STABLE HOMOTOPY VERSION

OF SEIBERG-WITTEN INVARIANT

M. Furuta

Max-Planck-Institut für Mathematik RIMS, Kyoto University

Abstract. In this paper we construct stable homotopy version of Seiberg-Witten invariant. The construction is given for families of any closed oriented 4-manifolds parameterized by compact spaces. As an application we show a divisibility of Seiberg-Witten invariant for non-simple type 4-manifolds.

§1. Introduction.

Let X be a 4-dimensional oriented closed manifold with a spin^{c}-structure. For simplicity we assume that the first Betti number b_{1} of X is 0. The Seiberg-Witten invariant is defined as the fundamental homology class of the moduli space of the monopoles associated with the spin^{c}-structure. Strictly speaking the invariant is well defined when (1) the rank b_{+} of a maximal positive definite subspace H^{+}(X) of H^{2}(X, R) is odd and larger than 1, and (2) one of the two orientations of H^{+}(X) is fixed. We denote b_{+} as 2p + 1. The formal dimension of the moduli space is even when b_{1} = 0 and b_{+} is odd. If we write 2d for the formal dimension, then the Seiberg-Witten invariant is valued in H_{2d}(CP^{∞}, Z), which is isomorphic to Z when d ≥ 0. The invariant is zero when d < 0. The following problem is due to E. Witten [7].

Problem. Is the Seiberg-Witten invariant zero when d > 0?

A 4-manifold satisfying this property is called a simple type manifold. In this paper we show that when d > 0, there is a restriction on the possible values of the Seiberg-Witten invariant. To state our result, we need to introduce the coefficients of the following Taylor expansion:

(\frac{log(1 − x)}{x})^{p} = ∑_{l=0}^{∞} a_{p,l}x^{l}.

Typeset by AMS-TEX

Theorem(1.1). When d > 0, the Seiberg-Witten invariant is divisible by the denominators of a_{p,1}, a_{p,2}, ⋯, and a_{p,d}.

Note that an integer is zero if and only if it is divisible by every nonzero integers. So integrality theorems like Theorem(1.1) could be regarded as a small step for the problem.

On the other hand, when b_{+} = 1 the Seiberg-Witten invariant depends on chambers. It is known, from the wall crossing formula, that there are examples which have non-zero Seiberg-Witten invariant for some positive d. The above divisibility still holds in this case for any chamber.

D. Ruberman pointed out to the author that the integrality of L-genus of the moduli space of monopoles implies a divisibility of the Seiberg-Witten invariant.

What we actually do is to give a refinement of the Seiberg-Witten invariant

as a certain stable homotopy class, which is constructed by using a finite dimensional approximation of the map defined by the monopole equation.

When X is spin and the spin^{c}-structure is the one derived from a spin structure of X, then the construction of the stable homotopy class is given in [5]. One purpose of the present paper is to show that the construction is extended to any spin^{c}-structure.

The second purpose of this paper is to give a definition of the refinement for a general setting. We consider the case when b_{1} is not necessarily zero. We also extend the construction to families of oriented closed 4-manifolds parameterized by compact spaces. The stable homotopy class is well defined for any such family. For example, it is defined for a family of homotopy 4-spheres.

The well-definedness of the stable homotopy class could be regarded as a topological version of an argument showing that renormalization groups preserve expectation values.

We obtain the original Seiberg-Witten invariant when we detect the stable homotopy class by using the ordinary cohomology theory. If we use K-theory instead of the ordinary cohomology theory, then we obtain a K-theory version of Seiberg-Witten invariant. The K-theory version is related to the original invariant through a map defined by Chern character with Todd class as correction term. Abstractly, the Chern character gives rise to two lattice structures on a certain single vector space over Q, and the two invariants, each of which sits on the corresponding lattice are identified with each other. Then measuring the difference of the two lattices implies a property of an integrality of the identified invariants. The difference is described by the Todd class which appears when the Thom classes in the two cohomology theories are compared. The formal power series mentioned above is essentially equal to the Todd class of a vector bundle over CP^{∞}. This kind of argument is standard in applications of K-theory ([1],[2]).

In [3] S. Bauer independently defined stable homotopy version of the Seiberg-Witten invariant and gave applications to topology of algebraic surfaces.

In Section 2, We recall the monopole equation. In Section 3, we give the definition of the refinement of the Seiberg-Witten invariant in Section 3 by using a finite dimensional approximation of the monopole equation. In Section 4, we use cohomology theories to explicitly detect the refined invariant and see that it is in fact a refinement. In Section 5, we compare the ordinary cohomology with the K-theory. We use the assumptions, b_{1}(X) = 0 and b_{+}(X) > 1, only in Sections 4 and 5.

§2. Monopole equation.

In this section we formulate the monopole equation for a spin^{c}-structure of X in a line following [5]. Let H be the quaternion numbers, Sp_{1} the group of the quaternions with norm 1 and S^{1} the intersection of Sp_{1} with C in H.

We define five Spin_{4}^{c}-modules _{−}H_{+}, _{+}H, _{−}H, _{+}H_{+} and C̃ as follows. As real vector spaces, the first four modules are just four copies of H. The actions of (q_{−}, q_{+}, z) ∈ Spin_{4}^{c} = (Sp_{1} × Sp_{1} × S^{1})/{(1, 1, 1,)(−1, −1, −1)} on a ∈ _{−}H_{+}, ϕ ∈ _{+}H, ψ ∈ _{−}H and ω ∈ _{+}H_{+} are defined by q_{−}aq_{+}^{−1}, q_{+}ϕz^{−1}, q_{−}ψz^{−1} and q_{+}ωq_{+}^{−1} respectively. The last one is a complex one-dimensional representation C̃ defined by the multiplication of z^{2}.

Let X be a closed 4-manifold. For a principal Spin_{4}^{c}-bundle P on X, we have five associated vector bundles T, S^{+}, S^{−}, Λ and L from the Spin_{4}^{c}-modules _{−}H_{+}, +H, _{−}H, _{+}H_{+} and C̃.

Suppose we are given a pair of a principal Spin_{4}^{c}-bundle P and an isomorphism T X ≅ T. Then, since T has a natural orientation and a natural Riemannian metric, the pair induces an orientation and a Riemannian metric of X. We call the homotopy class of (P, T X ≅ T) a spin^{c}-structure.

The Spin_{4}^{c}-equivariant map _{−}H_{+} ×_{+}H → _{−}H defined by (a, ϕ) ↦ aϕ induces the Clifford multiplication C: T ⊗ S^{+} → S^{−}. Similarly the Spin_{4}^{c}-equivariant map −H_{+}×_{−}H_{+} → _{+}H_{+} defined by (a, b) ↦ āb induces a twisted Clifford multiplication C̄: T ⊗ T → Λ.

Since _{−}H_{+} ⊕ C̃ is a faithful representation of the Lie algebra of Spin_{4}^{c}, a pair of metric connections on T and on L induces a principal connection on P. We use the Riemannian connection on T = T X and a fixed connection A_{0} on L. Then we have the covariant derivatives ∇_{1} on Γ(S^{+}) and ∇_{2} on Γ(T). Let D_{1} and D_{2} be the twisted Dirac operators

D_{1} = C∇_{1} : Γ(S^{+}) → Γ(S^{−}) and D_{2} = C̄∇_{2} : Γ(T) → Γ(Λ).

Let D be the direct sum of D_{1} and D_{2}:

D = D_{1} ⊕ D_{2} : Γ(S^{+} ⊕ T) → Γ(S^{−} ⊕ Λ).

Let Q be a quadratic map from S^{+}⊕T to S^{−}⊕Λ induced from the Spin_{4}^{c}-equivariant map

+H ×_{−}H_{+} → _{−}H ×_{+}H_{+}, (ϕ, a) ↦ (aϕi, ϕiϕ̄).

We shall consider the nonlinear map

D + Q: V → W,

where V is the L_{4}^{2}-completion of Γ(S^{+}⊕T) and W is the L_{3}^{2}-completion of Γ(S^{−}⊕Λ̃).

Remark. We can identify the imaginary part of Γ(Λ) with the self-dual 2-forms ([5] Section 2 Remark(1)). Let F_{A_{0}}^{+} be the self-dual part of the curvature of A_{0}. Then an element v of V is called monopole when (D + Q)v + F_{A_{0}}^{+} = 0.

Next, we consider the symmetry of the map D + Q under a group action of Harm(X, S^{1}) which is defined as the kernel of the composition of the exterior derivative d: Map(X, S^{1}) → Γ(T) and D_{2} : Γ(T) → Γ(Λ). Here we identify T with its dual T^{∗} by using the Riemannian metric and we regard Map(X, S^{1}) as a group by using the multiplication of S^{1}. Then Harm(X, S^{1}) consists of the harmonic maps from X to S^{1}. (See [5] Section 2 Remark(2).) Note that the center of Spin_{4}^{c} is S^{1} = {1} × {1} × S^{1} and it naturally acts on each fiber of S^{+} and S^{−}. We can identify the connected component of Harm(X, S^{1}) containing 1 with the center of Spin_{4}^{c}. Then Harm(X, S^{1}) acts from the right on S^{+} and S^{−} by the right multiplication. We want to define an action of Harm(X, S^{1}) on T and Λ so that D + Q is Harm(X, S^{1})-equivariant. When we locally write e^{if} for an element of Harm(X, S^{1}), we have:

D_{1}(ϕe^{if}) = C∇_{1}(ϕe^{if}) = C((∇_{1}ϕ)e^{if} + df ⊗ ϕe^{if}i) = (D_{1}ϕ)e^{if} + dfϕe^{if}i,
D_{2}(a − df) = D_{2}a − D_{2}df = D_{2}a,
Q(ϕe^{if}, a − df) = (a − df)ϕe^{if}i ⊕ ϕe^{if}ie^{−if} ϕ̄ = aϕe^{if}i − dfϕe^{if}i ⊕ ϕiϕ̄

and hence

(D + Q)(ϕe^{if}, a − df) = (D_{1}ϕ + aϕi)e^{if} ⊕ D_{2}a + ϕiϕ̄

Now we define the action of Harm(X, S^{1}) on Λ as trivial action. The action on T is defined by looking at the above relation. Note that we have the exact sequence

1 ⟶ S^{1} ⟶ Harm(X, S^{1}) ⟶ H^{1}(X, Z) → 0,

and df is the image of e^{if} written in terms of harmonic 1-form. The action on T is defined through the additive action of H^{1}(X, Z) if H^{1}(X, Z) is identified with the harmonic 1-forms with integral periods.

On the other hand, it is easy to check that D_{1}, D_{2}, and Q commute with the S^{1}-actions.

We decompose V and W into L^{2}-direct sums:

V = H^{1}(X, R) ⊕ V̄ , W = H^{0}(X, R) ⊕ W̄ ,

where H^{i}(X, R) is the space of harmonic i-forms. Then the image of D + Q is contained in W̄ ([5]). We regard D + Q as a family of maps from V̄ to W̄ parameterized by H^{1}(X, R). The action of Harm(X, S^{1}) on V preserves the direct sum. We decomposed this action into two parts corresponding to the direct summands. First, fix a splitting of the above exact sequence and identify Harm(X, S^{1}) with the product H^{1}(X, Z) × S^{1}. The action of H^{1}(X, Z) on V preserves the orthogonal decomposition and the action is free on H^{1}(X, R) since it is just given by translation.

Now we have a family of S^{1}-equivariant maps from V̄ to W̄ parameterized by the torus H^{1}(X, R)/H^{1}(X, Z). For this parameter space we use the notation T_{0}. The actions of S^{1} on V̄ and W̄ are just the restrictions of its actions on V and W respectively.

We shall construct a finite dimensional approximation of this family. The Seiberg-Witten invariant is defined by using the approximation.

Remark. We can identify Γ(T) with the space of connections on L by the correspondence a ↦ A_{0} + ai. The moduli space of S^{1}-connections on L which has the same curvature with A_{0} is parameterized by T_{0} = H^{1}(X, R)/H^{1}(X, Z). The space of all the gauge-equivalence classes of connections on L is a product of T_{0} and an infinite dimensional vector space. The Γ(T)-part of the above decomposition of V is identified with this product structure if we consider only the kernel of D_{2} in order to take a slice for the gauge group action.

§3. Finite dimensional approximation.

We have a family of maps D + Q : V̄ → W̄. Here everything is parameterized continuously by the finite dimensional torus T_{0}, but we suppress the notation for its parameter.

Since the zero set of D + Q is compact ([6]), we can take a large R so that D + Q does not have zero on the sphere of radius R in V̄. Since the parameter space is compact, we can take R uniformly.

We construct a finite dimensional approximation of this family. It is a non-linear analogue of the construction of index for a family of Fredholm operators [2].

step 1

Let W̄_{λ} be the subspace of W̄ spanned by eigenspaces of DD^{∗} with eigenvalues less than or equal to λ. Similarly we define V̄_{λ} by using the eigen-decomposition with respect to D^{∗}D. Let p_{λ} : W̄ → W̄_{λ} be the orthogonal projection.

In [5], it is shown that for large enough λ, D + p_{λ}Q is a good finite dimensional approximation of D + Q in that it does not vanish on the finite dimensional sphere in V̄_{λ} of radius R centered in 0 while the image of this sphere is contained in the finite dimensional vector space W̄_{λ}. We denote this sphere as S_{R}(V̄_{λ}).

The proof of [5] implies that we can take λ uniformly again. However the orthogonal projection p_{λ} : W̄ → W̄_{λ} does not vary continuously with respect to the parameter. Actually the space W̄_{λ} itself may jump. It is necessary to modify the projection so that we have a continuous family of maps.

Let β : (−1, 0) → [0, ∞) be a compact-supported smooth non-negative cut-off function whose integral over (−1, 0) is 1. For each λ > 1, let p̃_{λ} : W̄ → W̄_{λ} be the smoothing of the projection defined by

∫_{−1}^{0} β(λ + t)p_{λ+t}dt

The composition of p̃_{λ} with the inclusion W̄_{λ} → W̄ varies continuously.

Then the proof in [5] can immediately be extended to obtain:

Lemma(3.1). For large λ, D + p̃_{λ}Q does not vanish on the sphere S_{R}(V̄_{λ}). Here λ can be taken uniformly with respect to the family.

step 2

Since W̄_{λ} does not vary continuously with respect to the parameter, we want to replace W̄_{λ} with a vector bundle W̄^{f}. We follow the procedure to define index for a family of Fredholm operators ([2]). We modify the argument slightly in order to consider a nonlinear term in Step 3.

Lemma(3.2). There is an S^{1}-equivariant vector bundle W̄^{f} over the parameter space T_{0} and an S^{1}-equivariant bundle homomorphism χ : W̄^{f} → W̄ which have the following properties.

(1) For each parameter, the image of χ contains W̄_{λ}.

(2) There is an S^{1}-equivariant homomorphism s : W̄ → W̄^{f} for each parameter, and the restriction of the composition χs on W̄_{λ} is the identity.

(3) There is an S^{1}-equivariant isomorphism from W̄^{f} to the product bundle T_{0} × (C^{b} ⊕ R^{c}) for some b and c.

Note that (1) is an immediate consequence of (2).

Proof. Take an open covering U_{i} of the parameter space T_{0} so that there is λ_{i} > λ which is not equal to the eigenvalues of DD^{∗} or D^{∗}D for every parameter in U_{i}. Then W̄_{λ_{i}} varies continuously for parameters in U_{i}. (This continuity is shown by using a min-max principle to characterize eigenspaces [4].) When we replace the open covering with a finer one, if necessary, we can assume that the family W̄_{λ_{i}} makes a trivial S^{1}-equivariant vector bundle over U_{i}. Fix a trivialization and let W̄_{i}^{f} be the obvious extension of this bundle over the whole parameter space T_{0}.

Now the construction of W̄^{f}, χ and s is as follows. We take the direct sum ⊕_{i}W̄_{i}^{f} for W̄^{f}. For parameters in U_{i}, let χ_{i} be the inclusion W̄_{λ_{i}} → W̄ and let s_{i} be the orthogonal projection W̄ → W̄_{λ_{i}}. They are defined only on U_{i}. Take a partition of unity {ρ_{i}} for the open covering and define χ and s as

χ = ∑_{i} ρ_{i}χ_{i}, s = ∑_{i} ρ_{i}s_{i}.

The right-hand-sides are well defined and they satisfy (1) and (2).

Consider the kernel of the surjective map

D + χ : V̄ ⊕ W̄^{f} → W̄ .

¿From (1) in Lemma (3.2) we can show that this map is always surjective. Hence the kernel has a constant dimension given by

dim V̄_{λ} − dim W̄_{λ} + rank W̄^{f} = index(D : V̄ → W̄) + rank W̄^{f}.

Hence V̄^{f} := Ker(D + ϕ) is an S^{1}-equivariant (finite rank) vector bundle over the parameter space.

Now we can replace the family of linear map D : V̄_{λ} → W̄_{λ} with the following continuous family D^{f}.

Lemma(3.3). The family

D^{f} : V̄^{f} → W̄^{f}, (v, e) ↦ e

depends continuously on the parameter space T_{0}.

The formal difference [V̄^{f}] − [W̄^{f}] gives the index of the family D : V̄_{λ} → W̄_{λ} ([2]).

step 3

Now we have the continuous family of linear maps D^{f} between finite dimensional vector spaces and the continuous family of nonlinear maps p̃_{λ}Q between infinite dimensional vector spaces. Let us define a continuous family of nonlinear maps Q^{f} between finite dimensional vector spaces as

Q^{f} : V̄^{f} → W̄^{f}, (v, e) ↦ −sp̃_{λ}Q.

Then a good finite dimensional approximation of D + Q is given by D^{f} + Q^{f} in the following sense. Fix an S^{1}-invariant metric on W̄^{f}. Let S_{(R,R_{1})}(V̄^{f}) be the topological sphere bundle over T_{0} defined as the boundary of the topological disk bundle

V̄^{f} ∩ (B_{R}(V̄) × B_{R_{1}}(W̄^{f})).

Lemma(3.4). For large R_{1}, D^{f}+Q^{f} does not vanish on the sphere bundle S_{(R,R_{1})}(V̄^{f}).

Proof. Take (v, e) in the intersection of V̄^{f} and B_{R}(V̄). Assume that (D^{f}+Q^{f})(v, e) = 0. It suffices to show that (1) v does not lie on the boundary sphere S_{R}(V̄) and that (2) e is bounded. ¿From the definition of V̄^{f}, we have Dv + χe = 0. ¿From the definition of D^{f} and Q^{f}, we have e − sp̃_{λ}Q(v) = 0. Since the image of p̃_{λ} is contained in W̄_{λ}, we obtain the following equation.

−Dv = χe = χsp̃_{λ}Q(v) = p̃_{λ}Q(v).

¿From Lemma(3.1), we can show that v is contained inside the disk B_{R}(V̄) and does not lie on the boundary sphere S_{R}(V̄). Moreover, since v is bounded, e = sχp̃_{λ}Q(v) is also bounded.

In the above argument we did not use any particular property of the parameter space T_{0} except its compactness. For a family of 4-manifolds X parameterized by a compact space K, we can obtain a finite approximation of the family of monopole equation, where the total parameter space is a bundle over K with fiber T_{0}. Strictly speaking, we have to fix a family of spin^{c}-structure. (Later we shall consider all the spin^{c}-structures.) Let us denote the family of spin^{c}-structures as c.

Now we define a Seiberg-Witten invariant as a certain stable homotopy class of a map. We define it for family of 4-manifolds. We do not assume any inequality for b_{+}(X) in this definition. For example it gives an invariant of homotopy 4-spheres, though we do not know how to detect it explicitly. Later on, when we detect this invariant by using cohomology theories, we will use an assumption for b_{+}(X).

Fix a family of Riemannian metric of the fiber which varies continuously in C^{∞}-topology. Let T_{0} be the total space of the fiber bundle over K with fiber T_{0} = H^{1}(X, R)/H^{1}(X, Z). This is a fiber bundle over K.

¿From the above argument, we can construct a finite approximation D^{f} + Q^{f} : V̄^{f} → W̄^{f} parameterized by T_{0} since the lemmas can be extended to any family of 4-manifolds parameterized by a compact set.

Let R be the trivial real 1-dimensional representation space of S^{1}, and C be the standard complex 1-dimensional representation space.

Definition(3.2). Let M(X) be the set of the isomorphism classes of the triples (E, F, f), where

(1) E is a trivial S^{1}-equivariant vector bundle over T_{0} whose fiber is a direct sum of finitely many R’s and C’s,

(2) F is an S^{1}-equivariant real finite-rank vector bundle over T_{0} and

(3) f is an S^{1}-equivariant bundle map from S(E) to S(F).

By using the extended lemmas and the finite approximation we can give the element (V̄^{f}, W̄^{f}, S(D^{f} + Q^{f})) in M(X). To define a topological invariant, which should be independent of the choices to construct the finite dimensional approximation, we need to take a quotient of M(X) by an equivalence relation. Before that, we recall the definition of join. Suppose S_{0} and S_{1} are some subsets of vector spaces V_{0} and V_{1} respectively. Assume that S_{0} and S_{1} does not contain any real line passing the origin. Then the join of S_{0} and S_{1} is defined to be the set of the points in the direct sum V_{0} ⊕V_{1} of the form (1−t)a_{0} ⊕ta_{1} for some a_{0} ∈ S_{0}, a_{1} ∈ S_{1} and t ∈ [0, 1]. Note that the fiber-wise join of two sphere bundles is topologically the sphere bundle of the direct sum of the associated vector bundles. We call it the join of the sphere bundles. Then, for two maps between sphere bundles, we can naturally construct the join of the maps between the joins of the sphere bundles.

Definition(3.3). Two elements (E_{0}, F_{0}, f_{0}) and (E_{1}, F_{1}, f_{1}) of M(X) are stable homotopic to each other if and only if there are two finite dimensional representation spaces G_{0} and G_{1} of S^{1} satisfying the following conditions.

(1) The two representations are direct sums of finitely many R’s and C’s.

(2) We regard G_{0} and G_{1} as trivial vector bundles over K. Then E_{0} ⊕ G_{0} is isomorphic to E_{1} ⊕ G_{1} and F_{0} ⊕ G_{0} is isomorphic to F_{1} ⊕ G_{1}.

(3) The join of f_{0} and the identity on S(G_{0}) is an S^{1}-equivariant bundle map from S(E_{0} ⊕ G_{0}) to S(F_{0} ⊕ G_{0}). Similarly we have an S^{1}-equivariant map from S(E_{1} ⊕ G_{1}) to S(F_{1} ⊕ G_{1}). Then, through the isomorphism in (2), the two joins are homotopic to each other.

Definition(3.4)(stable homotopy version of Seiberg-Witten invariant).

(1) Let M(X) be the set of all the stable homotopy classes

of M(X).

(2) Define SW (X, c) to be the stable homotopy class of (V̄^{f}, W̄^{f}, S(D^{f} + Q^{f})).

Remark. The set M(X) has a natural structure of Abelian semi-group.

One of the purposes of this paper is to show the well-definedness of SW (X, c). The proof is parallel to that of the well-definedness of the index of family of Fredholm operators.

Theorem(3.5). The stable homotopy class SW (X, c) ∈ M(X) is

independent of the family of Riemannian metric and other choices necessary to define it, and hence it gives a topological invariant of the pair (X, c).

Proof.

Step 1

First we fix the family of Riemannian metrics and show that the stable homotopy class does not depend on other choices. Let λ_{0} and λ_{1} be two large real numbers. Suppose χ_{j} : W̄_{j}^{f} → W̄ and s_{j} : W̄ → W̄_{j}^{f} (j = 0, 1) are the maps satisfying the condition of Lemma(3.2). We compare these two and assume for simplicity that the other data are the same for these two cases. (The other data can be treated using an argument similar to that in Step 2.) We have two finite dimensional approximations D_{j}^{f} + Q_{j}^{f} : V̄_{j}^{f} → W̄_{j}^{f} (j = 0, 1). For j = 0, 1 we denote the triple (V̄_{j}^{f}, W̄_{j}^{f}, S(D_{j}^{f} + Q_{j}^{f})) by (E_{j}, F_{j}, f_{j}). Recall that F_{j} = W̄_{j}^{f} (j = 0, 1) are trivial vector bundles. Take F_{1} and F_{0} for G_{0} and G_{1} respectively.

We consider a finite dimensional approximation parameterized by t ∈ [0, 1] defined as follows. For each t ∈ [0, 1], we use

χ(t) = tχ_{0} + (1 − t)χ_{1} : W̄_{0}^{f} ⊕ W̄_{1}^{f} → W̄ ,

and

s(t) = ts_{0} + (1 − t)s_{1} : W̄ → W̄_{0}^{f} ⊕ W̄_{1}^{f}.

to obtain the finite approximation D^{f}(t) + Q^{f}(t) : V̄^{f}(t) → W̄^{f}(t). From this construction, D^{f}(j) + Q^{f}(j) is equal to the direct sum of D_{j}^{f} + Q_{j}^{f} and the identity of G_{j}. This implies that the joins of f_{j} and the identity of S(G_{j}) for j = 0, 1 are homotopic to each other, and hence the two stable homotopy classes are the same.

Step 2

Suppose we have two choices for the family of Riemannian metrics. Then We can connect them continuously as a family parameterized by T_{0} × [0, 1]. Since the construction of finite dimensional approximation works for any family parameterized by a compact space, we can construct a family of finite dimensional approximations. Hence the homotopy classes of the finite dimensional approximations for t = 0 and t = 1 are the same.

We can easily extend the above construction for all the spin^{c}-structures at once.

So far, the symmetry of all the spaces and maps has been S^{1}. If we collect all the spin^{c} structures, however, then we have a symmetry of P in_{2}, where P in_{2} is the normalizer of S^{1} in Sp_{1} and is generated by S^{1} and j.

Let ι be the involution of Spin_{4}^{c} = (Sp_{1}×Sp_{1}×S^{1})/{±1} defined by ι(q_{−}, q_{+}, z) = (q_{−}, q_{+}, z^{−1}). Then any representation of Spin_{4}^{c} can be twisted by ι. Let_{−}H′_{+},_{+}H′, −H′, _{+}H′_{+} and C̃′ be the the twisting of _{−}H_{+}, _{+}H, _{−}H, _{+}H_{+} and C̃. Since the S^{1}-component of Spin_{4}^{c} acts trivially on _{−}H_{+} and C̃, the Spin_{4}^{c}-modules _{−}H′_{+} and C̃′ are canonically isomorphic to _{−}H_{+} and C̃ respectively. However we introduce another isomorphisms below and it is convenient to distinguish them.

Let _{−}j_{+}, _{+}j, _{−}j, _{+}j_{+} and j̃ be the Spin_{4}^{c}-equivariant homomorphisms defined by:

−j_{+} : _{−}H_{+} → _{−}H′_{+}, a ↦ −a,
+j : _{+}H → _{+}H′, ϕ ↦ ϕj
−j : _{−}H → _{−}H′, ψ ↦ ψj
+j_{+} : _{+}H_{+} → _{+}H′_{+}, ω ↦ −ω
j̃ : C̃ → C̃′, t ↦ t̄.

Let Spin^{c}(X) be the set of spin^{c} structures on X, i.e., Spin_{4}^{c}(X) is the set of all the equivalent classes of pair of an Spin_{4}^{c}-bundle P and an isomorphism T = P ×_{Spin_{4}^{c}} _{−}H_{+} ≅ T X. It is well known, from a simple argument of obstruction theory, that Spin_{4}^{c}(X) is an affine space over H^{2}(X, Z). The twisting of ι induces an involution on Spin^{c}(X). For an Spin_{4}^{c}-bundle P, let P′ := P ×_{ι} Spin_{4}^{c} be the twisting of P. We use the notation S^{+}′ etc. to denote the vector bundles associated with P′. Then the five homomorphisms above give five bundle homomorphisms +j : S^{+} → S^{+}′ etc. Note that the connections on L correspond to the connections on L′ = L^{−1} bijectively through ι. Take a pair of connections A_{0} and A′_{0} on P and P′ corresponding to each other by this bijection. We use the notations D′ and Q′ etc. to denote the object associated with P′. It is easy to see the following:

Lemma(3.6). The two maps D + Q : V → W and D′ + Q : V′ → W′ correspond to each other via the maps _{+}j etc.

For each spin^{c} structure, we want to fix a connection A_{0} so that they are compatible with the involution. One minor problem occurs when the involution on Spin^{c}(X) has a fixed point. In that case we have to take A_{0} so that it is compatible with the involution. We need the next lemma.

Lemma(3.7). If a spin^{c}-structure is preserved by the involution, then the spin^{c}- structure is reduced to a spin structure and hence L is topologically trivial. Then we can take a trivial connection for A_{0}.

Proof. Suppose we have an isomorphism f : P → P′. Since P′ is identified with P as a set, we can consider the fixed point P_{0} of P. The fixed-point set of the action of ι on Spin_{4}^{c} is the subgroup Spin_{4}. This implies that P_{0} is a principal Spin_{4}-bundle and hence the structure group of P is reduced to Spin_{4}. The rest of the statement is obtained immediately.

We take the trivial flat connection as A_{0} for the spin^{c}-structure which can be reduced to a spin structure. Then the map D + Q : V → W and its finite dimensional approximation has a natural P in_{2} symmetry [5]. We choose A_{0} for each spin^{c}-structure and assume that they are compatible with the involution in the following sense: (1) if the spin^{c}-structure comes from an spin structure, then A_{0} is a trivial connection and (2) if not, then A_{0} and A′_{0} correspond to each other by the involution. By using this choice we have a family D^{f} : V̄^{f} → W̄^{f} parameterized by Spin^{c}(X) × T_{0}. ¿From the bundle homomorphisms _{+}j etc we define self maps J on the families {V̄ } and {W̄ }. This action is not an involution, but its order is 4. This map together with the obvious S^{1}-action gives rise to a P in_{2} action on the families.

Let R̃ and H be the following real representation space of P in_{2}: R̃ is the unique non-trivial 1-dimensional representation and H is the space of quaternions on which P in_{2} acts as right multiplication.

Proposition(3.8). We can take a finite approximation of D : V̄ → W̄ so that W̄^{f} is a trivial vector bundle whose fiber is a direct sum of finitely many R̃’s and H, and the family D^{f} : V̄^{f} → W̄^{f} parameterized by Spin^{c}(X) × T_{0} has an P in_{2} symmetry.

Proof. We can just repeat the construction of the finite dimensional approximation with the action of P in_{2}. Let T̃_{0} be the total space of the fiber bundle over K with fiber Spin^{c}(X)×T_{0}. Since the argument is almost the same except for replacement of S^{1}, R, C and T_{0} by P in_{2}, R̃, H and T̃_{0} respectively, we omit the details. Lemma(3.7) implies that this family is preserved by the action of J. Since this family already has the S^{1}-symmetry, we totally have a symmetry of P in_{2} which is generated by S^{1} and J.

Remark. The above argument using Lemma(3.7) is not actually necessary to define the action of P in_{2}. If we use the original monopole equation D +Q+F_{A_{0}} = 0, then it does not depend on the choice of A_{0} and we can easily define the P in_{2}-action which preserves this equation.

We now formulate the invariant without fixing a spin^{c}-structure. Suppose X is a fiber bundle with fiber X over K, where X is a closed oriented 4-manifold. Fix a family of Riemannian metric of the fiber which varies continuously in C^{∞}- topology. Recall that T̃_{0} is the total space of the fiber bundle over K with fiber Spin^{c}(X)×T_{0}. We regard it as a P in_{2}-equivariant fiber bundle over K, where P in_{2}- action on T_{0} = H^{1}(X, R)/H^{1}(X, Z) factors through multiplication of {±1} = ⟨J⟩ while its action on Spin^{c}(X) also factors through the involution.

¿From the above argument we can construct a finite dimensional approximation

D^{f} + Q^{f} : V̄^{f} → W̄^{f} parameterized by T̃_{0}, which now has a P in_{2}-symmetry.

Parallel to Definition(3.2), we use the notation M̃(X) to denote the set of the isomorphism classes of the triples (E, F, f), where

(1) E is a trivial P in_{2}-equivariant vector bundle over T̃_{0} whose fiber is a direct sum of finitely many R̃’s and H’s,

(2) F is a P in_{2}-equivariant real finite-rank vector bundles over T̃_{0} and

(3) f is a P in_{2}-equivariant bundle map from S(E) to S(F).

Similarly we define the set of stable homotopy classes M̃(X) by using the obvious equivalence relation similar to Definition(3.3).

The rest of the argument is also quite parallel to the previous argument. We define SW (X) to be the stable homotopy class of (V̄^{f}, W̄^{f}, S(D^{f} + Q^{f})). Then we obtain:

Theorem(3.9). The stable homotopy class SW (X) ∈ M̃(X) is independent of the family of Riemannian metric and other choices to define it, and hence it gives a topological invariant of X

§4. Evaluation of the Seiberg-Witten invariant using cohomology theory

In this section we show how to detect the stable homotopy version of the Seiberg-Witten invariant in the simplest case. We consider a single oriented closed 4manifold X satisfying b_{1}(X) = 0 and b_{+}(X) > 1. We also assume that b_{+}(X) is odd and write b_{+}(X) as 2p+1. Moreover we fix a spin^{c}-structure c and consider the stable homotopy class SW (X, c). In this case the finite dimensional approximation is just an S^{1}-equivariant map between two representation spaces of S^{1}. ¿From an index calculation, the map is defined as a map from C^{a+m} ⊕ R^{n} to C^{m} ⊕ R^{2p+1+n}, where a = (c_{1}(L)^{2}−sign(X))/8, and m and n are some non-negative integers. When p is larger than or equal to a, it is easy to see that there is only one homotopy class of S^{1}-equivariant map from S(C^{a+m} ⊕ R^{n}) to S(C^{m} ⊕ R^{2p+1+n}). In the rest of this section we assume that p < a.

Let f_{0} be an S^{1}-equivariant map between spheres representing the class SW (X). By taking suspension, we can extend f_{0} to be an S^{1}-equivariant map from the pair (B(C^{a+m}⊕R^{n}), S(C^{a+m}⊕R^{n})) to the pair (B(C^{m}⊕R^{2p+1+n}), S(C^{m}⊕R^{2p+1+n})). The S^{1}-equivariant map f preserves the fixed point sets of the S^{1}-action. The restriction of f_{0} on the fixed point sets is a map from the pair (B(R^{n}), S(R^{n})) to the pair (B(R^{2p+1+n}), S(R^{2p+1+n})). If we perturb this map slightly, the origin of B(R^{2p+1+n}) does not lie on the image of B(R^{n}). The perturbation is not unique. However, since we are assuming that 2p + 1 > 1, the relative homotopy class of the perturbed map from the pair (B(R^{n}), S(R^{n})) to the pair (B(R^{2p+1+n})\{0}, S(R^{2p+1+n})) is uniquely determined. Since S(R^{2p+1+n}) is a retraction of B(R^{2p+1+n}) \ {0}, the map can be perturbed further to a map from the pair (B(R^{n}), S(R^{n})) to the pair (S(R^{2p+1+n}), S(R^{2p+1+n})). ¿From homotopy extension property we can extend the perturbation to a perturbation of f_{0} supported in the interior of B(C^{a+m} ⊕ R^{n}). Average it linearly by using the S^{1}-action, then we can assume that the perturbed map f_{1} is still S^{1}-equivariant.

Now f_{1} is an S^{1}-equivariant map from the pair (B(C^{a+m}⊕R^{n}), S(C^{a+m}⊕R^{n})∪ B(R^{n})) to the pair (B(C^{m} ⊕ R^{2p+1+n}), S(C^{m} ⊕ R^{2p+1+n})). We replace these data in the following way just for convenience.

(1) As for the first pair of the spaces (B(C^{a+m} ⊕ R^{n}), S(C^{a+m} ⊕ R^{n}) ∪ B(R^{n})), this is homotopically equivalent to the pair of a disk bundle and its sphere bundle. In fact if we write S(C^{a+m}) for the sphere in the disk B(C^{a+m}) centered in 0 with half radius, then the complement of S(C^{a+m} ⊕ R^{n}) ∪ B(R^{n}) in B(C^{a+m} ⊕ R^{n}) is a trivial disk bundle over S(C^{a+m}) with fiber R^{n+1}. We use the notation S to denote S(C^{a+m}) and E to denote this trivial bundle.

(2) As for the second pair of the spaces (B(C^{m} ⊕ R^{2p+1+n}), S(C^{m} ⊕ R^{2p+1+n})), we multiply it by S in order to regard every space as an S^{1}-equivariant bundle over S. We use the notation F to denote the trivial vector bundle over S whose fiber is C^{m} ⊕ R^{2p+1+n}.

(3) As for the S^{1}-equivariant map between the pairs, we use, instead of f_{1}, the bundle map f over S induced from f_{1}. We need to use homotopy extension property to construct f.

The geometric data we have now is the S^{1}-equivariant homotopy class of the S^{1}-equivariant bundle map f from the disk bundle of E to the disk bundle of F which preserves their boundaries. Since the S^{1}-action is free on S, we can divide everything by S^{1} to get a bundle map f̄ between disk bundles associated with S ×_{S^{1}} E and S ×_{S^{1}} F over S̄ = S/S^{1}:

f̄: (B(Ē), S(Ē)) → (B(F̄), S(F̄)).

This is the final geometric data we shall use. Instead of using maps between pairs, we could formulate everything by using maps between Thom spaces.

It is easy to show that a certain stable homotopy class of this map is well defined for X with b_{1} = 0 and b_{+} > 1. Since we will not use this well-definedness, we omit the details.

Suppose h is a multiplicative generalized cohomology theory for which Ē and F̄ are orientable. We shall use this h to detect the stable homotopy class.

First we need to fix the orientations of Ē and F̄. We shall explain it later for the ordinary cohomology and the K-theory. We shall need some extra geometric data to define the orientations. Fixing the orientations of Ē and F̄ implies, by definition, that h^{∗}(B(Ē), S(Ē)) h^{∗}(B(F̄), S(F̄)) are free h^{∗}(S̄)-modules generated by given classes τ h ; Ē and τ h ; F̄ respectively.

Definition(4.1). Suppose X satisfies b_{1} = 0 and b_{+} > 0. Let c be a spin^{c}-structure of X. (We also assume that certain data necessary to define the orientations are given.) Then the h-version of Seiberg-Witten invariant k^{h}(X, c) of (X, c) is defined by using the following relation:

k^{h}(X, c)τ h ; Ē = f̄^{∗}τ h ; F̄.

Remark. The above k^{h}(X, c) is defined as an element of h^{∗}(S̄). However this cohomology group itself depend on various choices to define f. Strictly speaking, we need to construct some inverse system of cohomology groups. It would be more systematic to use homology rather than cohomology. We shall discuss this point later for the ordinary cohomology.

Ordinary cohomology

Let k^{H}(X, c) be the Seiberg-Witten invariant for ordinary cohomology. To define it, we need the orientations of Ē and F̄. If we change both of these orientations, the invariant does not change. We only need the orientation of the formal difference [E]−[F] which is an element of K(point). Since complex vector spaces have natural orientations, we obtain:

Lemma(4.2). Each choice of the orientation of [H^{0}(X, R)]−[H^{1}(X, R)]+[H^{+}(X, R)] gives the required orientation to define k^{H}(X, c).

Proof. The formal difference [E] − [F] is equal to [H^{1}(X, R)] − [H^{+}(X, R)] modulo some formal difference of complex vector spaces.

Since we assumed that b_{1} = 0, it is not necessary to put H^{1}(X, R). However we put it so that the statement is generalized to any X or any family of 4-manifolds. We fix one of the choices of the orientations.

The degree of k^{H}(X, c) is given by

deg k^{H}(X, c) = deg τ H ; F̄−deg τ H ; Ē = dim F −dim E = (2m+2p+1+n)−(n+1) = 2m+2p.

Since S̄ is a complex projective space of complex dimension a+m−1, its cohomology ring is given by

H^{∗}(S̄, Z) = Z[α], α^{a+m} = 0

where α is the first Chern class of hyperplane line bundle. The above relation is the only one satisfied by α.

Recall that we are assuming that a > p. The relation between k^{H}(X, c) and the usual Seiberg-Witten invariant is given by the following definition and lemma.

Definition(4.3). Define k(X, c) ∈ Z by

k^{H}(X, c) = k(X, c)α^{m+p}.

Lemma(4.4). Perturb f if necessary and assume that the equation f(v) = 0 is transversal. Then the quotient of the space of the solutions {v|f(v) = 0} divided by S^{1} is an oriented closed submanifold of S̄ × E and its fundamental class is equal to k(X, c) × [generator] in H_{2d}(S̄ × E) = Z for d = a − p − 1.

It is not hard to see that the characterization of k(X, c) in the above lemma implies that k(X, c) is equal to the usual Seiberg-Witten invariant.

The dimension 2d of {v|f(v) = 0}/S^{1} is calculated as follows.

2d = dim S̄+dim E−dim F = 2(a+m−1)+(n+1)−(2m+2p+1+n) = 2(a−p−1).

K-cohomology

Let k^{K}(X, c) be the Seiberg-Witten invariant for K-cohomology. To define it, we need the orientations of Ē and F̄. It suffices to give S^{1}-invariant weak complex structures on E and F. Here a weak-complex structure of a real vector space is an obvious equivalence class of the complex structure of the direct sum of the vector space and R^{l} for some l. (“Weak-spin^{c}-structure” would be enough.) It is necessary to give weak-complex structures only on the S^{1}-invariant parts. Then we obtains

Lemma(4.5). Each choice of the homotopy class of weak-complex structures on H^{1}(X, R) and H^{+}(X, R) gives the required orientation to define k^{K}(X, c).

Since we assumed that b_{1} = 0, it is not necessary to give a weak-complex structure of H^{1}(X, R). A choice of homotopy class of the weak-complex structure of a real vector space is equivalent to a choice of orientation of the space. However if we consider family of 4-manifolds, we would need to fix more than the the choice of orientation to define the K-theory version of Seiberg-Witten invariant, though we do not deal with that case in this paper.

We fix one of the choices of the orientations of H^{+}(X, R) and hence a homotopy class of weak-complex structure on it. For simplicity we assume that n is odd so that the S^{1}-invariant parts have a complex structure which is compatible with the given choice. We can always choose such a representative in the stable homotopy class.

Let L be the hyperplane line bundle on the complex projective space S̄. Then K^{∗}(S̄) (∗ = 0, 1) is given by

K^{∗}(S̄) = Z[ξ], ξ^{a+m} = 0

where ξ is the K-theoretic Euler class of hyperplane line bundle L. More explicitly ξ = 1 − [L^{∗}]. The degree of ξ is 0. The above relation is the only one satisfied by ξ. (Here we use the above convention of K-theoretic Euler class so that we have ch_{1}(ξ) = α.)

The K-theory Seiberg-Witten invariant is a polynomial in ξ with integral coefficient.

k^{K}(X, c) ∈ Z[ξ].

In the next section we show that the integrality of each coefficient gives a divisibility of k(X, c).

§5. Divisibility.

Theorem(1.1) is immediately shown from the next theorem.

Theorem(5.1). Let a_{p,l} be the coefficient of x^{l} in the Taylor expansion of (\frac{log(1−x)}{x})^{p}. Then we have the following relation in K(S̄) ⊗ Q.

k^{K}(X, c) = k(X, c)ξ^{m+p} ∑_{l} a_{p,l}ξ^{l}.

Proof of Theorem(1.1) admitting Theorem(5.1). Since k^{K}(X, c) is a polynomial in ξ with integral coefficient, k(X, c)a_{p,n} is an integer as long as ξ^{m+p+n} is not zero, i.e., m + p + n < m + a. This inequality is equivalent to n ≤ a − p − 1 = d.

To show Theorem we compare the K-cohomology and the ordinary cohomology by using the Chern character. We use the following lemma.

Lemma(5.2). Suppose a complex vector bundle G over S̄ has a splitting into a direct sum of line bundles:G = ⊕_{l}L_{l}. We use the notation α_{i} to denote c_{1}(L_{i}) and ξ_{i} to denote 1 − [L_{i}^{∗}]. Then we have

ch(ξ_{i}) = 1 − e^{−x_{i}}, ch(log(1 − ξ_{i})) = x_{i},

ch(τ_{G}^{K}) ∏_{i} \frac{x_{i}}{1 − e^{−x_{i}}} = τ_{G}^{H}

and

ch(τ_{G}^{K} ∏_{i} \frac{log(1 − ξ_{i})}{ξ_{i}}) = τ_{G}^{H}.

Note that x_{i} and ξ_{i} are nilpotent in H^{∗}(S̄, Z) and in K^{∗}(S̄) respectively. However the expressions of the terms in the above equalities are well defined by using Taylor expansion. All the equalities should be understood in H^{∗}(S̄, Q). The first two equalities follows from the definition. The third one is the well-known relation that gives a characterization of the Todd genus. The fourth equality is equivalent to the third one.

Proof of Theorem(5.1). ¿From Lemma we have

ch(τ K ; Ē) = τ H ; Ē

and

ch(τ K ; F̄(\frac{log(1 − ξ)}{ξ})^{m}) = τ H ; F̄.

Hence we have

ch(k^{K}(X, c)(\frac{log(1 − ξ)}{ξ})^{m}) = k^{H}(X, c).

On the other hand, from k^{H}(X, c) = k(X, c)α^{p+m}, we have

ch(k(X, c)(log(1 − ξ))^{p+m}) = k^{H}(X, c).

Since the Chern character is injective on S̄, we obtain

k^{K}(X, c)(\frac{log(1 − ξ)}{ξ})^{m} = k(X, c)(log(1 − ξ))^{p+m},

which implies Theorem 5.1.

Acknowledgments: The author would like to thank Stefan Bauer for informing him about his recent works. The author is grateful to the Max-Planck-Institut für Mathematik in Bonn for its hospitality.

STABLE HOMOTOPY VERSION OF SEIBERG-WITTEN INVARIANT 17

References

[1] M. F. Atiyah, K-theory, Benjamin, New York, 1967.

[2] M. F. Atiyah and I. M. Singer, The index of elliptic operators IV, Ann. of Math. 93 (1971), 119–138.

[3] S. Bauer, Connected sums of 4-manifolds, in preparation.

[4] R. Courant and D. Hilbert, Methods of mathematical physics I, John Wiley & Sons, New York, 1962.

[5] M. Furuta, Monopole equation and the 11/8-conjecture, preprint.

[6] P. B. Kronheimer and T. S. Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Letters 1 (1994), 797–808.

[7] E. Witten, Monopoles and four-manifolds, Math. Res. Letters 1 (1994), 769–796.

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   E-mail address: furuta@mpim-bonn.mpg.de     furuta@kurims.kyoto-u.ac.jp