γ to the zero-locus of a section of an obstruction bundle γ^{∗}ΞΞ/S^{1} over the complex projectivization of a bundle N_{𝔱}(Ξ, 𝔰) → M_{𝔰}. Thus, in §4.2, we calculate pullbacks of the classes μ_{p} and μ_{c} by γ to ℙN_{𝔱}(Ξ, 𝔰). In §4.3 we compute the total Segre class—the formal inverse of the total Chern class as defined in Lemma 4.10—of the virtual normal bundle N_{𝔱}(Ξ, 𝔰) of M_{𝔰} and the Euler class of the obstruction bundle γ^{∗}Ξ/S^{1}. Finally, in §4.4 we perform the actual computation and complete the proofs of Theorems 1.1, 1.2, 1.4, and Corollary 1.5.

4.1. A definition of the Seiberg-Witten invariants. In this subsection we give a definition of the Seiberg-Witten invariants for a closed, smooth four-manifold X; we allow b_{1}(X) ≥ 0 and b_{2}^{+}(X) ≥ 1.

Recall that X is equipped with an orientation for which b_{2}^{+}(X) > 0 and that we have fixed an orientation for H^{1}(X; ℝ) ⊕ H^{+}(X; ℝ): the Seiberg-Witten moduli spaces are then oriented according to the conventions of [46, §6.6] (or see our §2.3). In [10, §2.4.2] we defined a universal complex line bundle,

𝕃_{𝔰} = C̃_{𝔰}^{0} ×_{G_{𝔰}} \underline{ℂ} → C_{𝔰}^{0} × X, (4.1)

where \underline{ℂ} = X × ℂ and the action of G_{𝔰} is given for s ∈ G_{𝔰}, x ∈ X and z ∈ ℂ by

(s, (B, Ψ), (x, z)) ↦ (s(B, Ψ), (x, s(x)^{−1}z)) . (4.2)

We then defined cohomology classes on C_{𝔰}^{0} by

μ_{𝔰} : H_{•}(X; ℝ) → H^{2−•}(C_{𝔰}^{0}; ℝ), μ_{𝔰}(α) = c_{1}(𝕃_{𝔰})/α, (4.3)

where α is either the positive generator x ∈ H_{0}(X; ℤ) or a class γ ∈ H_{1}(X; ℝ).

If z ∈ 𝔹(X) is a monomial α_{1} ⋯ α_{p} with α_{i} ∈ H_{0}(X; ℤ) or H_{1}(X; ℝ), then it has total degree deg(z) = ∑_{i=1}^{p} deg(α_{i}). If z = x^{m}γ_{1} ⋯ γ_{n} ∈ 𝔹(X), we set

μ_{𝔰}(z) = \underbrace{μ_{𝔰}(x) ⌣ ⋯ ⌣ μ_{𝔰}(x)}_{m times} ⌣ μ_{𝔰}(γ_{1}) ⌣ ⋯ ⌣ μ_{𝔰}(γ_{n}), (4.4)

and define μ_{𝔰}(z) for arbitrary elements z ∈ 𝔹(X) by ℝ-linearity.

Let X̃ = X#\overline{ℂℙ}^{2} be 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 §4.5 for a more detailed explanation of the relation between spin^{c} and spin^{u} structures on X and X̃, as well as the blow-up formula for Seiberg-Witten invariants, which we shall invoke below.) One easily checks that dim M_{𝔰^{±}}(X̃) = dim M_{𝔰}(X), where the Seiberg-Witten moduli spaces M_{𝔰}(X) and M_{𝔰^{±}}(X̃) are defined in [10, Equation (2.55) & Lemma 3.12] with perturbation η = F^{+}(A_{Λ}). Here, 2A_{Λ} is the fixed connection on det(V^{+}) ≅ det(Ṽ^{+}) and 𝔱̃ = (ρ̃, Ṽ) is the spin^{u} structure on X̃ defined in Lemma 4.19, with c_{1}(𝔱̃) = c_{1}(𝔱), p_{1}(𝔱̃) = p_{1}(𝔱) − 1, and w_{2}(𝔱̃) = w_{2}(𝔱) + PD[e]. Now

c_{1}(𝔰^{±}) − c_{1}(𝔱̃) = 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 g on X and related metrics on the connect sum X̃—the moduli spaces M_{𝔰^{±}}(X̃) contain no zero-section pairs [46, Proposition 6.3.1]. Thus, for our choice of generic perturbations, the moduli spaces M_{𝔰^{±}}(X̃) are compact, oriented, smooth manifolds.

Noting that 𝔹(X) ≅ 𝔹(X̃), we define the Seiberg-Witten invariants for (X, 𝔰) as an ℝ-linear function (1.7) by setting

SW_{X,𝔰}(z) = ⟨μ_{𝔰^{+}}(z), [M_{𝔰^{+}}(X̃)]⟩, (4.5)

with SW_{X,𝔰}(z) = 0 when deg(z) ≠ dim M_{𝔰}(X). The blow-up formula for Seiberg-Witten invariants (Theorem 4.20) implies that

⟨μ_{𝔰}(z), [M_{𝔰}(X)]⟩ = ⟨μ_{𝔰^{±}}(z), [M_{𝔰^{±}}(X̃)]⟩, (4.6)

when the pairing on the left is well-defined, that is, when M_{𝔰}(X) contains no zero-section monopoles. For example, with our version of the Seiberg-Witten equations [10, Equation (2.55)], this situation arises when c_{1}(𝔰)−Λ ∈ H^{2}(X; ℤ) is not a torsion class and thus M_{𝔰}(X) contains no zero-section pairs if the metric g is generic and b_{2}^{+}(X) > 0 [46, Proposition 6.3.1]. Therefore, our definition of the Seiberg-Witten invariants coincides with the usual one [46] in this case, but has the advantage that it is valid even when c_{1}(𝔰) − Λ is torsion and one cannot perturb the Seiberg-Witten equations by a generic two-form η (see Remark 2.14 in [10]). When b_{2}^{+}(X) > 1, the pairing on the right-hand-side of equation (4.5) is independent of the metric, [46, Lemma 6.7.1].

When b_{2}^{+}(X) = 1, however, the pairing on the right-hand-side of definition (4.5) depends on the period point ω(g̃) of the metric g̃ on X̃, as in the case of the Donaldson invariants (see §3.4.2). To explain this dependence when the Seiberg-Witten moduli spaces are defined as in [10, §2.3] with the perturbation parameters η = F^{+}(A_{Λ}) described above, we note that the moduli space M_{𝔰^{+}}(g̃) contains zero-section pairs if and only if the period point ω(g̃) lies on the (c_{1}(𝔰^{+}) − Λ)-wall in the positive cone of H^{2}(X̃; ℝ), that is

ω(g̃) ⌣ (c_{1}(𝔰^{+}) − Λ) = 0. (4.7)

When ω(g̃) does not lie on the wall, the pairing in definition (4.5) may depend on the sign of ω(g̃) ⌣ (c_{1}(𝔰^{+}) − Λ). The chambers for the Seiberg-Witten invariants of M_{𝔰^{+}} are thus connected components of the complement of the (c_{1}(𝔰^{+}) − Λ)-wall in the positive cone of H^{2}(X̃; ℝ), which we call (c_{1}(𝔰^{+}) − Λ)-chambers.

By an argument which is the same as the one we gave for the Donaldson invariants in §3.4.2, if c_{1}(𝔰) − Λ is not torsion then each (c_{1}(𝔰) − Λ)-chamber in the positive cone of H^{2}(X; ℝ) is contained in a unique (c_{1}(𝔰^{+})−Λ)-chamber in the positive cone of H^{2}(X̃; ℝ) ≅ ℝ[e]⊕H^{2}(X; ℝ), the related chamber. The Seiberg-Witten invariant associated to a (c_{1}(𝔰)− Λ)-chamber is then defined by evaluating the pairing in equation (4.5) with a metric whose period point lies in the related (c_{1}(𝔰^{+}) − Λ)-chamber.

Suppose w_{2}(X)−Λ (mod 2) is good, in the sense of Definition 3.20. For any spin^{c} structure 𝔰 over X, we have c_{1}(𝔰) ≡ w_{2}(X) (mod 2) and so c_{1}(𝔰)−Λ (mod 2) is good. Then c_{1}(𝔰)−Λ is not torsion and the Seiberg-Witten invariants for M_{𝔰} depend only on the metric g through the (c_{1}(𝔰) − Λ)-chamber for ω(g).

If w_{2}(X) − Λ (mod 2) is not good, then c_{1}(𝔰) − Λ may be torsion and in this situation

ω(g̃) ⌣ (c_{1}(𝔰^{+}) − Λ) = ω(g̃) ⌣ PD[e],

so the sign of the cup-product would depend on the sign of ω(g̃) ⌣ PD[e], which converges to zero as the neck is stretched. Hence, the definition (4.5) of the Seiberg-Witten invariant in this case requires a more delicate analysis of the sign of ω(g̃) ⌣ PD[e] as the length of the neck converges to infinity [59], which we shall not consider here.

Thus, when b_{2}^{+}(X) = 1, we shall assume that w_{2}(X) − Λ (mod 2) is good. Since

w ≡ w_{2}(X) − Λ (mod 2),

this coincides with the constraint we used to define the Donaldson invariants in §3.4.2. We now compare the Donaldson and Seiberg-Witten chamber structures:

Lemma 4.1. Let 𝔱 be a spin^{u} structure on a four-manifold X with b_{2}^{+}(X) = 1, where w is an integral lift of w_{2}(𝔱) and w (mod 2) is good, and c_{1}(𝔱) = Λ. Then there is a one-to-one correspondence between the set of (w, p_{1}(𝔱))-walls and the set of (c_{1}(𝔰) − Λ)-walls, where M_{𝔰} is contained in the space of ideal PU(2) monopoles, ∪_{ℓ=0}^{∞} (M_{𝔱_{ℓ}} × Sym^{ℓ}(X)).

Proof. A (w, p_{1}(𝔱))-wall is defined by class α ∈ H^{2}(X; ℤ) with α ≡ w (mod 2) and α^{2} = p_{1}(𝔱) + 4ℓ for ℓ ≥ 0 (see equation (3.33)). Because α is an integral lift of w_{2}(𝔱), the class Λ − α is characteristic. Hence, there is a spin^{c} structure 𝔰 with c_{1}(𝔰) = Λ − α. By Lemma 3.32 and the identity α^{2} = p_{1}(𝔱) + 4ℓ, a spin^{u} structure 𝔱_{ℓ} with c_{1}(𝔱_{ℓ}) = Λ and p_{1}(𝔱_{ℓ}) = p_{1}(𝔱) + 4ℓ admits a splitting 𝔱_{ℓ} = 𝔰 ⊕ 𝔰′ for any such spin^{c} structures. Conversely, given a spin^{c} structure 𝔰 with M_{𝔰} contained in the space of ideal monopoles, Lemma 3.32 implies that (c_{1}(𝔰) − Λ)^{2} = p_{1}(𝔱) + 4ℓ and c_{1}(𝔰) − Λ ≡ w_{2}(𝔱) (mod 2), so the (c_{1}(𝔰) − Λ)-wall is a (w, p_{1}(𝔱))-wall.

Remark 4.2. Lemma 4.1 implies that in formulas such as (1.20) and (1.21) derived using the cobordism M_{𝔱}^{∗,0}/S^{1} to compare Donaldson and Seiberg-Witten invariants, if a period point ω(g) crosses a wall for the Donaldson invariant, it will also cross a wall for one of the Seiberg-Witten invariants in the formula; thus both sides of the identity will change.

4.2. Pullbacks of cohomology classes to the link of a stratum of reducibles. We now compute the pullbacks of the cohomology classes μ_{p}(z) and μ_{c} by γ : N_{𝔱}^{ε}(Ξ, 𝔰)/S^{1} → C_{𝔱}/S^{1}, the restriction of the S^{1}-equivariant embedding defined in [10, Equation (3.44)] to the ε-sphere of the bundle N_{𝔱}(Ξ, 𝔰). The result will be expressed in terms of the Seiberg-Witten μ_{𝔰}-classes and one additional cohomology class:

Definition 4.3. Let ν ∈ H^{2}(ℙN_{𝔱}(Ξ, 𝔰); ℤ) be the negative of the first Chern class of the S^{1} bundle N_{𝔱}^{ε}(Ξ, 𝔰) → ℙN_{𝔱}(Ξ, 𝔰). Restricted to each fiber of ℙN_{𝔱}(Ξ, 𝔰), the class ν is the positive generator of the cohomology. With the conventions of [24, §3.1], the class ν is the first Chern class of the line bundle O_{ℙN_{𝔱}(Ξ,𝔰)}(1), the dual of the tautological bundle.

Let Ñ_{𝔱}(Ξ, 𝔰) → M̃_{𝔰} be the pullback of N_{𝔱}(Ξ, 𝔰) by the projection M̃_{𝔰} → M_{𝔰}. To compute the pullbacks by γ of the cohomology classes μ_{p}(β) to ℙN_{𝔱}(Ξ, 𝔰), we first compute the pullback of the universal bundle 𝔽_{𝔱} defined in equation (3.2).

Lemma 4.4. Let 𝔱 be a spin^{u} structure which admits a splitting 𝔱 = 𝔰 ⊕ 𝔰 ⊗ L. Assume M_{𝔰} contains no zero-section pairs. Let γ : N_{𝔱}^{ε}(Ξ, 𝔰) ↪ C_{𝔱} be the embedding constructed in [10, §3.5.3], and let Ñ_{𝔱}^{ε}(Ξ, 𝔰) denote the ε-sphere bundle of Ñ_{𝔱}(Ξ, 𝔰). Then, we have an isomorphism of SO(3) bundles over ℙN_{𝔱}(Ξ, 𝔰) × X,

(γ × id_{X})^{∗}𝔽_{𝔱} ≅ Ñ_{𝔱}^{ε}(Ξ, 𝔰) ×_{G_{𝔰}×S^{1}} (i\underline{ℝ} ⊕ L), (4.8) where s ∈ G_{𝔰} and e^{iθ} ∈ S^{1} act on Ñ_{𝔱}^{ε}(Ξ, 𝔰) × (i\underline{ℝ} ⊕ L) by

((B, Ψ, η), f ⊕ z) ↦ (s(B, Ψ, e^{iθ}η), f ⊕ s^{−2}e^{iθ}z) , (4.9)

where (B, Ψ, η) ∈ Ñ_{𝔱}(Ξ, 𝔰), (B, Ψ) ∈ M̃_{𝔰}, and f ⊕ z ∈ i\underline{ℝ} ⊕ L.

Proof. Since V = W ⊕ W ⊗ L, where 𝔰 = (ρ, W) and 𝔱 = (ρ, V), we have an isomorphism of SO(3) bundles 𝔤_{𝔱} ≅ i\underline{ℝ} ⊕ L and the definition (3.2) of 𝔽_{𝔱} yields isomorphism of SO(3) bundles over C_{𝔱}^{∗} × X,

𝔽_{𝔱} ≅ C̃_{𝔱}^{∗} ×_{G_{𝔱}×S^{1}} (i\underline{ℝ} ⊕ L).

From [10, §3.5.4] we recall that the embedding γ : N_{𝔱}(Ξ, 𝔰) → C_{𝔱} lifts to a map

γ̃ : Ñ_{𝔱}(Ξ, 𝔰) → C̃_{𝔱},

which is G_{𝔰} equivariant when s ∈ G_{𝔰} acts on C̃_{𝔱} via the embedding

ϱ : G_{𝔰} ↪ G_{𝔱}, s ↦ ϱ(s) = s id_{W} ⊕ s^{−1}id_{W ⊗L}, (4.10)

while s acts on the base M̃_{𝔰} by the usual action of G_{𝔰} and on the fibers of Ñ_{𝔱}(Ξ, 𝔰) → M̃_{𝔰} by the action on L_{k}^{2}(Λ^{1} ⊗ L) ⊕ L_{k}^{2}(W^{+} ⊗ L) induced by the isomorphisms 𝔤_{𝔱} ≅ i\underline{ℝ} ⊕ L and V = W ⊕ W ⊗ L and the action of ϱ(s) on V.

We also recall from [10, §3.5.4] that the map γ̃ is S^{1} equivariant with respect to the action on the complex fibers of Ñ_{𝔱}(Ξ, 𝔰) by scalar multiplication and the trivial action on the base M̃_{𝔰}, while S^{1} acts on C̃_{𝔱} through

(4.11) ϱ_{L} : S^{1} × V → V, where ϱ_{L}(e^{iθ}) = id_{W} ⊕ e^{iθ} id_{W ⊗L}.

Therefore, we have an isomorphism of SO(3) bundles:

(γ̃ × id_{X})^{∗} (C̃_{𝔱}^{∗} ×_{G_{𝔱}} (i\underline{ℝ} ⊕ L)) ≅ Ñ_{𝔱}(Ξ, 𝔰) ×_{G_{𝔰}} (i\underline{ℝ} ⊕ L).

We obtain (γ̃ × id_{X})^{∗}𝔽_{𝔱} on the left above after we take the S^{1} quotient, with S^{1} acting on C̃_{𝔱} through complex multiplication on V and trivially on 𝔤_{𝔱}. Given

[A, Φ, f, z] = [γ̃(B, Ψ, η), f, z] ∈ C̃_{𝔱}^{∗} ×_{G_{𝔱}} (i\underline{ℝ} ⊕ L),

and noting that

e^{iθ} id_{V} = ϱ(e^{iθ})ϱ_{L}(e^{2iθ}), e^{iθ} ∈ S^{1}, (4.12)

then we can identify the pull-back of the S^{1} action:

[e^{iθ}(A, Φ), f, z] = [e^{iθ}γ̃(B, Ψ, η), f, z]
= [ϱ(e^{iθ})ϱ_{L}(e^{2iθ})γ̃(B, Ψ, η), f, z] (Equation (4.12))
= [ϱ(e^{iθ})γ̃(B, Ψ, e^{2iθ}η), f, z] (see [10, §3.5.4])
= [γ̃(B, Ψ, e^{2iθ}η), ϱ(e^{−iθ})(f, z)]
= [γ̃(B, Ψ, e^{i2θ}η), f, e^{2iθ}z] (see [10, §3.4.2]).

The final equality follows from the observation that the action of s ∈ G_{𝔰} induced by the embedding ϱ : G_{𝔰} → G_{𝔱}, the homomorphism Ad : Aut(V) → 𝔰𝔲(V), the projection 𝔰𝔲(V) → 𝔤_{𝔱}, and the isomorphism 𝔤_{𝔱} ≅ i\underline{ℝ} ⊕ L, is given by (f, z) ↦ (f, s^{−2}z); see [10, §3.4.2] for details.

Lemma 4.4 shows that we can compute γ^{∗}p_{1}(𝔽_{𝔱}) ∈ H^{4}(ℙN_{𝔱}(Ξ, 𝔰) × X; ℝ) once we know the Chern class of the line-bundle component of the SO(3) bundle (4.8):

Lemma 4.5. Continue the hypotheses of Lemma 4.4 and let ν be the cohomology class in Definition 4.3. Then the complex line bundle,

Ñ_{𝔱}^{ε}(Ξ, 𝔰) ×_{G_{𝔰}×S^{1}} L → ℙN_{𝔱}(Ξ, 𝔰) × X, (4.13)

has first Chern class

π_{ℙN}^{∗}ν + 2(π_{𝔰} × π_{X})^{∗}c_{1}(𝕃_{𝔰}) + π_{X}^{∗}c_{1}(L) ∈ H^{2}(ℙN_{𝔱}(Ξ, 𝔰) × X; ℤ),

where π_{ℙn}, π_{𝔰} and π_{X} are the projections from ℙN_{𝔱}(Ξ, 𝔰) × X to ℙN_{𝔱}(Ξ, 𝔰), M_{𝔰}, and X, respectively, and 𝕃_{𝔰} is the universal Seiberg-Witten line bundle (4.1).

Proof. The projection Ñ_{𝔰}^{ε}(Ξ, 𝔰) → ℙN_{𝔱}(Ξ, 𝔰) is a principal G_{𝔰} × S^{1} bundle, where S^{1} acts by scalar multiplication on the fibers of Ñ_{𝔱}(Ξ, 𝔰). One has an isomorphism of G_{𝔰} × S^{1} bundles

Ñ_{𝔱}^{ε}(Ξ, 𝔰) ≅ N_{𝔱}^{ε}(Ξ, 𝔰) ×_{M_{𝔰}} M̃_{𝔰}, (4.14)

defined, for (B, Ψ) ∈ M̃_{𝔰} and (B, Ψ, η) ∈ Ñ_{𝔱}^{ε}(Ξ, 𝔰), by the map

(B, Ψ, η) ↦ ([B, Ψ, η], (B, Ψ)) ,

where the square brackets indicate equivalence modulo G_{𝔰}. Applying the isomorphism (4.14) to the G_{𝔰} × S^{1} bundle (4.13) yields an isomorphism of complex line bundles,

Ñ_{𝔱}^{ε}(Ξ, 𝔰) ×_{G_{𝔰}×S^{1}} L ≅ (N_{𝔱}^{ε}(Ξ, 𝔰) ×_{M_{𝔰}} (M̃_{𝔰} ×_{G_{𝔰}} L))/S^{1} (4.15)

where, as in definition (4.9), an element s ∈ G_{𝔰} acts on M̃_{𝔰}×L by (B, Ψ, z) ↦ (s(B, Ψ), s^{−2}z) and S^{1} acts diagonally on N_{𝔱}^{ε}(Ξ, 𝔰) × L.

The isomorphism in [10, Equation (6.68)] gives

M̃_{𝔰} ×_{G_{𝔰}} L ≅ 𝕃_{𝔰}^{⊗2} ⊗ π_{X}^{∗}L. (4.16)

Substituting the isomorphism (4.16) into equation (4.15) yields an isomorphism of complex line bundles,

Ñ_{𝔱}^{ε}(Ξ, 𝔰) ×_{G_{𝔰}×S^{1}} L ≅ (N_{𝔱}^{ε}(Ξ, 𝔰) ×_{M_{𝔰}} 𝕃_{𝔰}^{⊗2} ⊗ π_{X}^{∗}L)/S^{1}. (4.17)

The proof is completed by applying Lemma 3.27 in [10] to compute the first Chern class of a fiber product with an S^{1} action, the observation that the S^{1} action in (4.17) is diagonal, and the fact that ν = −c_{1}(N_{𝔱}^{ε}(Ξ, 𝔰)).

The reduction of (γ × id_{X})^{∗}𝔽_{𝔱} in Lemma 4.4, the computation in Lemma 4.5, and the fact that c_{1}(L) = c_{1}(𝔱) − c_{1}(𝔰) give the following expression for (γ × id_{X})^{∗}p_{1}(𝔽_{𝔱}).

Corollary 4.6. Continue the hypotheses and notation of Lemmas 4.4 and 4.5. Then,

\begin{aligned}(γ × id_{X})^{∗}p_{1}(𝔽_{𝔱}) & = (π_{ℙN}^{∗}ν + 2(π_{𝔰} × π_{X})^{∗}c_{1}(𝕃_{𝔰}) + π_{X}^{∗}(c_{1}(𝔱) − c_{1}(𝔰)))^{2} \\ & ∈ H^{2}(ℙN_{𝔱}(Ξ, 𝔰) × X; ℤ). \end{aligned} (4.18)

We compute the pullbacks of the cohomology classes μ_{p}(β) in C_{𝔱}^{∗,0}/S^{1} to ℙN_{𝔱}(Ξ, 𝔰):

Corollary 4.7. Continue the hypotheses of Lemma 4.4. Let {γ_{i}} be a basis for H_{1}(X; ℤ)/ Tor and let {γ_{i}^{∗}} be the dual basis for H^{1}(X; ℤ). Then if x ∈ H_{0}(X; ℤ) is the positive generator, γ ∈ H_{1}(X; ℝ), h ∈ H_{2}(X; ℝ), and [Y] ∈ H_{3}(X; ℝ), the pullbacks of the cohomology classes μ_{p}(β) in H^{•}(C_{𝔱}^{∗,0}/S^{1}; ℝ) by the embedding γ : ℙN_{𝔱}(Ξ, 𝔰) ↪ C_{𝔱}^{∗,0}/S^{1} to cohomology classes γ^{∗}μ_{p}(β) in H^{•}(ℙN_{𝔱}(Ξ, 𝔰); ℝ) are given by

\begin{aligned}γ^{∗}μ_{p}([Y]) & = ∑_{i=1}^{b_{1}(X)} ⟨(c_{1}(𝔰) − c_{1}(𝔱)) ⌣ γ_{i}^{∗}, [Y]⟩ μ_{𝔰}(γ_{i}), \\ γ^{∗}μ_{p}(h) & = \frac{1}{2}⟨c_{1}(𝔰) − c_{1}(𝔱), h⟩ (2μ_{𝔰}(x) + ν) − 2 ∑_{i<j}⟨γ_{i}^{∗} ⌣ γ_{j}^{∗}, h⟩μ_{𝔰}(γ_{i}γ_{j}), \\ γ^{∗}μ_{p}(γ) & = − ∑_{i=1}^{b_{1}(X)}⟨γ_{i}^{∗}, γ⟩ (2μ_{𝔰}(x) + ν)) ⌣ μ_{𝔰}(γ_{i}), \\ γ^{∗}μ_{p}(x) & = −\frac{1}{4} (2μ_{𝔰}(x) + ν)^{2} , \end{aligned} (4.19)

where we have written μ_{𝔰}(β) for the pullback of this class to ℙN_{𝔱}(Ξ, 𝔰).

Proof. Recall from [10, Lemma 2.24] that

c_{1}(𝕃_{𝔰}) = μ_{𝔰}(x) × 1 + ∑_{i=1}^{b_{1}(X)} μ_{𝔰}(γ_{i}) × γ_{i}^{∗} ∈ H^{2}(C_{𝔰}^{0} × X; ℝ).

The identities (4.19) then follow from equation (4.18), the definition (3.3) of the cohomology classes μ_{p}(β) = −\frac{1}{4}p_{1}(𝔽_{𝔱})/β, and standard computations (compare the proof of [7, Proposition 5.1.21]).

Finally, we compute the pullback of the class μ_{c} in C_{𝔱}^{∗,0}/S^{1} to ℙN_{𝔱}(Ξ, 𝔰):

Lemma 4.8. Continue the hypotheses of Lemma 4.4 and let ν be the cohomology class in Definition 4.3. Then

γ^{∗}μ_{c} = ν ∈ H^{•}(ℙN_{𝔱}(Ξ, 𝔰); ℝ). (4.20)

Proof. We compute c_{1}(γ^{∗}𝕃_{𝔱}), where 𝕃_{𝔱} is the line bundle (3.10) with c_{1}(𝕃_{𝔱}) = μ_{c}, so

𝕃_{𝔱} = (C_{𝔱}^{∗,0} × ℂ)/S^{1},

with circle action given by

(e^{iθ}, ([A, Φ], z)) → (e^{iθ}[A, Φ], e^{2iθ}z) = (ϱ_{L}(e^{2iθ})[A, Φ], e^{2iθ}z) , (4.21)

where ϱ_{L} is given by definition (4.11) and the preceding equality follows from the relation (4.12) between the circle actions. The embedding γ : N_{𝔱}^{ε}(Ξ, 𝔰) → C_{𝔱}^{∗,0} is S^{1} equivariant with respect to scalar multiplication on the fibers of N_{𝔱}(Ξ, 𝔰) and the action induced by ϱ_{L} : S^{1} ×V → V on C_{𝔱}^{∗,0}. Hence, by definition of 𝕃_{𝔱} and the S^{1} equivariance of γ, we obtain an isomorphism of complex line bundles

γ^{∗}𝕃_{𝔱} ≅ (N_{𝔱}^{ε}(Ξ, 𝔰) × ℂ)/S^{1},

where the circle acts by

(e^{iθ}, ([B, Ψ, η], z)) ↦ ([B, Ψ, e^{iθ}η), e^{iθ}z).

Therefore, γ^{∗}μ_{c} = γ^{∗}c_{1}(𝕃_{𝔱}) = c_{1}(γ^{∗}𝕃_{𝔱}) = ν, as desired.

Corollary 4.7 completes the proof of Lemma 3.15:

Proof of Assertions 2 (a), (b) and (c) in Lemma 3.15. Because the class ν is non-trivial on the fiber of the projection ℙN_{𝔱}(Ξ, 𝔰) → M_{𝔰}, the closures of the geometric representatives V(β), V(γ), and V(x) dual to the cohomology classes μ_{p}(β), μ_{p}(γ), and μ_{p}(x) must contain each point in ι(M_{𝔰}). Hence, the closures V̄(β), V̄(γ), and V̄(x) contain ι(M_{𝔰}).

4.3. Euler and Segre classes. In [10, Equation (3.48)], the homology class of the link is given by

[L_{𝔱,𝔰}] = e(γ^{∗}(Ξ/S^{1})) ∩ [ℙN_{𝔱}(Ξ, 𝔰)],

where Ξ/S^{1} is an obstruction bundle over an open neighborhood of ι(M_{𝔰}) ⊂ C_{𝔱}, as defined in [10, Theorem 3.19]. In this section, we compute the Euler class of this obstruction bundle and then compute the Segre classes of N_{𝔱}(Ξ, 𝔰) in order to relate intersection pairings on ℙN_{𝔱}(Ξ, 𝔰) with pairings on M_{𝔰}.

The obstruction bundle Ξ is given by

Ξ ≅ U × ℂ^{r_{Ξ}} → U,

where (see [10, Theorem 3.19]) U is a neighborhood of M_{𝔰} in C_{𝔱} and the S^{1} action is given by, for [A, Φ] ∈ U, z ∈ ℂ^{r_{Ξ}}, and ϱ_{L} is the map (4.11),

([A, Φ], z) → (ϱ_{L}(e^{iθ})[A, Φ], e^{iθ}z) . (4.22)

Because the embedding γ : N_{𝔱}^{ε}(Ξ, 𝔰) → U ⊂ C_{𝔱} is S^{1} equivariant with respect to scalar multiplication on the fibers of N_{𝔱}(Ξ, 𝔰) and the action induced by the map ϱ_{L} on C_{𝔱}, we have an isomorphism:

γ^{∗} (Ξ/S^{1}) ≅ N_{𝔱}^{ε}(Ξ, 𝔰) ×_{(S^{1},×−1)} ℂ^{r_{Ξ}},

where the factor −1 indicates that the S^{1} action in equation (4.22) is diagonal. Thus, we can calculate the Euler class of γ^{∗}Ξ/S^{1}:

Lemma 4.9. The vector bundle γ^{∗}Ξ/S^{1} has Euler class e(γ^{∗}Ξ/S^{1}) = ν^{r_{Ξ}}, where ν is the cohomology class in Definition 4.3, r_{Ξ} = rank_{ℂ} Ξ, and

[L_{𝔱,𝔰}] = ν^{r_{Ξ}} ∩ [ℙN_{𝔱}(Ξ, 𝔰)],

where L_{𝔱,𝔰} is given the complex orientation of Definition 2.7 and [ℙN_{𝔱}(Ξ, 𝔰)] is given the orientation defined by the orientation of T M_{𝔰} and the complex orientation of the fibers.

Proof. The obstruction section vanishes transversely, so its zero locus, L_{𝔱,𝔰} is Poincaré dual to e(γ^{∗}Ξ/S^{1}) = ν^{r_{Ξ}} by [3, Proposition 12.8]. Note that the top Chern class is the Euler class associated to the complex orientation of the fibers of a complex vector bundle [43, Lemma 14.1 & Definition, p. 158].

Although the definition of Segre classes is well-known [24, p. 69], we include a definition here via the following lemma in order to make our conventions clear.

Lemma 4.10. Let N be a complex rank-r vector bundle with Chern classes c_{i} = c_{i}(N) over an oriented, real m-dimensional manifold M. Let N^{ε} ⊂ N be the associated ε-sphere bundle. Define Segre classes s_{i} = s_{i}(N) ∈ H^{2i}(X; ℤ) by the relation

(1 + c_{1} + c_{2} + ⋯ + c_{r})(s_{0} + s_{1} + ⋯) = 1. (4.23)

Let π : ℙ(N) → M be the projectivization of N and h the negative of the first Chern class of the bundle N^{ε} → ℙ(N). Then, for any α ∈ H^{m−2i}(M; ℤ),

⟨h^{r+i−1} ⌣ π^{∗}α, [ℙ(N)]⟩ = ⟨s_{i} ⌣ α, [M]⟩, (4.24)

where ℙ(N) is given the orientation arising from that of M and the complex orientation of the fibers of π.

Proof. The cohomology ring of ℙ(N) is given by [3, Equation (20.7)],

H^{•}(ℙ(N); ℤ) = π^{∗}H^{•}(M; ℤ)[h]/(h^{r} + π^{∗}c_{1}h^{r−1} + ⋯ + π^{∗}c_{r}).

We then have h^{r} = − ∑_{i=1}^{r} π^{∗}c_{i}h^{r−i}. Suppose α ∈ H^{m}(M; ℤ), so i = 0 in the assertion of the lemma, and α is dual to ⟨α, [M]⟩p ∈ H_{0}(M; ℤ), where p is a point. Then

⟨h^{r−1} ⌣ π^{∗}α, [ℙ(N)]⟩ = ⟨h^{r−1}, [π^{−1}(p)]⟩⟨α, [M]⟩
= ⟨h^{r−1}, ℂℙ^{r−1}]⟩⟨α, [M]⟩ = ⟨α, [M]⟩.

Because s_{0}(N) = 1 by equation (4.23), equation (4.24) holds for i = 0. We now use induction on i and consider α ∈ H^{m−2i}(M, ℤ):

⟨h^{r+i−1} ⌣ π^{∗}α, [ℙ(N)]⟩ = −⟨(∑_{j=1}^{i} π^{∗}c_{j}h^{r+i−1−j}) ⌣ π^{∗}α, [ℙ(N)]⟩

= − ∑_{j=1}^{i}⟨c_{j}s_{i−j} ⌣ α, [M]⟩ = ⟨s_{i} ⌣ α, [M]⟩.

The last equality follows from the identity (∑_{i} c_{i})(∑_{j} s_{j}) = 1, by equating degrees. This gives the desired relation.

Next we compute the Segre classes of N_{𝔱}(Ξ, 𝔰). Recall from [10, Equation (3.72)] that

\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} ((2c_{1}(𝔱) − c_{1}(𝔰))^{2} − σ) , \end{aligned} (4.25)

where n_{s} = n′_{s} + n′′_{s} is the index of the elliptic “normal deformation operator” for the Seiberg-Witten stratum ι(M_{𝔰}) ⊂ M_{𝔱}.

Lemma 4.11. Suppose that for all α, α′ ∈ H^{1}(X; ℤ) one has α ⌣ α′ = 0. Let 𝔱 be a spin^{u} structure with 𝔱 = 𝔰 ⊕ 𝔰 ⊗ L and assume M_{𝔰} contains no zero-section pairs. Then the bundle N_{𝔱}(Ξ, 𝔰) → M_{𝔰} has Segre classes

s_{i}(N_{𝔱}(Ξ, 𝔰)) = μ_{𝔰}(x)^{i} ∑_{j=1}^{i} 2^{j}\binom{−n′_{s}}{j}\binom{−n′′_{s}}{i − j}, i = 0, 1, 2, …. (4.26)

Proof. With the hypothesis on H^{1}(X; ℤ), Corollary 3.30 in [10] asserts that N_{𝔱}(Ξ, 𝔰) has total Chern class

c(N_{𝔱}(Ξ, 𝔰)) = (1 + 2μ_{𝔰}(x))^{n′_{s}}(1 + μ_{𝔰}(x))^{n′′_{s}}.

As described in Lemma 4.10, the total Segre class s = s_{0} +s_{1} +s_{2} +⋯ is the formal inverse of the total Chern class c = 1 + c_{1} + c_{2} + ⋯, so

s(N_{𝔱}(Ξ, 𝔰)) = (1 + 2μ_{𝔰}(x))^{−n′_{s}}(1 + μ_{𝔰}(x))^{−n′′_{s}}.

The lemma follows by computing the formal power series expansions for the above expression, using equation (2) in Lemma 4.16 to simplify before multiplying the two series.

Remark 4.12. The assumption α ⌣ α′ = 0 in Lemma 4.11 is used to simplify the expression for the Chern character of N_{𝔱}(Ξ, 𝔰) computed in [10, Theorem 3.29]. Without this assumption, the universal expression of the Segre classes in terms of the Chern character given in [39, Equation (2.11)] and Theorem 3.29 of [10] still show that the Segre classes of N_{𝔱}(Ξ, 𝔰) are expressible in terms of the μ_{𝔰}-classes, though not as explicitly.

4.4. Computation of the intersection pairing. We now compute intersection pairings with L_{𝔱,𝔰} of the type encountered in the cobordism formula (3.70). Some combinatorial factors appearing in this computation can be expressed in terms of the Jacobi polynomials [29, §8.96], which are defined by

P_{n}^{a,b}(ξ) = \frac{1}{2^{n}} ∑_{k} \binom{a + n}{n − k}\binom{n + b}{k}(ξ − 1)^{k}(ξ + 1)^{n−k}, ξ ∈ ℂ. (4.27)

Functional relations, relations with other special functions, and the generating function for the Jacobi polynomials can be found in [29, pp. 1034–1035].

Theorem 4.13. Let X be a four-manifold with α ⌣ α′ = 0 for every α, α′ ∈ H^{1}(X; ℤ), and let Ω be a homology orientation. Let 𝔰 and 𝔱 be a spin^{c} and spin^{u} structure on X for which 𝔱 = 𝔰 ⊕ 𝔰 ⊗ L, and assume M_{𝔰} contains no zero-section pairs. Give L_{𝔱,𝔰} the complex orientation, determined by the orientation for M_{𝔰} fixed by the homology orientation Ω as in Definition 2.5. Let z ∈ 𝔸(X) and let δ_{c} be a non-negative integer satisfying

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

If z = z′Y for some Y ∈ H_{3}(X; ℤ) and z′ ∈ 𝔸(X), then

⟨μ_{p}(z) ⌣ μ_{c}^{δ_{c}}, [L_{𝔱,𝔰}]⟩ = 0. (4.28)

If z = x^{δ_{0}}ϑh^{δ_{2}}, where h ∈ H_{2}(X; ℝ), ϑ ∈ Λ^{δ_{1}}(H_{1}(X; ℝ)), and x ∈ H_{0}(X; ℤ) is the positive generator, then d_{s}(𝔰) ≡ δ_{1} (mod 2) and if we set

2d = d_{s}(𝔰) − δ_{1}, (4.29)

then

\begin{aligned}⟨μ_{p}(z) ⌣ μ_{c}^{δ_{c}}, [L_{𝔱,𝔰}]⟩ & = (−1)^{δ_{0}+δ_{1}}2^{−δ_{2}−2δ_{0}}C_{χ,σ} (deg(z), δ_{c}, d_{a}(𝔱), d_{s}(𝔰), δ_{1}) \\ & × ⟨μ_{𝔰}(x^{d}ϑ), [M_{𝔰}]⟩ ⟨c_{1}(𝔰) − c_{1}(𝔱), h⟩^{δ_{2}}, \end{aligned} (4.30)

where

C_{χ,σ} (deg(z), δ_{c}, d_{a}(𝔱), d_{s}(𝔰), δ_{1}) = (−2)^{d}P_{d}^{a,b}(0),

for

a = δ_{c} − d − 1 and b = \frac{1}{2} (deg(z) − d_{a}(𝔱) − d_{s}(𝔰)) − \frac{1}{4} (χ + σ) .

If d = 0, then C_{χ,σ} = 1.

Remark 4.14. If δ_{1} > d_{s}(𝔰), then the pairing ⟨μ_{𝔰}(x^{d}ϑ), [M_{𝔰}]⟩ vanishes and so the pairing (4.30) also vanishes.

Proof of Theorem 4.13. By the multilinearity of the pairing, we can assume that ϑ is a monomial, ϑ = γ_{1} ⋯ γ_{δ_{1}}, where {γ_{1}, …, γ_{δ_{1}}} is a subset of a basis for H_{1}(X; ℤ)/ Tor. Extend it to a basis and let {γ_{i}^{∗}} be a dual basis for H^{1}(X; ℝ), so ⟨γ_{i}^{∗}, γ_{j}⟩ = δ_{ij}.

Suppose z = z′Y for Y ∈ H_{3}(X; ℤ) and z′ ∈ 𝔸(X). The expression for μ_{p}([Y]) in equation (4.19) is a sum of terms of the form

⟨(c_{1}(𝔰) − c_{1}(𝔱)) ⌣ γ_{i}^{∗}, [Y]⟩μ_{𝔰}(γ_{i}) = ⟨(c_{1}(𝔰) − c_{1}(𝔱)) ⌣ γ_{i}^{∗} ⌣ PD[Y], [X]⟩μ_{𝔰}(γ_{i}),

which vanish by our hypothesis on H^{1}(X; ℤ). This yields identity (4.28).

The integers δ_{i} and δ_{c} satisfy

\begin{aligned}2δ_{2} + 3δ_{1} + 4δ_{0} + 2δ_{c} & = deg(z) + 2δ_{c} = dim(M_{𝔱}^{∗,0}/S^{1}) − 1 \\ & = d_{s}(𝔰) + 2n′_{s} + 2n′′_{s} − 2. \end{aligned} (4.31)

Thus δ_{1} ≡ d_{s}(𝔰) (mod 2) and d = \frac{1}{2}(d_{s}(𝔰) − δ_{1}) is an integer. We use Corollary 4.7, the comments following Lemma 3.23 in [10], and Lemma 4.9 to write the pairing in equation (4.30) as

\begin{matrix}⟨μ_{p}(h^{δ_{2}}ϑx^{δ_{0}})μ_{c}^{δ_{c}}, [L_{𝔱,𝔰}]⟩ \\ = 2^{−δ_{2}}⟨c_{1}(𝔰) − c_{1}(𝔱), h⟩^{δ_{2}} \\ × ⟨(2μ_{𝔰}(x) + ν)^{δ_{2}} (∏_{k=1}^{δ_{1}}(−2μ_{𝔰}(x) − ν)μ_{𝔰}(γ_{k})) (−\frac{1}{4}(2μ_{𝔰}(x) + ν)^{2})^{δ_{0}} ν^{δ_{c}}, \\ ν^{r_{Ξ}} ∩ [ℙN_{𝔱}(Ξ, 𝔰)]⟩ . \end{matrix} (4.32)

Write C_{1} = (−1)^{δ_{1}+δ_{0}}2^{−δ_{2}−2δ_{0}}⟨c_{1}(𝔰) − c_{1}(𝔱), h⟩^{δ_{2}} and define

δ_{p} = \frac{1}{2}(deg(z) − δ_{1}) = δ_{2} + δ_{1} + 2δ_{0}. (4.33)

The pairing (4.32) is then equal to

\begin{matrix}C_{1} ⟨(2μ_{𝔰}(x) + ν)^{δ_{p}}μ_{𝔰}(ϑ)ν^{δ_{c}+r_{Ξ}}, [ℙN_{𝔱}(Ξ, 𝔰)]⟩ \\ = C_{1}⟨∑_{i=0}^{δ_{p}} 2^{i}\binom{δ_{p}}{i}μ_{𝔰}(x^{i}ϑ)ν^{δ_{p}+δ_{c}+r_{Ξ}−i}, [ℙN_{𝔱}(Ξ, 𝔰)]⟩. \end{matrix} (4.34)

Then μ_{𝔰}(x^{i}ϑ) ∈ H^{d_{s}(𝔰)−2(d−i)}(M_{𝔰}; ℤ) and

δ_{p} + δ_{c} + r_{Ξ} − i = δ_{2} + δ_{1} + 2δ_{0} + δ_{c} + r_{Ξ} − i (as δ_{p} = δ_{2} + δ_{1} + 2δ_{0})
= n′_{s} + n′′_{s} + r_{Ξ} + \frac{1}{2}(d_{s}(𝔰) − δ_{1}) − 1 − i (Equation (4.31))
= n′_{s} + n′′_{s} + r_{Ξ} + (d − i) − 1 (as d = \frac{1}{2}(d_{s}(𝔰) − δ_{1})).

We use the preceding equation, the Segre class relation (4.24), and the formulas (4.26) for the Segre classes s_{i}(N_{𝔱}(Ξ, 𝔰)) to calculate

\begin{aligned}⟨μ_{𝔰}(x^{i}ϑ)ν^{δ_{p}+δ_{c}+r_{Ξ}−i}, [ℙN_{𝔱}(Ξ, 𝔰)]⟩ & = ⟨μ_{𝔰}(x^{i}ϑ)s_{d−i}, [M_{𝔰}]⟩ \\ & = ∑_{j=0}^{d−i} 2^{j}\binom{−n′_{s}}{j}\binom{−n′′_{s}}{d − i − j} ⟨μ_{𝔰}(x^{d}ϑ), [M_{𝔰}]⟩ . \end{aligned} (4.35)

Writing C_{2} = C_{1}⟨μ_{𝔰}(x^{d}ϑ), [M_{𝔰}]⟩ and substituting the formula (4.35) into equation (4.34) yields a simplified expression for that pairing:

\begin{matrix}C_{2} ∑_{i=0}^{d} 2^{i}\binom{δ_{p}}{i} ∑_{j=0}^{d−i} 2^{j}\binom{−n′_{s}}{j}\binom{−n′′_{s}}{d − i − j} \\ = C_{2} ∑_{i=0}^{d} ∑_{j=0}^{d−i} 2^{i+j}\binom{δ_{p}}{i}\binom{−n′_{s}}{j}\binom{−n′′_{s}}{d − i − j}. \end{matrix} (4.36)

If we write u = i + j, then the pairing (4.36) becomes

\begin{aligned}C_{2} ∑_{u=0}^{d} ∑_{i=0}^{u} 2^{u}\binom{δ_{p}}{i}\binom{−n′_{s}}{u − i}\binom{−n′′_{s}}{d − u} & = C_{2} ∑_{u=0}^{d} 2^{u}\binom{−n′′_{s}}{d − u} ∑_{i=0}^{u} \binom{δ_{p}}{i}\binom{−n′_{s}}{u − i} \\ & = C_{2} ∑_{u=0}^{d} 2^{u}\binom{−n′′_{s}}{d − u}\binom{δ_{p} − n′_{s}}{u}, \end{aligned} (4.37)

where the second equality follows from the Vandermonde convolution identity (see equation (5) in Lemma 4.16). Equation (4.30) will follow from equation (4.37) and Lemma 4.18 which expresses the last sum in equation (4.37) in terms of the Jacobi polynomial given by C_{χ,σ} (deg(z), δ_{c}, d_{a}(𝔱), d_{s}(𝔰), δ_{1}). This completes the proof of the theorem.

Remark 4.15. The proof of Theorem 4.13 implies that Conjecture 3.1 of [8] holds for levelzero reducibles, even without the assumption that α ⌣ α′ = 0 for every α, α′ ∈ H^{1}(X; ℤ). We only used this condition on H^{1}(X; ℤ) in equation (4.35) in order to apply the Segre class computations of Lemma 4.11. If the condition on H^{1}(X; ℤ) is is omitted, then—as noted in Remark 4.12—the Segre classes can still be computed in terms of μ_{𝔰}-classes, though less explicitly. In the general situation, the pairing (4.30) could still be expressed in terms of a pairing of μ_{𝔰}-classes with M_{𝔰} and, when c_{1}(𝔰) is not a basic class, the pairing (4.30) would be zero.

Before proving the relation between the combinatorial expression in equation (4.37) and the Jacobi polynomial used at the end of the proof of Theorem 4.13, it is convenient to collect the combinatorial identities we shall need here. For a ∈ ℝ and n ∈ ℕ, define

(a)_{n} = a(a + 1) ⋯ (a + n − 1), and (a)_{0} = 1. (4.38)

We then have:

Lemma 4.16. [38, p. 9] Let a, b ∈ ℝ and let k, n ∈ ℕ be non-negative integers. Then:

(1)  (a)_{k} = (−1)^{k}(1 − a − k)_{k},
(2)  \binom{a}{k} = \frac{(−1)^{k}(−a)_{k}}{k!} ,
(3)  (a)_{n−k} = \frac{(−1)^{k}(a)_{n}}{(1 − a − n)_{k}} , n ≥ k,
(4)  (n − k)! = \frac{n!}{(−1)^{k}(−n)_{k}} ,

(5)  ∑_{i=0}^{u} \binom{a}{i}\binom{b}{u − i} = \binom{a + b}{u}.

Identity (5) in Lemma 4.16 (the Vandermonde convolution identity) follows by comparing coefficients in binomial expansions of the two sides of the identity (x + y)^{a}(x + y)^{b} = (x + y)^{a+b}.

Remark 4.17. Note that Identity (2) in Lemma 4.16 allows one to extend the definition of a binomial coefficient \binom{n}{r} to the case when n ≤ 0. If r = 0, the identity \binom{n}{r} = 1 still holds.

We now prove the relation between the combinatorial expression in equation (4.37) and the Jacobi polynomial.

54 PAUL M. N. FEEHAN AND THOMAS G. LENESS

Lemma 4.18. Continue the notation of Theorem 4.13. Then

∑_{u=0}^{d} 2^{u}\binom{−n′′_{s}}{d − u}\binom{δ_{p} − n′_{s}}{u} = (−2)^{d}P_{d}^{a,b}(0),

where δ_{p} is defined in equation (4.33) and

a = δ_{c} − d − 1 and b = \frac{1}{2} (deg(z) − d_{a}(𝔱) − d_{s}(𝔰)) − \frac{1}{4} (χ + σ) .

Proof. We first recall that the hypergeometric functions [29, §9.10] are defined by

2F_{1}(a, b; c; ξ) = ∑_{k=0}^{∞} \frac{(a)_{k}(b)_{k}}{(c)_{k}k!} ξ^{k}, ξ ∈ ℂ. (4.39)

We shall use the following identities [38, Equation (23), p. 40 & Equation (2), p. 274]:

\begin{aligned}_{2}F_{1}(−m, b; c; ξ) & = \frac{(b)_{m}(−1)^{m}ξ^{m}}{(c)_{m}}_{2}F_{1}(−m, 1 − m − c; 1 − m − b; ξ^{−1}), \\ P_{n}^{a,b}(ξ) & = \frac{(−1)^{n}(b + 1)_{n}}{n!}_{2}F_{1}(−n, n + a + b + 1; b + 1; \frac{1}{2}(1 + ξ)). \end{aligned} (4.40)

By equation (2) of Lemma 4.16, the combinatorial expression in equation (4.37) can be written as

\begin{aligned}C(n′′_{s}, n′_{s}, δ_{p}, d) & := ∑_{u=0}^{d} 2^{u}\binom{−n′′_{s}}{d − u}\binom{δ_{p} − n′_{s}}{u} \\ & = ∑_{u=0}^{d} 2^{u}(−1)^{d−u} \frac{(n′′_{s})_{d−u}}{(d − u)!}(−1)^{u} \frac{(n′_{s} − δ_{p})_{u}}{u!} . \end{aligned} (4.41)

Applying equation (3) of Lemma 4.16 to (n′′_{s})_{d−u} and equation (4) to (d − u)! yields

\begin{aligned}C(n′′_{s}, n′_{s}, δ_{p}, d) & = (−1)^{d} ∑_{u=0}^{d} 2^{u} \frac{(−1)^{u}(n′′_{s})_{d}(−1)^{u}(−d)_{u}(n′_{s} − δ_{p})_{u}}{(1 − n′′_{s} − d)_{u}(d)!u!} \\ & = \frac{(−1)^{d}(n′′_{s})_{d}}{(d)!} ∑_{u=0}^{d} \frac{(−d)_{u}(n′_{s} − δ_{p})_{u}}{(1 − n′′_{s} − d)_{u}u!} 2^{u} \\ & = \frac{(−1)^{d}(n′′_{s})_{d}}{(d)!}_{2}F_{1}(−d, n′_{s} − δ_{p}; 1 − n′′_{s} − d; 2). \end{aligned} (4.42)

Substituting the first identity in equation (4.40) into equation (4.42) gives

C(n′′_{s}, n′_{s}, δ_{p}, d) = \frac{(n′′_{s})_{d}2^{d}(n′_{s} − δ_{p})_{d}}{(d)!(1 − n′′_{s} − d)_{d}}_{2}F_{1}(−d, n′′_{s}; δ_{p} − n′_{s} − d + 1; \frac{1}{2}). (4.43)

By substituting equation (1) from Lemma 4.16 into equation (4.43) we obtain

C(n′′_{s}, n′_{s}, δ_{p}, d) = \frac{2^{d}(−1)^{d}(n′_{s} − δ_{p})_{d}}{(d)!}_{2}F_{1}(−d, n′′_{s}; δ_{p} − n′_{s} − d + 1; \frac{1}{2}). (4.44)

Applying the second identity in (4.40) to equation (4.44) yields

C(n′′_{s}, n′_{s}, δ_{p}, d) = \frac{2^{d}(n′_{s} − δ_{p})_{d}(d)!}{(d)!(δ_{p} − n′_{s} − d + 1)_{d}} P_{d}^{n′′_{s}+n′_{s}−δ_{p}−1,δ_{p}−n′_{s}−d}(0). (4.45)

By applying equation (1) from Lemma 4.16, we can then simplify the right-hand side of equation (4.45) to give

C(n′′_{s}, n′_{s}, δ_{p}, d) = (−2)^{d}P_{d}^{n′′_{s}+n′_{s}−δ_{p}−1,δ_{p}−n′_{s}−d}(0). (4.46)

Then, the equality

d_{s}(𝔰) + 2n′_{s} + 2n′′_{s} − 2 = dim(M_{𝔱}^{∗,0}/S^{1}) − 1 = deg(z) + 2δ_{c} = 2δ_{p} + δ_{1} + 2δ_{c}

and d_{s}(𝔰) = 2d + δ_{1} imply

n′′_{s} + n′_{s} − δ_{p} − 1 = δ_{c} − d − 1. (4.47)

The definition (4.25) of n′_{s} implies that

n′_{s} = \frac{1}{2}d_{a}(𝔱) + \frac{1}{4}(χ + σ),

which, together with the identities d = \frac{1}{2}(d_{s}(𝔰) − δ_{1}) and δ_{p} = \frac{1}{2}(deg(z) − δ_{1}), yields

δ_{p} − n′_{s} − d = \frac{1}{2} (deg(z) − d_{a}(𝔱) − d_{s}(𝔰)) − \frac{1}{4} (χ + σ) . (4.48)

Substituting equations (4.47) and (4.48) into equation (4.46) then completes the proof.

4.5. A blow-up formula for Seiberg-Witten link pairings. Our formula (4.30) in Theorem 4.13 for pairings with Seiberg-Witten links [L_{𝔱,𝔰}] only applies when M_{𝔰} contains no zero-section pairs. In the same vein, our formula (3.70) in Theorem 3.33 for #(V̄(z) ∩ M̄_{κ}^{w}) in terms of pairings with [L_{𝔱,𝔰}] only applies when w_{2}(𝔱) ≡ w (mod 2) and w (mod 2) is good; when 𝔱 = 𝔰 ⊕ 𝔰′ and w_{2}(𝔱) is good, then M_{𝔰} contains no zero-section pairs. The definition (3.31) of the Donaldson invariants D_{X}^{w}(z) incorporates the blow-up formula in order to remove any such constraint on w. Therefore, we derive a “blow-up” formula for the Seiberg-Witten link pairings which, together with equations (3.70) and (4.30), will allow us to compute D_{X}^{w}(z) for arbitrary w ∈ H^{2}(X; ℤ).

As before, we let X̃ = X#\overline{ℂℙ}^{2} be 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). We first need to relate spin^{c} structures on X with those on X̃. Because \overline{ℂℙ}^{2} is simply-connected, the following map is a bijection:

Spin^{c}(\overline{ℂℙ}^{2}) → \{(2k − 1) PD[e] : k ∈ ℤ\} ⊂ H^{2}(\overline{ℂℙ}^{2}; ℤ), 𝔰 ↦ c_{1}(𝔰).

Let 𝔰_{2k−1} denote the spinc structure on \overline{ℂℙ}^{2} with c_{1}(𝔰_{2k−1}) = (2k − 1) PD[e]. By the discussion in [52, §12.4], a spin^{c} structure 𝔰 on X and 𝔰_{2k−1} on \overline{ℂℙ}^{2} can be spliced together to yield a spin^{c} structure 𝔰#𝔰_{2k−1} on X̃ with

c_{1}(𝔰#𝔰_{2k−1}) = c_{1}(𝔰) + (2k − 1) PD[e]. (4.49)

Moreover, every spin^{c} structure on X̃ can be realized in this way. The dimensions of the Seiberg-Witten moduli spaces are related by

\begin{aligned}d_{s}(𝔰#𝔰_{2k−1}) & = \frac{1}{4} (c_{1}(𝔰#𝔰_{2k−1})^{2} − 2χ̃ − 3σ̃) \\ & = d_{s}(𝔰) − k(k − 1), \end{aligned} (4.50)

where χ̃ = χ + 1 denotes the Euler characteristic of X̃ and σ̃ = σ − 1 is the signature of X̃. We now define a spin^{u} structure 𝔱̃ on X̃ related to a spin^{u} structure 𝔱 on X, and relate reducible PU(2) monopoles in M̄𝔱̃ to those in M̄_{𝔱}.

Lemma 4.19. Let 𝔱 be a spin^{u} structure on X with the property that reducible PU(2) monopoles in M̄_{𝔱} appear only in the top level M_{𝔱}. For X̃ = X#\overline{ℂℙ}^{2}, let e ∈ H_{2}(X̃; ℤ) be the exceptional class, and let PD[e] ∈ H^{2}(X̃; ℤ) be its Poincaré dual. Then we have:

1. There is a spin^{u} structure 𝔱̃ on X̃ satisfying

c_{1}(𝔱̃) = c_{1}(𝔱), p_{1}(𝔱̃) = p_{1}(𝔱) − 1, and w_{2}(𝔱̃) ≡ w_{2}(𝔱) + PD[e] (mod 2).

2. The reducible PU(2) monopoles in M̄𝔱̃ appear only in the top level, M𝔱̃, and are defined by spin^{c} structures 𝔰^{±} on X̃ with c_{1}(𝔰^{±}) = c_{1}(𝔰) ± PD[e], where ι(M_{𝔰}) ⊂ M_{𝔱}.

3. Suppose we relax the assumption that reducible PU(2) monopoles in M̄_{𝔱} appear only in the top level M_{𝔱}, to the assumption that reducible PU(2) monopoles in M̄_{𝔱} with non-trivial Seiberg-Witten functions appear only in the top level M_{𝔱}. Then reducible PU(2) monopoles in M̄𝔱̃ with non-trivial Seiberg-Witten functions appear only in the top-level M𝔱̃.

Proof. Suppose 𝔱 = (ρ, V). By Lemma 2.3 in [10], we may assume V = W ⊗ E, where 𝔰 = (ρ, W) is a spin^{c} structure on X and E → X is a complex rank-two vector bundle. Let Ẽ → X̃ be the complex, rank-two bundle with c_{1}(Ẽ) = c_{1}(E) + PD[e] and c_{2}(Ẽ) = c_{2}(E), and let 𝔰̃ = (ρ̃, W̃) be the spin^{c} structure on X̃ with c_{1}(𝔰̃) = c_{1}(𝔰) − PD[e]. Then set Ṽ = W̃ ⊗ Ẽ and 𝔱̃ = (ρ̃, Ṽ), and observe that 𝔱̃ has the desired characteristic classes.

By the neck-stretching argument described in [19], the only non-empty Seiberg-Witten moduli spaces on X̃ are defined by spin^{c} structures 𝔰#𝔰_{2k−1}, where M_{𝔰} is non-empty. Since ι(M_{𝔰}) ⊂ M_{𝔱} by hypothesis, equation (3.63) implies that c_{1}(𝔰) obeys

(c_{1}(𝔰) − c_{1}(𝔱))^{2} = p_{1}(𝔱).

To see which Seiberg-Witten moduli spaces ι(M_{𝔰#𝔰_{2k−1}}) can be contained in M̄_{𝔱} and in which level, we need to check the corresponding equation for c_{1}(𝔰#𝔰_{2k−1}). Equation (4.49) and the relations between the characteristic classes of 𝔱 and 𝔱̃ yields

(c_{1}(𝔰#𝔰_{2k−1}) − c_{1}(𝔱̃))^{2} = (c_{1}(𝔰) + (2k − 1) PD[e] − c_{1}(𝔱))^{2}
= p_{1}(𝔱) − (2k − 1)^{2}
= p_{1}(𝔱̃) − 4k(k − 1).

Restricted to integers, the function −4k(k −1) takes its maximum value at k = 0 and k = 1. Hence, only the spaces ι(M_{𝔰#𝔰_{2k−1}}) with k = 0, 1, appear in M̄𝔱̃, as all other spin^{c} structures 𝔰#𝔰_{2k−1} would require an SO(3) bundle with Pontrjagin class smaller than p_{1}(𝔱̃).

Theorem 4.20. [50, Theorem 3.2], [19, Theorem 1.4] Let X be a four-manifold, and let X̃ = X#\overline{ℂℙ}^{2} denote its blow-up, with exceptional class e ∈ H_{2}(X̃; ℤ). If b_{2}^{+}(X) > 1, then for each spin^{c} structure 𝔰̃ on X̃ with d_{s}(𝔰̃) ≥ 0 and each z ∈ 𝔹(X) ≅ 𝔹(X̃), we have

SW_{X̃,𝔰̃}(z) = SW_{X,𝔰}(x^{m}z), (4.51)

where 𝔰 is the spin^{c} structure induced on X by restriction, and 2m = d_{s}(𝔰) − d_{s}(𝔰̃). If b_{2}^{+}(X) = 1 and c_{1}(𝔰) − Λ is not torsion, there is a one-to-one correspondence between (c_{1}(𝔰)−Λ)-chambers in the positive cone of H^{2}(X; ℝ) and (c_{1}(𝔰̃)−Λ)-chambers in the positive cone of H^{2}(X̃; ℝ), and the above relation holds provided both invariants are calculated in related chambers.

Remark 4.21. The presence of the class Λ in the hypotheses of Theorem 4.20 when describing the chambers arises because of the nature of the fixed perturbation used in our definition of the Seiberg-Witten moduli spaces; see the discussion in §4.1.

Lemma 4.19, the blow-up formula for Seiberg-Witten invariants (Theorem 4.20), and Theorem 3.33 then yield the following “blow-up” formula for Seiberg-Witten link pairings:

Proposition 4.22. Continue the hypotheses and notation of Theorem 4.13 leading to equation (4.30), except we omit the requirement that M_{𝔰} contains no zero-section pairs and define z as given below. Let X̃ = X#\overline{ℂℙ}^{2} be the blow-up, let e ∈ H_{2}(X̃; ℤ) be the exceptional class, and let PD[e] ∈ H^{2}(X̃; ℤ) be its Poincaré dual. Let z = x^{δ_{0}}ϑh^{δ_{2}−k} ∈ 𝔸(X) ⊂ 𝔸(X̃). Let 𝔱 and 𝔱̃ be related spin^{u} structures on X and X̃, respectively, as in Lemma 4.19. Then, for k even,

(−1)^{o𝔱̃(w+PD[e],𝔰^{+})} ⟨μ_{p}(e^{k+1}z) ⌣ μ_{c}^{δ_{c}}, [L_{𝔱̃,𝔰^{+}}]⟩
+(−1)^{o𝔱̃(w+PD[e],𝔰^{−})} ⟨μ_{p}(e^{k+1}z) ⌣ μ_{c}^{δ_{c}}, [L_{𝔱̃,𝔰^{−}}]⟩
= (−1)^{o_{𝔱}(w,𝔰)+δ_{0}+δ_{1}}2^{−δ_{2}−2δ_{0}}C_{χ,σ} (deg(z) + 2k, δ_{c}, d_{a}(𝔱), d_{s}(𝔰), δ_{1})
× SW_{X,𝔰}(x^{d}ϑ)⟨c_{1}(𝔰) − c_{1}(𝔱), h⟩^{δ_{2}−k},

(4.52)

while the left-hand side is zero if k is odd or z is replaced by z′Y and Y ∈ H_{3}(X; ℤ).

Proof. The result follows by applying equation (4.30) in Theorem 4.13 to the links L_{𝔱̃,𝔰^{±}} of M_{𝔰^{±}}(X̃) in M𝔱̃(X̃), together with the following observations.

The vanishing result in the case z = z′Y follows immediately from equation (4.28).

Because d_{s}(𝔰^{±}) = d_{s}(𝔰) by equation (4.50), and d_{a}(𝔱̃) = d_{a}(𝔱) + 2 by equation (3.21) (noting that p_{1}(𝔱) = p_{1}(𝔱̃) − 1 from Lemma 4.19), and χ̃ + σ̃ = χ + σ, we have:

C_{χ̃,σ̃}(deg(ze^{2k+1}), δ_{c}, d_{a}(𝔱̃), d_{s}(𝔰^{±}), δ_{1}) = C_{χ,σ} (deg(z) + 2k, δ_{c}, d_{a}(𝔱), d_{s}(𝔰), δ_{1}) . (4.53)

The proof (see [19, §4]) of the blow-up formula, Theorem 4.20, gives an identity

⟨μ_{𝔰^{+}}(x^{d}ϑ), [M_{𝔰^{+}}(X̃)]⟩ = ⟨μ_{𝔰^{−}}(x^{d}ϑ), [M_{𝔰^{−}}(X̃)]⟩ ,

and thus our definition (4.5) of the Seiberg-Witten invariants yields

⟨μ_{𝔰^{±}}(x^{d}ϑ), [M_{𝔰^{±}}(X̃)]⟩ = SW_{X,𝔰}(x^{d}ϑ).

Noting that c_{1}(𝔱̃) = c_{1}(𝔱) by Lemma 4.19, the product

⟨c_{1}(𝔰^{±}) − c_{1}(𝔱̃), e⟩^{k+1}⟨c_{1}(𝔰^{±}) − c_{1}(𝔱), h⟩^{δ_{2}−k}

appearing in equation (4.30) can be simplified to

⟨c_{1}(𝔰^{±}) − c_{1}(𝔱̃), e⟩^{k+1}⟨c_{1}(𝔰^{±}) − c_{1}(𝔱̃), h⟩^{δ_{2}−k} = (∓1)^{k+1}⟨c_{1}(𝔰) − c_{1}(𝔱), h⟩^{δ_{2}−k}. (4.54)

From its definition (3.66), the orientation term is given by

\begin{aligned}o𝔱̃(w + PD[e], 𝔰^{±}) & = \frac{1}{4} (w + PD[e] − c_{1}(𝔱̃) + c_{1}(𝔰^{±}))^{2} \\ & = \frac{1}{4} (w − c_{1}(𝔱) + c_{1}(𝔰))^{2} − \frac{1}{4}(1 ± 1)^{2} \\ & = o_{𝔱}(w, 𝔰) − \frac{1}{4}(1 ± 1)^{2}. \end{aligned} (4.55)

Hence, applying equation (4.30) to the pair L_{𝔱̃,𝔰^{±}}, using equations (4.54) and (4.55) to compute the sign differences between the pairings with L_{𝔱̃,𝔰^{+}} and L_{𝔱̃,𝔰^{−}}, and using equation (4.53) to relate the constants yields

(−1)^{o𝔱̃(w+PD[e],𝔰^{+})} ⟨μ_{p}(e^{k+1}z) ⌣ μ_{c}^{δ_{c}}, [L_{𝔱̃,𝔰^{+}}]⟩

+(−1)^{o𝔱̃(w+PD[e],𝔰^{−})} ⟨μ_{p}(e^{k+1}z) ⌣ μ_{c}^{δ_{c}}, [L_{𝔱̃,𝔰^{−}}]⟩

= ((−1)(−1)^{k+1} + 1) (−1)^{o_{𝔱}(w,𝔰)+δ_{0}+δ_{1}}2^{−δ_{2}−1−2δ_{0}}C_{χ,σ} (deg(z) + 2k, δ_{c}, d_{a}(𝔱), d_{s}(𝔰), δ_{1})

× SW_{X,𝔰}(x^{d}ϑ)⟨c_{1}(𝔰) − c_{1}(𝔱), h⟩^{δ_{2}−k}.

Since (−1)(−1)^{k+1} +1 = 2 if k is even and is zero if k is odd, the preceding equation reduces to the desired formula (4.52).

4.6. Proofs of main theorems. Combining Theorems 3.33 and 4.13 and a brief discussion will complete the proofs of Theorems 1.1, 1.2, 1.4, and Corollary 1.5:

Proof of Theorem 1.2. By hypothesis we have w − Λ ≡ w_{2}(X) (mod 2) and the invariants D_{X}^{w}(z) are zero unless deg(z) obeys the constraint (1.4). Let p ∈ ℤ satisfy p ≡ w^{2} (mod 4) and recall that—see the paragraph following equation (2.20) in [10]—we may choose a spin^{u} structure 𝔱 on X for which

c_{1}(𝔱) = Λ, p_{1}(𝔱) = p, and w_{2}(𝔱) ≡ w (mod 2).

Then d_{a}(𝔱) = −2p − \frac{3}{2}(χ + σ) and n_{a}(𝔱) = \frac{1}{4}(p + Λ^{2} − σ), by equation (3.21).

From equation (3.64), a reducible PU(2) monopole in M̄_{𝔱} defined by a spin^{c} structure 𝔰 lies in the level M_{𝔱_{ℓ}} × Sym^{ℓ}(X), where ℓ = ℓ(𝔱, 𝔰) and

4ℓ(𝔱, 𝔰) = (c_{1}(𝔱) − c_{1}(𝔰))^{2} − p_{1}(𝔱).

But c_{1}(𝔱) = Λ and p_{1}(𝔱) = −\frac{1}{2}d_{a}(𝔱) − \frac{3}{4}(χ + σ) by equation (3.21), so the definition (1.12) of r(Λ) and r(Λ, c_{1}(𝔰)) implies that

\begin{aligned}4ℓ(𝔱, 𝔰) & = \frac{1}{2}d_{a}(𝔱) + (Λ − c_{1}(𝔰))^{2} + \frac{3}{4}(χ + σ) \\ & = \frac{1}{2}d_{a}(𝔱) − r(Λ, c_{1}(𝔰)) \\ & ≤ \frac{1}{2}d_{a}(𝔱) − r(Λ). \end{aligned} (4.56)

Hence, when d_{a} ≤ 2r(Λ), the strata ι(M_{𝔰}) with non-trivial Seiberg-Witten functions SW_{X,𝔰} can only appear in the top level M_{𝔱} of M̄_{𝔱} (if at all), where they correspond to splittings 𝔱 = 𝔰 ⊕ 𝔰′.

From equation (3.21), the stratum ι(M_{κ}^{w}) has real codimension 2n_{a}(𝔱) in M_{𝔱}, with

\begin{aligned}4n_{a}(𝔱) & = p_{1}(𝔱) + c_{1}(𝔱)^{2} − σ \\ & = −\frac{1}{2}d_{a}(𝔱) − \frac{3}{4}(χ + σ) + Λ^{2} − σ \\ & = −\frac{1}{2}d_{a}(𝔱) − \frac{1}{4}(7χ + 11σ) + Λ^{2} + χ + σ \\ & = −\frac{1}{2}d_{a}(𝔱) + i(Λ), \end{aligned} (4.57)

where the second equality follows from equation (3.21) and the final one by definition (1.11) of i(Λ). Thus, our hypotheses on deg(z) and Λ imply that n_{a}(𝔱) > 0 in Cases (a) and (b) below (where d_{a} = deg(z)), and also in Case (c) (where d_{a} < deg(z)), recalling that deg(z) is denoted by 2δ, for δ ∈ \frac{1}{2}ℤ, in the hypotheses of Theorem 1.2.

Therefore, provided w (mod 2) is good, we can apply Theorem 3.33 to the cobordism M_{𝔱}^{∗,0}. To eliminate this last constraint on w when b_{2}^{+}(X) > 1, we shall instead apply Theorem 3.33 to the cobordism M∗,0 ; 𝔱̃, where 𝔱̃ is the related spin^{u} structure on the blow-up X̃ = X#\overline{ℂℙ}^{2} produced by Lemma 4.19. When b_{2}^{+}(X) = 1, we assume that w ≡ w_{2}(X) − Λ (mod 2) is good so that the Donaldson and Seiberg-Witten invariants are well-defined in this case (see §3.4.2 and §4.1) and M_{𝔱} contains no zero-section pairs.

From Lemma 4.19, Seiberg-Witten strata ι(M_{𝔰^{±}}) with non-trivial invariants appear only in the top level M𝔱̃ of M̄𝔱̃ if and only if Seiberg-Witten strata ι(M_{𝔰}) ⊂ M_{𝔱} with non-trivial invariants appear only in the top level M_{𝔱} of M̄_{𝔱}. Since X is “effective” by hypothesis, we may assume Conjecture 3.34 holds. Also,

n_{a}(𝔱̃) = n_{a}(𝔱) > 0,

using equation (4.57) and the facts that p_{1}(𝔱̃) = p_{1}(𝔱) − 1 and c_{1}(𝔱̃) = c_{1}(𝔱) by Lemma 4.19 and σ(X̃) = σ(X) − 1. Hence, Corollary 3.35 applies to M∗,0 ; 𝔱̃.

Theorem 1.2 now follows by applying Proposition 4.22 in conjunction with the relation (3.70) for the cobordism M̄∗,0 ; 𝔱̃. Equation (3.70) (with the additional hypothesis of Corollary 3.35) gives

# (V̄(ez) ∩ M̄_{κ+1/4}^{w+PD[e]}(X̃))

= −2^{1−n_{a}} ∑_{{𝔰: 𝔰⊕𝔰′=𝔱}} ((−1)^{o𝔱̃(w+PD[e],𝔰^{+})}⟨μ_{p}(ez) ⌣ μ_{c}^{n_{a}−1}, [L_{𝔱̃,𝔰^{+}}]⟩

+(−1)^{o𝔱̃(w+PD[e],𝔰^{−})}⟨μ_{p}(ez) ⌣ μ_{c}^{n_{a}−1}, [L_{𝔱̃,𝔰^{−}}]⟩) .

(4.58)

From Theorem 3.33 we see that we need to to consider the following cases, when n_{a} > 0:

(a) deg(z) = d_{a} < 2r(Λ),

(b) d_{a} = 2r(Λ) and deg(z) = d_{a},

(c) d_{a} = 2r(Λ) and d_{a} < deg(z) ≤ d_{a} + 2n_{a} − 2.

Case (a). The condition n_{a} > 0 is equivalent to δ < i(Λ), since n_{a} = \frac{1}{8}(2i(Λ) − d_{a}) by equation (4.57) and deg(z) = d_{a} = 2δ in this case.

Using the definition (3.31) of the Donaldson invariants and using c_{1}(𝔱) = Λ in Theorem 3.33 and Corollary 3.35 yields

D_{X}^{w}(z) = 0, for deg(z) < 2r(Λ). (4.59)

This proves Case (a). Case (b). The condition n_{a} > 0 is again equivalent to δ < i(Λ), since deg(z) = d_{a} = 2δ in this case. We also have deg(z) = 2r(Λ).

Using the definition (3.31) of the Donaldson invariants, applying our blow-up formula (4.52) (with k = 0) for link pairings to equation (4.58), and using c_{1}(𝔱) = Λ yields

\begin{aligned}D_{X}^{w}(z) & = 2^{1−n_{a}}2^{−δ_{2}−2δ_{0}}(−1)^{δ_{0}+δ_{1}+1} \\ & × ∑_{\{𝔰: 𝔰⊕𝔰′=𝔱\}}(−1)^{o_{𝔱}(w,𝔰)}C_{χ,σ} (deg(z), δ_{c}, 2r(Λ), d_{s}(𝔰), δ_{1}) \\ & × SW_{X,𝔰}(x^{d}ϑ)⟨c_{1}(𝔰) − Λ, h⟩^{δ_{2}}. \end{aligned} (4.60)

Note that although we assume δ_{c} = n_{a}−1 in Case (b), we allow δ_{c} ≥ n_{a}−1 in the above sum, as δ_{c} > n_{a} − 1 in Case (c). The inequality (4.56) implies that the subset of 𝔰 ∈ Spin^{c}(X) giving a splitting 𝔱 = 𝔰 ⊕ 𝔰′ coincides with the subset for which r(Λ, c_{1}(𝔰)) = r(Λ). Hence, the sum in (4.60) is over the same subset of Spin^{c}(X) as that in equation (1.14).

We simplify the sign factor (−1)^{o_{𝔱}(w,𝔰)} in equation (4.60) by writing

o_{𝔱}(w, 𝔰) = \frac{1}{4} (w − Λ + c_{1}(𝔰))^{2} (from definition (3.66))

= \frac{1}{2}c_{1}(𝔰) · (w − Λ) + \frac{1}{4} ((w − Λ)^{2} + c_{1}(𝔰)^{2}) . (4.61)

Because c_{1}(𝔰) and Λ − w are characteristic, we have c_{1}(𝔰)^{2} ≡ (Λ − w)^{2} ≡ σ (mod 8). Thus, equation (4.61) yields

o_{𝔱}(w, 𝔰) ≡ \frac{1}{2}c_{1}(𝔰) · (w − Λ) + \frac{1}{2}σ (mod 2)

≡ \frac{1}{2} (w^{2} + c_{1}(𝔰) · (w − Λ)) + \frac{1}{2} (σ − w^{2}) (mod 2). (4.62)

Substituting equation (4.62) for o_{𝔱}(w, 𝔰) (mod 2) into equation (4.60) implies that the power of (−1) in that formula for D_{X}^{w}(z) becomes

(−1)^{δ_{0}+δ_{1}+1}(−1)^{\frac{1}{2} (σ−w^{2})}(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·(w−Λ))},

matching the power of (−1) appearing in equation (1.14).

Equation (4.57) gives n_{a}(𝔱) = \frac{1}{4}(i(Λ) − δ) and so the power of 2 in equation (4.60) for D_{X}^{w}(z) then becomes

2^{1− \frac{1}{4} (i(Λ)−δ)}2^{−δ_{2}−2δ_{0}},

matching the power of 2 appearing in equation (1.14).

Finally we simplify the expression for the constant C_{χ,σ} (deg(z), δ_{c}, 2r(Λ), d_{s}(𝔰), δ_{1}). Equation (4.57), the equality deg(z) + 2δ_{c} = d_{a} + 2n_{a} − 2 and the assumption that d_{a} = 2r(Λ) gives

δ_{c} = \frac{1}{2} (d_{a} + 2n_{a} − 2 − deg(z)) = \frac{1}{4} (3r(Λ) + i(Λ)) − \frac{1}{2} deg(z) − 1.

(Note that this holds without the assumption δ_{c} = n_{a} − 1.) Then, by the expression for C_{χ,σ} in Lemma 4.18,

C_{χ,σ} (deg(z), δ_{c}, 2r(Λ), d_{s}(𝔰), δ_{1}) = H_{χ,σ}(Λ^{2}, deg(z), d_{s}(𝔰), δ_{1}),

where the function H is defined in equation (1.15). This completes the proof of the formula (1.14) in Case (b).

The result mentioned in Remark 1.3 for z = z′Y can be proved by the same argument, noting that z as described there is intersection-suitable in the sense of Lemma 3.17 and that the pairings with L_{𝔱̃,𝔰^{±}} all vanish by Proposition 4.22. Case (c). Continue to assume d_{a} = 2r(Λ), so the reducibles (with non-trivial Seiberg-Witten functions) can lie in the top level (but not in any lower level). This case follows in exactly the same way as Case (b), except that we now use equation (3.71) in place of equation (3.70) when deg(z) lies in the range

d_{a} < deg(z) ≤ d_{a} + 2n_{a} − 2, (4.63)

so we obtain non-trivial relations among the Seiberg-Witten invariants from the cobordism. Using n_{a} = \frac{1}{8}(2i(Λ) − d_{a}), the upper bound in equation (4.63) becomes

d_{a} + 2n_{a} − 2 = d_{a} + \frac{1}{4}(2i(Λ) − d_{a}) − 2
= \frac{3}{4}d_{a} + \frac{1}{2}i(Λ) − 2
= \frac{3}{2}r(Λ) + \frac{1}{2}i(Λ) − 2
= \frac{1}{2}(3r(Λ) + i(Λ)) − 2.

Thus, our pair of inequalities reduces to

2r(Λ) < deg(z) ≤ r(Λ) + \frac{1}{2}(r(Λ) + i(Λ)) − 2. (4.64)

Therefore we obtain a non-trivial relation amongst the Seiberg-Witten invariants and a vanishing result for Donaldson invariants when the constraint (4.64) on Λ^{2} and deg(z) holds.

Proof of Theorem 1.4. By hypothesis, Λ · c_{1}(𝔰) = 0 for all 𝔰 ∈ Spin^{c}(X) with SW_{X}(𝔰) ≠ 0, so from equation (1.12) for r(Λ, c_{1}(𝔰)) we have

r(Λ, c_{1}(𝔰)) = −c_{1}(𝔰)^{2} − Λ^{2} − \frac{3}{4}(χ + σ)
= −(2χ + 3σ) − Λ^{2} − \frac{3}{4}(χ + σ)
= −(χ + σ) − Λ^{2} + c(X)
= r(Λ),

using the definition of c(X) (see §1.1) and the definition (1.12) of r(Λ), and the formula (§1.1) for c_{1}(𝔰)^{2} when X has SW-simple type. A reducible PU(2) monopole in M̄_{𝔱} defined by a splitting 𝔱_{ℓ} = 𝔰 ⊕ 𝔰′ appears in level

ℓ(𝔱, 𝔰) = \frac{1}{8}(d_{a}(𝔱) − 2r(Λ, c_{1}(𝔰))) = \frac{1}{8}(d_{a}(𝔱) − 2r(Λ)),

and thus all reducibles appear in the same level of M̄_{𝔱}. Hence, the sum over 𝔰 ∈ Spin^{c}(X) with r(Λ, c_{1}(𝔰)) = r(Λ) can be replaced by a sum over 𝔰 ∈ Spin^{c}(X) when Λ ∈ B^{⊥}. We write deg(z) = 2δ, as in the hypothesis of the theorem.

Case (a). In this situation, δ < r(Λ), δ < i(Λ), and

D_{X}^{w}(z) = 0, when 0 ≤ δ < r(Λ),

by Theorem 1.2.

Case (b). In this situation, δ = r(Λ) and δ < i(Λ). We can further simplify the formula (1.14). First, using i(Λ) = 2c(X) − r(Λ) = 2c(X) − δ, δ_{2} = δ − 2m, and δ_{0} = m, the power of 2 in equation (1.14) becomes

2^{1− \frac{1}{4} (2c(X)−2δ)−δ} = 2^{1− \frac{1}{2} (c(X)+δ)},

matching the power of 2 in equation (1.19). The power of (−1) in equation (1.14) is

(−1)^{m+1}(−1)^{\frac{1}{2} (σ−w^{2})}(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·w)},

since δ_{0} = m and c_{1}(𝔰) · Λ = 0, and also matches that in (1.19). Finally, d_{s}(𝔰) = 0 because we assume X is SW-simple type, so the constant H_{χ,σ}(·) is equal to one and thus equation (1.19) follows from equation (1.14).

Case (c). Using deg(z) = 2δ, r(Λ) + i(Λ) = c(X), and equation (1.17) for r(Λ), the constraint (4.64) simplifies to

2r(Λ) < 2δ ≤ r(Λ) + c(X) − 2. (4.65)

The vanishing relation follows from Case (c) in Theorem 1.2. This completes the proof.

Proof of Corollary 1.5. We consider the last case of Theorem 1.4, where δ and Λ^{2} obey the constraints (4.65) and so

r(Λ) < c(X) − 2.

Therefore, the choice of r(Λ) < c(X) − 2 giving the largest possible integer δ (admitting a non-trivial vanishing relation) is r(Λ) = c(X) − 4, achieved when Λ = Λ_{0} with Λ_{0}^{2} = 4 − (χ + σ). By hypothesis, such a class Λ_{0} ∈ B^{⊥} exists. Therefore, the pair (4.65) of inequalities constrains

δ = c(X) − 3.

Thus, using z = x^{m}h^{δ−2m} with 0 ≤ m ≤ [δ/2], we obtain for w_{0} ∈ H^{2}(X; ℤ) with w_{0} −Λ_{0} ≡ w_{2}(X) (mod 2)

∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w_{0}^{2}+c_{1}(𝔰)·w_{0})}SW_{X}(𝔰)⟨c_{1}(𝔰) − Λ_{0}, h⟩^{d} = 0, 0 ≤ d ≤ c(X) − 3. (4.66)

Indeed, if c(X) − 3 is even, then we may choose m = δ/2 to obtain the above relation with d = 0 and, as we explain shortly, the relation for d = 0 also holds when c(X) − 3 is odd. Hence, the degree-d terms in the Taylor expansion of SW_{X}^{w_{0}}(h)e^{−⟨Λ,h⟩} about h = 0 are zero for 0 ≤ d ≤ c(X) − 3 and so the same holds for SW_{X}^{w_{0}}(h).

If w is any integral lift of w_{2}(X), write w = w + Λ_{0} − Λ_{0} and observe that

SW_{X}^{w}(h) = (−1)^{\frac{1}{2} (Λ_{0}^{2}−2w·Λ_{0})}SW_{X}^{w+Λ_{0}}(h).

Thus SW_{X}^{w}(h) vanishes to the same order as SW_{X}^{w_{0}}(h) with w_{0} = w+Λ_{0} and this completes the proof, aside from the remark below on the case of odd c(X) − 3.

When c(X) − 3 is odd, it only remains to show that the relation (4.66) still holds when d = 0. We choose Λ_{1} ∈ B^{⊥} with Λ_{1}^{2} = 6 − (χ + σ) and r(Λ_{1}) = c(X) − 6, so that (4.65) allows δ_{1} = c(X) − 4, which must be even and thus, taking m = δ_{1}/2 yields

∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w_{1}^{2}+c_{1}(𝔰)·w_{1})}SW_{X}(𝔰) = 0,

for any w_{1} ∈ H_{2}(X; ℤ) with w_{1} − Λ_{1} ≡ w_{2}(X). Writing w_{1} = w_{0} + Λ_{1} − Λ_{0} and combining

SW_{X}^{w_{1}}(h) = (−1)^{\frac{1}{2} ((Λ_{1}−Λ_{0})^{2}+2w_{0}·(Λ_{1}−Λ_{0}))}SW_{X}^{w_{0}}(h).

with the previous vanishing result yields the relation (4.66) when d = 0.

Proof of Theorem 1.1. We assume without loss that c(X) > 0. From Theorem 1.4 we know that D_{X}^{w}(x^{m}h^{δ−2m}) = 0 if δ < r(Λ) and that the first potentially non-zero invariant is given by equation (1.19) when δ = r(Λ). The cobordism method constrains Λ^{2} by requiring that δ < i(Λ). Hence, from the graphs of r(Λ) and i(Λ) as functions of Λ^{2} (see Figure 1 in [8]) one sees that these two lines meet for Λ_{0} ∈ H^{2}(X; ℤ) with Λ_{0}^{2} = −(χ + σ), at which point r(Λ_{0}) = c(X) = i(Λ_{0}). Therefore, we choose Λ^{2} to give the largest possible δ = r(Λ) < c(X). We also take Λ ∈ B^{⊥}, to simplify the formula (1.19) and as the SW-basic classes c_{1}(𝔰) are characteristic, this gives (for B non-empty) Λ · c_{1}(𝔰) = 0 and Λ · c_{1}(𝔰) ≡ Λ^{2} (mod 2), so that Λ^{2} is even. Thus we want to choose Λ ∈ B^{⊥} with smallest even value of Λ^{2} > −(χ + σ), namely

Λ^{2} = 2 − (χ + σ), (4.67)

because χ + σ is even (in fact, divisible by four since b_{1}(X) = 0 and b_{2}^{+}(X) is odd). By hypothesis, Λ exists and the formula (1.17) for r(Λ) and the definition of c(X) yield

δ = r(Λ) = −Λ^{2} + c(X) − (χ + σ)
= c(X) − 2.

Therefore Theorem 1.4 and the fact that δ = c(X) − 2 = r(Λ) yield

D_{X}^{w}(x^{m}h^{d−2m}) = 0, 0 ≤ d < δ and 0 ≤ m ≤ [d/2]. (4.68)

When m = 0, the power of (−1) in equation (1.19) simplifies to

(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·w)},

as \frac{1}{2}(σ−w^{2}) ≡ 1 (mod 2). Indeed, to see this note that w−Λ is characteristic, so (w−Λ)·Λ ≡ Λ^{2} (mod 2), and as χ + σ ≡ 0 (mod 4), we have

w^{2} = (w − Λ)^{2} + 2(w − Λ) · Λ + Λ^{2}

≡ σ + Λ^{2} (mod 4)

≡ σ + 2 (mod 4) (by Equation (4.67)).

(4.69)

The power of 2 in equation (1.19), when δ = c(X) − 2, becomes

2^{2−c(X)},

as we expect from Witten’s formula (1.6).

From equation (4.68), the invariants D_{X}^{w}(h^{d}) and D_{X}^{w}(xh^{d−2}) are zero when d < δ = c(X) − 2 (while the method of this article does not allow us to compute the invariants when d ≥ δ + 4), so (compare equation (1.5))

D_{X}^{w}(h) ≡ 0 (mod h^{δ}),
D_{X}^{w}(h) ≡ \frac{1}{δ!}D_{X}^{w}(h^{δ}) + \frac{1}{2(δ−2)!}D_{X}^{w}(xh^{δ−2}) (mod h^{δ+2}). (4.70)

For a monomial z ∈ 𝔸(X), the invariant D_{X}^{w}(z) is zero unless deg(z) ≡ −2w^{2} − \frac{3}{2}(χ + σ) (mod 8). Therefore, as δ ≡ w^{2} − \frac{3}{4}(χ + σ) (mod 4), the invariants D_{X}^{w}(h^{δ+2}) and D_{X}^{w}(xh^{δ}) are necessarily zero and the next potentially non-zero invariant of higher degree in h would be D_{X}^{w}(xh^{δ+2}).

For the terms D_{X}^{w}(h^{δ}), equation (1.19) yields

\frac{1}{δ!}D_{X}^{w}(h^{δ}) = 2^{2−c(X)} ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·w)}SW_{X}(𝔰)\frac{1}{δ!}⟨c_{1}(𝔰) − Λ, h⟩^{δ}. (4.71)

Since SW_{X}^{w−Λ}(h) ≡ 0 (mod h^{δ}) by Corollary 1.5, we have

(4.72) SW_{X}^{w}(h) = (−1)^{\frac{1}{2} (Λ^{2}+2(w−Λ)·Λ)}SW_{X}^{w−Λ}(h) ≡ 0 (mod h^{δ}).

Therefore, using equation (4.71) and noting that the terms in e^{\frac{1}{2} Q(h,h)} and e^{⟨−Λ,h⟩} of lowest degree in h are both 1 and the lowest-degree non-zero term in SW_{X}^{w}(h) has degree δ in h by equation (4.72), we see that

\frac{1}{δ!}D_{X}^{w}(h^{δ}) = [2^{2−c(X)}SW_{X}^{w}(h)e^{⟨−Λ,h⟩}]_{δ} = [2^{2−c(X)}e^{\frac{1}{2} Q(h,h)}SW_{X}^{w}(h)]_{δ} , (4.73)

where [ ·]_{δ} denotes the term of degree δ in h in the power series.

For the term D_{X}^{w}(xh^{δ−2}), equation (1.19) yields

D_{X}^{w}(xh^{δ−2}) = −2^{2−c(X)} ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·w)}SW_{X}(𝔰)⟨c_{1}(𝔰) − Λ, h⟩^{δ−2}

= − [2^{2−c(X)}SW_{X}^{w}(h)e^{⟨−Λ,h⟩}]_{δ−2} · (δ − 2)!
= 0,

(4.74)

where the final equality follows from the fact that the term in e^{⟨−Λ,h⟩} of lowest degree in h is 1 and the terms in SW_{X}^{w}(h) of degree δ − 2 or lower in h are zero. Combining equations (4.70), (4.73), and (4.74) thus completes the proof.

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Department of Mathematics, Ohio State University, Columbus, OH 43210

   E-mail address: feehan@math.ohio-state.edu
   URL: http://www.math.ohio-state.edu/∼feehan/

Current address: Max Planck Institut für Mathematik, Vivatsgasse 7, Bonn, D-53111, Germany

   E-mail address: feehan@mpim-bonn.mpg.de

Department of Mathematics, Florida International University, Miami, FL 33199

   E-mail address: lenesst@fiu.edu
   URL: http://www.fiu.edu/∼lenesst/