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Floer homology for 2-torsion instanton invariants

We construct a variant of Floer homology groups and prove a gluing formula for a variant of Donaldson invariants. As a corollary, the variant of Donaldson invariants is non-trivial for connected sums of 4-manifolds which satisfy a condition for Donaldson invariants. We also show a non-existence result of compact, spin 4-manifolds with boundary some homology 3-spheres and with certain intersection forms.


Introduction
In this paper, we construct a variant of Floer homology groups and prove a gluing formula for a variant of Donaldson invariants introduced in [13].
As S. K. Donaldson showed in [3,4,5,6], the moduli spaces of instantons have crucial information about the topology of closed, oriented, smooth 4-manifolds X. Mainly there are two methods to draw out the information from the moduli spaces. First one is to describe the structures of the singular points and the ends of the moduli spaces [3,4]. The description gives strong restrictions to the realization of unimodular quadratic forms as the intersection form of a smooth 4-manifold. Second one is to count the number of points of 0-dimensional moduli spaces or to integrate cohomology classes over moduli spaces [5,6]. This gives differential-topological invariants of 4-manifolds which distinguish different smooth structures on the same topological 4-manifold.
It is hard to compute Donaldson invariants in general. However Floer theory gives us a way to compute Donaldson invariants Ψ X when X has a decomposition X = X 0 ∪ Y X 1 for some compact 4-manifolds X 0 and X 1 with boundary Y andȲ . Here Y is a closed, oriented 3manifold andȲ is Y with opposite orientation. A. Floer introduced instanton homology groups HF * (Y ) for homology 3-spheres Y in [10]. Floer's groups allow us to generalize Donaldson invariants for compact, oriented 4-manifolds X 0 whose boundaries are homology 3-spheres Y . The relative invariant Ψ X 0 is an element of HF * (Y ), which is defined by counting the number of points of the moduli spaces of instantons overX 0 = X 0 ∪ (Y × R ≥0 ) or integrating cohomology classes of moduli spaces . Assume that a closed 4-manifold X has a decomposition X = X 0 ∪ Y X 1 for some homology 3-sphere Y and two compact 4-manifolds X 0 , X 1 with boundary Y ,Ȳ and with b + (X 0 ), b + (X 1 ) > 1. Here b + stands for the dimension of a maximal positive subspace of the intersection form on H 2 (X). Then we have the relative invariants Ψ X 0 ∈ HF * (Y ), Ψ X 1 ∈ HF * (Ȳ ). There is a natural pairing < , >: HF * (Y ) ⊗ HF * (Ȳ ) −→ Q, and we have a relation (1) Ψ X =< Ψ X 0 , Ψ X 1 > .
Floer homology groups also enable us to generalize the results on the non-existence of closed 4-manifolds with b + = 0 and with non-standard intersection forms. Donaldson [7] showed that if HF 1 (Y ) = 0 then there is a restriction to the realization of unimodular quadratic forms as the intersection form of a 4-manifold X 0 with boundary Y and with b + = 0. The proof involves the description of the singular points and the ends of the moduli spaces of instantons overX 0 as in the case of closed 4-manifolds.
On the other hand, there are some variants of Donaldson invariants. R. Fintushel and R. Stern defined a variant of Donaldson invariants for closed, simply connected, spin 4-manifolds in [9]. They used a 2-torsion cohomology class u 1 ∈ H 1 (M P ; Z 2 ) of moduli spaces M P which was originally defined by Donaldson in [4]. In [1], S. Akbulut, T. Mrowka and Y. Ruan extended the construction of u 1 to non-spin 4-manifolds and they showed a universal constraint of Donaldson invariants for non-spin 4-manifolds. Using u 1 for non-spin 4-manifolds, the author [13] defined a variant Ψ u 1 X of Donaldson invariants for non-spin 4-manifolds , which can be regarded as an extension of Fintushel-Stern's invariant to non-spin 4-manifolds. A remarkable feature of these variants is that they are non-trivial for connected sums of the form X = Y #S 2 × S 2 in general.
In [12], it was announced (without proof) that K. Fukaya, M. Furuta and H. Ohta showed a non-existence result of compact, spin 4-manifolds with b + = 1 and with some intersection form when their boundaries are diffeomorphic to some homology 3-spheres. This is a generalization of a result in [4]. In [12], Furuta explained that a variant of Floer homology groups and an extension of Ψ u 1 X to compact, spin 4-manifolds with boundary a homology 3-sphere are used in the proof.
The main theme of this paper is to construct a variant of Floer homology groups and to extend the relation (1) to Ψ u 1 X . As a corollary, we obtain a non-vanishing result for connected sums of some 4-manifolds. Our construction can be applied to not only the case when 4manifolds are spin but also the case when 4-manifold are non-spin. Moreover we will write down a proof of Fukaya-Furuta-Ohta's non-existence result using the variant of Floer homology groups. In the proof, we make use of the structure of the ends of some moduli spaces as in [4].
In [13] the author showed that the variant of Donaldson invariants is non-trivial for 2CP 2 #CP 2 . As we will explain in Section 3, this calculation is suggesting that we need to take into account the following aspects in the construction of the variant of Floer homology groups: • The trivial flat connection on Y should play an important role in the gluing formula for Ψ u 1 X . • We need a similar construction to Fukaya's extension [11] of Floer homology groups.
The organization of this paper is as follows. In Section 2, we recall Fukaya's construction. We will basically follow [11] and [2], however, with some modification. In Section 3, making use of techniques developed in Section 2, we will introduce a variant I (a)

Fukaya-Floer homology
Let X be a closed, oriented 4-manifold with b + > 1 and take a U(2)-bundle P over X and assume that the dimension of the moduli space of instantons on P is 2d for some integer d ≥ 0. We can associate cohomology classes µ([Σ]) of degree 2 on the moduli space to homology classes [Σ] ∈ H 2 (X; Z). Here Σ is a closed, oriented surface embedded in X which represent the homology class. The cohomology classes µ([Σ]) are the first Chern classes of the determinant line bundles of a family of twisted∂ operators on Σ. Roughly speaking Donaldson invariants are the evaluations of cup products µ([Σ 1 ]) ∪ · · · ∪ µ([Σ d ]) on the moduli spaces.
Suppose that we have a decomposition X = X 0 ∪ Y X 1 and that the surfaces Σ l are split into two surfaces with boundary γ l ∼ = S 1 along Y . We consider how to recover the Donaldson invariants of X from relative invariants of X 0 and X 1 . Fukaya's construction [11] provide us suitable homology groups to define the relative invariants. In the construction, we need sections of determinant line bundles of families of twisted∂ operators on γ l × R which behave nicely on the ends of moduli spaces of instantons over Y × R and satisfy certain transversality conditions. We call such sections admissible. The main point of this section is the construction of admissible sections of the determinant line bundles (Proposition 2.8).
2.1. Outline of proof of gluing formula. In this subsection, we give a sketch of the construction of a gluing formula for Donaldson invariants.
Let X be a closed, oriented, simply connected smooth 4-manifolds with b + > 1 and odd, and P be a U(2)-bundle over X with w 2 (P ) non-trivial. We write P det for the U(1)-bundle over X induced by P , and fix a connection a det on P det . We take a Riemannian metric on g on X and suppose that the virtual dimension of M P is 2d with d ≥ 0. Here M P = M P (g) is the moduli space of instantons on P which induce the connection a det on P det . The moduli space M P is a smooth manifold for generic metrics g, and a choice of an orientation of the space H + g (X) of self-dual harmonic 2-forms on X orients M P . The moduli space M P gives us the Donaldson invariant Ψ X,P : H 2 (X; Z) ⊗d −→ Q. This is defined as follows. Let [Σ] be a class in H 2 (X; Z). We denote byB * Σ the space of gauge equivalence classes of framed, irreducible connections on the restriction P | Σ of P to Σ which are compatible with a det | Σ . We have the determinant line bundlẽ Σ ) is the∂ operator twisted by the rank-two complex vector bundle E associated to P and a square root K 1 2 Σ of the canonical line bundle of Σ, and∂ * A is the adjoint. There is a natural action of SU(2) onB * Σ , and the action of {±1} ⊂ SU(2) is trivial. Hence we have the SO(3) = SU(2)/ ± 1 action onB * Σ . The quotient spaceB * Σ /SO(3) is the space B * Σ of irreducible connections on P | Σ which are compatible with a det | Σ . Since the action of {±1} ⊂ SU(2) onL ⊗2 Σ is trivial, we have the line bundle Note that L ⊗2 Σ may not be a square of a genuine line bundle. Take a section s Σ of L ⊗2 Σ and denote the zero locus by V Σ . We define M P ∩ V Σ by Since the restriction of elements of M P to Σ may not be irreducible, this is not well-defined. But we can avoid this problem by replacing Σ with a small neighborhood ν(Σ) of Σ. (See [8]. ) Take d homology classes [Σ 1 ], . . . , [Σ d ] ∈ H 2 (X; Z). We can show that the intersection We can see that this is independent of the choices of metric and sections.
Suppose that X has a decomposition X = X 0 ∪ Y X 1 . Choose a Riemannian metric g Y on Y and let g 0 , g 1 be Riemannian metrics on X 0 , X 1 whose restrictions to Y are equal to g Y . Assume that the dimension of the moduli space M P is 2d with 0 ≤ d ≤ 3. Take homology classes [Σ 1 ], . . . , [Σ d ] of X represented by surfaces Σ 1 , . . . , Σ d which intersect with Y . Put Σ ′ l = Σ l ∩ X 0 , Σ ′′ l = Σ l ∩ X 1 and γ l = Σ l ∩ Y . We assume that γ l are diffeomorphic to S 1 for all l. We show how to compute Ψ X ([Σ 1 ], . . . , [Σ d ]) from data of X 0 , X 1 briefly under this situation. We use some facts about instantons which can be found in [7]. Take a sequence {T α } ∞ α=1 of positive real numbers which diverges to infinity. We have manifolds X α = X 0 ∪ (Y × [−T α , T α ]) ∪ X 1 which are diffeomorphic to X. The Riemannian metrics g Y , g 0 , g 1 induce Riemannian metrics g α on X α . Take instantons [A α ] ∈ M P (g α ).
From this formula, formal sums The formal sums ψ X 0 , ψ X 1 depend on the metrics and sections. We will define a boundary map such that the composition ∂ • ∂ is identically zero, and show that ∂ψ X 0 = 0 and the class Ψ The formula (4) implies This is the gluing formula.

Fukaya's construction.
Let Y be a closed, oriented 3-manifold. Take a Riemannian metric g Y on Y and a U(2)-bundle Q over Y , and fix a connection a det on the U(1)-bundle Q det induced by Q. We consider connections on Q which induce the fixed connection a det on Q det . Let A Q be the space of connections on Q with fixed determinant and G Q be the space of automorphisms of Q of determinant 1. The Chern-Simons functional is an S 1 -valued functional on the quotient space B Q = A Q /G Q . The critical points are the gauge equivalence classes of projectively flat connections. If the Hessian of the Chern-Simons functional at a projectively flat connection is non-generate, we say that the projectively flat connection is non-degenerate. For simplicity, we will refer projectively flat connections over Y with fixed determinant as "flat connections". Throughout this section we assume the following hypothesis.
Hypothesis 2.1. All flat connections on Q are irreducible and non-degenerate.
We are always able to perturb the Chern-Simons functional such that any critical points are non-degenerate. See [10], [7].
Let R(Y ) = R(Y, Q) be the set of gauge equivalence classes of flat connections on Q. It follows from Hypothesis 2.1 that R(Y ) is a finite set. We define a Z 8 -grading function δ Y on R(Y ) as follows. Let π be the projection from Y × R to Y . Fix a flat connection ρ 0 on Q. For each flat connection ρ on Q, choose a connection A 0 = A 0 (ρ, ρ 0 ) over Y × R, which is compatible with π * a det , such that Then we have an operator Here g Q is the bundle of trace free, skew adjoint, endomorphisms of the rank-two complex vector bundle E associated with Q. Under Hypothesis 2.1, this operator is a Fredholm operator and we have the numerical index ind D A 0 ∈ Z. We set We can show that this depends only on the gauge equivalence class of ρ (and ρ 0 ).
We give the precise definition of (6). To do this, we introduce some spaces of connections on Y × R and γ l × R. For flat connections ρ, σ on Q, take a smooth connection A 0 = A 0 (ρ, σ) on π * Q as before. Let τ > 0 be a small positive number and we set The wighted Sobolev space L 2,τ 4 is defined as follows. Take a function W τ on Y × R such that W τ (y, t) > 0 for ∀ (y, t) ∈ Y × R, W τ (y, t) = e τ |t| for |t| > 1.
For a smooth, compact supported section f , we define the weighted L 2 4 -norm by The weighted Sobolev space L 2,τ 4 is the completion of the space of smooth, compact supported sections. Note that since ρ, σ are irreducible, every connection in A * Y ×R (ρ, σ) is irreducible. By Hypothesis 2.1, we need not to introduce the weighted Sobolev space for the construction of the moduli space. However we need the weighted Sobolev space to define the determinant line bundles as explained below.
We introduce a gauge group acting on the space of connections. We put Y ×R (ρ, σ) be the moduli space of instantons with limits ρ, σ. We can perturb the instanton equation such that the operators D A are surjective for all [A] ∈ M 0 Y ×R (ρ, σ) ( [10,7]). For simplicity, we always assume the following.
Under this hypothesis, the moduli space M 0 Y ×R (ρ, σ) is smooth of expected dimension. Next we introduce spaces of connections and gauge groups over Γ l = γ l ×R. Let A l = A l (ρ, σ) be the restriction of the fixed connection A 0 (ρ, σ) to Γ l . We set We denote the quotient space A Γ l (ρ, σ)/G 0 Γ l byB Γ l (ρ, σ). Note that the restrictions of ρ, σ to γ l may be reducible, and hence some connections in A Γ l (ρ, σ) are reducible.
We define the determinant line bundle overB Γ l (ρ, σ). We need a spin structure on Γ l . Since H 1 (Γ l ; Z 2 ) is isomorphic to Z 2 , there are two spin structures on each Γ l (up to isomorphism). We fix a spin structure on γ l which represent the trivial class in the 1-dimensional spin bordism group. This spin structure induces a spin structure on Γ l . We use this spin structure. (We will explain the reason why we take this spin structure in Remark 2.3 below.) The spin structure induces a square root K 1 2 Γ l of the canonical line bundle K Γ l . For connections A ∈ A Γ l (ρ, σ) we have the twisted∂ operators are the weighted Sobolev spaces with weight function W ′ τ > 0 such that W ′ τ (y, t) = e τ t for |t| > 1. We do not take the absolute value of t in the exponent this time. The operators are Fredholm operators for small τ > 0. Since we have the universal bundles we obtain complex line bundles Letr l be the map fromB * Y ×R (ρ, σ) toB Γ l (ρ, σ) defined by restricting connections to Γ l . Then we have a natural action of Z 2 × Z 2 on the pull-backr * lL ⊗2 Γ l and the action of the diagonal Z 2 = {±(1, 1)} is trivial. Hence we get the line bundle Remark 2.3. We explain the reason why we choose the spin structure on Γ l induced by the spin structure on γ l representing the trivial class of 1-dimensional spin bordism group Ω spin 1 = Z 2 . When we prove the gluing formula for Donaldson invariants, we consider Γ l as a neck of a closed surface in a closed 4-manifold and we need the restriction of a spin structure on the closed surface, which is used to define Donaldson invariants, to Γ l . This spin structure is the spin structure induced by the spin structure on γ l which represent the trivial element in Ω spin To define (6) we need sections s l = s l (ρ, σ) of L ⊗2 l (ρ, σ) which behave nicely on the ends of the moduli spaces and satisfy suitable transversality conditions. We briefly recall some basic definitions and facts which are relevant to the end of the moduli spaces. (See [7] for details.) For a real number T let c T be the translation We call a sequence T of real numbers T 1 < · · · < T r−1 a translation vector.
. . , z s i ] and δ z l are the delta functions.
). It follows from the additivity of the index of the operator D A that if the dimension of M 0 Y ×R (ρ, σ) is less than 8 then Z i are empty for all i. In Section 2 and 3, we only consider the case when dim M 0 Y ×R (ρ, σ) < 8 and hence Z i are always empty. In Section 4, we will Analise the end of moduli spaces under the situation where Z i are not empty.
The end of the moduli spaces are described by gluing maps. We consider the case when Z i are empty. Let U 1 , . . . , U r be precompact, open sets of M 0 Y ×R (ρ, ρ(1)), . . . , M 0 Y ×R (ρ(r − 1), σ). Then we have a gluing map We also have a gluing map which covers Gl. For T = (T 1 , . . . , T r−1 ) with T i > T 0 , we write Gl T for the restriction of Gl Using these definitions and facts, we state the properties of sections of the line bundles which are required to define (6).
Definition 2.7. Let s l be sections of the line bundles L ⊗2 l (ρ, σ). Under the above situation, we say that Here · is the norm on L ⊗2 l (ρ, σ) induced by the L 2,(−τ,τ ) -norms on the spaces of sections of K The following proposition is the key in this paper.
is transverse and compact. Hence the intersection is a finite set.
Here we introduce the following definition. Definition 2.9. If sections s l (ρ, σ) of L ⊗2 l (ρ, σ) have the properties in Proposition 2.8, we call them admissible.
The proof of Proposition 2.8 will be given in the following two subsections. In this subsection, we assume that we have admissible sections s l (ρ, σ) and define the boundary operator for the Fukaya-Floer homology groups.
Let d be an integer with 1 ≤ d ≤ 3. For subsets L 1 ⊂ L 2 ⊂ {1, . . . , d} and flat connections are finite by the property (c) in Proposition 2.8. Hence we can count the number of points in the intersections. They numbers give the definition of (6). More precisely we need to attach a sign ±1 to each point. However we mention nothing about signs. (The main purpose of this paper is to construct variants of Floer homology groups for 2-torsion instanton invariants. They are defined over Z 2 and we do not need signs for the construction. ) and choose subsets L 1 , L 2 of {1, . . . , d} with |L 1 | = β 1 , |L 2 | = β 2 . Then we put We prove the following.
This is given by counting the number of the ends of 1-dimensional moduli spaces. Let . Hence there is a number i(l) ∈ {1, . . . , r} such that i(l) ]) = 0 for each l ∈ L 2 \L 1 . This means that l ∈ L 2 \L 1 lies in L(i(l)). Therefore we have , ρ(i); L(i)), the intersection are not empty. The transversality condition (b) in Proposition 2.8 implies that for each i. From the additivity of the index, we have Therefore we get r ≤ 2. We assumed that r > 1, so r = 2.
Put L := L(1) L 1 . Using (b), (c) in Proposition 2.8, we can easily see that for some precompact open neighborhoods are transverse and that and the first intersection is transverse. The condition (a) in Proposition 2.8 means that s l (ρ, σ) and s ′ l are close to each other on the end of the moduli space. Hence for T 1 large enough, the intersection is transverse and consists of a single point which is close to Then M ′ is compact and if we perturb sections s l (ρ, σ) outside of neighborhoods of the triples , M ′ becomes a smooth manifold of dimension 1. Moreover there is a natural identification By counting the number of ∂M ′ with signs, we get and this gives ∂ • ∂ = 0 as required.
First, for flat connections ρ, σ with dim M 0 Y ×R (ρ, σ) < 8, we take locally finite open covers with U λ precompact as follows. We will use a partition of unity associated with the open cover to construct admissible sections.  (1)) × · · · × Λ(ρ(r − 1), σ), we have a gluing map This is a precompact, open set of M 0 Y ×R (ρ, σ). We write N for the complement of the union of the sets U λ,n : Lemma 2.13. Under the above notations and assumptions, N is compact.
. Note that Z i are empty since the dimension of the reduced moduli space is less than 8. If r > 1, it follows from Proposition 2.6 that [A α ′ ] lie in the image of Gl λ for large α ′ and some λ. This is a contradiction to the fact that [A α ′ ] are in the complement N of the images of the gluing maps.
To prove Lemma 2.14, we need to remove extra open sets from the open cover {U λ } λ∈Λ(ρ,σ) . We consider the condition Then {f λ } λ∈Λ(ρ,σ) is a partition of unity associated with {U λ } Λ(ρ,σ) . We show that this partition of unity has the property of Lemma 2.14. It is sufficient to show the following: Lemma 2.15. Let I, J, (λ, n) ∈ Λ(ρ, σ) be as in Lemma 2.14. Then there is a positive integer N = N(λ, n I ) such that U λ,n satisfies (10) if n j > N for all j ∈ J.
The proof will be given in Subsection 2.4. Here we prove that we can perturb the sections s l (ρ, σ) such that the sections become admissible, assuming Lemma 2.16. We show the next lemma to do this.
Hence we have Using this inequality, we show This is a contradiction since we assumed that dim M 0 Y ×R (ρ, σ) ≤ 2|L|. We have obtained (12). By the hypothesis of induction, s l (ρ(i 0 − 1), ρ(i 0 )) satisfy (b) in Proposition 2.8. Hence the inequality (12) means that In the first lemma, we consider the situation where the connections [A α ] can be written as gluing of three instantons. For example, such a situation occurs if [A α ] are given by ). for some [B α i ] ∈ U λ α i (i = 1, 2, 3) and S α j > T λ α (j = 1, 2). The first lemma is the following. Lemma 2.18. There is a subsequence {[A α ′ ]} α ′ which satisfies the following.
We can show similar statements to this lemma when we assume |λ α | = ℓ with ℓ ≥ 4. As we will see later, these statements imply that when [A α ] splits into two instantons there are no terms with |λ| ≥ 3 in (9) for large α .
In the second lemma, we consider the difference of two gluing maps. Let , ρ(i)) for i = 1, 2. (Here ρ(0) = ρ, ρ(2) = σ.) Then we have two gluing maps . When α is sufficiently large, we can write Lemma 2.19. Under the above notations, we have As before, let {[A α ]} α a sequence of instantons which converges to a pair ( ). In the third lemma, we consider the behavior of [A α i (λ)]. Lemma 2.20.
First we prove the following claim.
It follows from this claim and (9) that (13) s for α > α 1 . For i = 1, 2 and δ > 0, we put The open covers {U λ } λ∈Λ(ρ(i−1),ρ(i)) are locally finite, hence if δ is sufficiently small then the numbers of open sets U λ with U λ ∩ U i,δ non-empty are finite for all i. We fix such δ > 0. By Lemma 2.20, there is a positive integer α δ such that if α > α δ , we have p=1 be the sets of open sets which intersect U i,δ for each i, and put α ′ δ := max{α 1 , α δ }. Then λ in (13) take the form of for α > α ′ δ . By Lemma 2.19, for any ε > 0, there exists α(p 1 , p 2 ; ε) such that if α > α(p 1 , p 2 ; ε), we have Here we prove Lemma 2.18 and 2.20. The proof of Lemma 2.19 will be given in the next subsection.

Proof of Lemma 2.18
Let for all i. We will show that r i = 1 for all i. This implies (1) and (2) since the oven covers {U λ } λ are locally finite.

Proof of Lemma 2.20
Let [A α ], [A α i ] and T α (λ) be as above. Suppose that (1) does not hold. Then we have a subsequence . By Hypothesis 2.1, the set of gauge equivalence classes of flat connections is finite. Hence we can suppose ρ α ′ (1) are independent of α ′ . We write ρ ′ (1) for ρ α ′ (1). As in the proof of Lemma 2.18, we can see that there is a subsequence (still denoted by , σ) and we have shown that (1) holds.
]. This is a contradiction.
2.5. Evaluation of difference of two gluing maps. In this subsection, we prove Lemma 2.19. We need to compare two gluing maps defined on different regions intersecting each other. To do this, we must explicitly see the construction of gluing maps. We give outline of the construction following [7] before we prove Lemma 2.19. First note that elements in M 0 Y ×R (ρ, σ) can be considered as elements in M Y ×R (ρ, σ) with center of mass 0. Here the center of mass of Let X 1 , X 2 be two copies of Y × R and for T > 0 put Gluing X 1 (2T ) and X 2 (2T ) through ϕ, we obtain a manifold X #(T ) = X 1 (2T ) ∪ ϕ X 2 (2T ). We have a natural identification between X #(T ) and Y × R such that (y, 3T /2) ∈ X 1 (2T ) ⊂ X #(T ) corresponds to (y, 0) ∈ Y × R. Let ρ = ρ(0), ρ(1), σ = ρ(2) be flat connections and choose [ Here A 1 , A 2 are instantons on X 1 , X 2 with center of mass 0. We write ρ(0), ρ(2) for ρ, σ respectively. Then we can write A ′ be the self-dual part of the curvature of the connection on π * g Q induced by A ′ . Then we have F + That is, A ′ is an almost instanton. To obtain a genuine instanton, we consider the equation F + A ′ +a = 0 for a small π * g Q -valued 1-form a. The equation can be rewritten as (16) d When T is large enough, we can construct a right inverse P = P T of D A ′ by using P 1 , P 2 and some cut-off functions. We can see the operator norm P of P is bounded by 2( P 1 + P 2 ). If we substitute a = P φ into (16), we get the equation We can see that for large T > 0 this equation has a unique solution φ = φ T with φ T → 0 as T → ∞. Hence for large T we get an instanton of the form precompact open sets, the construction can be applied to all ([A 1 ], [A 2 ]) ∈ U 1 × U 2 and we obtain the gluing map More precisely, we must slightly translate A #(T ) to make the center of mass be 0. Let m T be the center of mass of A #(T ) . Then the translate c * −m T (A #(T ) ) is an instanton with center of mass 0. The precise definition of the gluing map is

We can easily show lim
T →∞ m T = 0 and it does not matter even if we assume that m T = 0 in the proof of Lemma 2.19. Therefore we will drop c * −m T from notations in the proof. Let γ ∼ = S 1 be a loop in Y and put Γ = γ × R. We see outline of the construction of the gluing map For simplicity of notations, we suppose that ∂ A i are surjective for i = 1, 2. Let Γ 1 , Γ 2 be copies of Γ and fix cut-off functions γ i = γ T i on Γ i with Here supp(γ i ) is the support of γ i , and Γ i (2T ), Γ #(T ) are defined as before. In the third equation, we consider γ i as functions on Γ #(T ) in the natural way. Take f i ∈ ker∂ A i for i = 1, 2 and put Then we can show that We may construct a right inverse Q T from right inverses Q i of∂ A i for large T . The difference Applying this construction to all connections in U 1 × U 2 , we get the map Gl.
When∂ A i are not compact, we can choose maps Γ l ) such that∂ A i ⊕ U i are surjective. Applying the above method to these operators, we obtain the gluing maps.

Proof of Lemma 2.19
Assume that we have a sequence Then we have two gluing maps -norms. From (19), (20) and (21), we have ). It follows from (15), (17) and (18) that Hence we obtain the required result.
From (22), we get Here · is the L 2,(−τ,τ ) norm. The first term on the right hand side is equal to

This is bounded by
Hence the first term goes to zero. The second term is equal to Thus we have Similarly Since and δ α converge to zero, we have ]. Hence we get Since ε is arbitrary, the norm of difference 1 ) goes to zero. Thus the limit of the second term is zero. Similarly the limits of the third and fourth terms are zero. Since 2.6. Well-definedness. A priori, Fukaya-Floer homology groups seem to depends on the choices of Riemannian metric on Y and sections of L ⊗2 l (ρ, σ). But we will prove that Fukaya-Floer homology does not depend on these choices up to canonical isomorphism. In the proof of well-definedness of usual Floer homology groups, we need the functorial property of Floer homology groups with respect to Riemannian bordisms. We shall generalize the functorial property to Fukaya-Floer homology groups and prove the well-definedness by using this property.
The main statement of this subsection is Proposition 2.23. Take two Riemannian metrics g 0 , g 1 on Y and two sets {s l (ρ, σ)} ρ,σ,l , {s ′ l (ρ, σ)} ρ,σ,l of admissible sections. We write HF F * (Y ; γ), HF F ′ * (Y ; γ) for Fukaya-Floer homology groups associated with the metrics and sections. Then we have a canonical isomorphism between HF F * (Y ; γ) and HF F ′ * (Y ; γ). Assume that we have the following data: • oriented, compact surfaces Σ l embedded in X with boundary γ l γ ′ l .
Lemma 2.24. Using the data (X,Ĝ,Σ l ,ŝ l ), we can define a homomorphism with the following property: • The homomorphism ζ is a chain map. That is, is independent of the metric G and the sectionsŝ l (ρ, σ).
The metrics G, G ′ naturally induce a metric G (T ) on X (T ) and we have surfaces Σ We can derive Proposition 2.23 from Lemma 2.24. Take two Riemannian metric g 0 , g 1 on Y and two sets . On the other hand, the metric G 0 = g 0 + dt 2 and the pull-backs p * (s l (ρ, σ)) of s l (ρ, σ) by . It follows from the construction of ζ we will see below that ζ ′′ * is the identity map. The third part of Lemma 2.24 implies We give outline of the proof that ζ has the properties stated in Lemma 2.24.
The first part follows from a similar discussion to that in the proof that ∂ • ∂ = 0. Take a generator [τ ] ∈ CF j−2β 1 −1 (Y 1 ). Then we have a cut-down moduli space MX (ρ, τ ; L 1 \L 0 ) with dimension 1. Counting the number of the end of this moduli space, we get To prove the second part, take other metricĜ ′ and sectionsŝ ′ l (ρ, σ) overX. Then we have another map ζ ′ : CF F * (Y 0 ; γ) → CF F * (Y 1 , ; γ ′ ). We will construct a chain homotopy Take a path {G s } 0≤s≤1 of metrics on X from G to G ′ , and put Then M (ρ, σ) is smooth for generic paths of metrics. We can define line bundles L ⊗2 l (ρ, σ) over M (ρ, σ) as usual. As in Subsection 2.3, for ρ, σ with dim M(ρ, σ) < 8, we can construct sections s l (ρ, σ) such that s(ρ, σ) are compatible with gluing maps, {s l (ρ, σ)} l satisfy the transversality conditions as in Proposition 2.8, and We can show the chain homotopy condition (25) by counting the number of the ends of the cut-down moduli spaces M (ρ, τ ; L 1 \L 0 ) for generators [τ ] ∈ CF j−2β 1 (Y 1 ). Thus we obtain the second part of Lemma 2.24.
The proof of the third part is essentially same as the proof of the gluing formula for Donaldson invariants which will be given in the next subsection, and we omit the proof here.
We give a remark on the dependence of HF F * (Y ; γ) on γ. It seems that HF F * (Y ; γ) depends only on the homology classes [γ l ] ∈ H 1 (Y ; Z). If γ l and γ ′ l are homologous, then we have oriented, compact, surfaces Σ l in Y × [0, 1] with boundary γ l γ ′ l . As above we can construct a linear map To prove that ζ Σ induces an isomorphism from HF F * (Y ; γ) to HF F * (Y ; γ ′ ), it is sufficient to show that the map ζ Σ has the properties as in Lemma 2.24. The most difficult part of the proof is to prove that the induced map between the Fukaya-Floer homology groups is independent of Σ.
Suppose that we have another bordism Σ ′ l from γ l to γ ′ l . We would like to show that the two maps between the Fukaya-Floer homology groups induced by ζ Σ and ζ Σ ′ are the same. A natural way to prove this is to use an isomorphism between LΣ 1 (ρ, σ) and LΣ′ 1 (ρ, σ) which is compatible with the gluing maps in an appropriate sense, if the isomorphism exists. Hence the problem reduces to the existence of the isomorphism. This can be regarded as a generalization of the fact that the index of a family of elliptic differential operators on a closed manifold depends only on the bordism class.
We can take a compact, oriented manifold W with corner, whose boundary is For a genericŝ, the zero locus W ofŝ is a compact, oriented manifold whose boundary is (26). 1] to W , we get a "bordism"Ŵ fromΣ l toΣ ′ l , i.e. W is an oriented manifold with boundaryΣ l Σ ′ l . If we mimic the proof of the invariance of the index of operators over closed manifolds with respect to bordism, it may be possible to prove that there is an isomorphism from LΣ l (ρ, σ) to LΣ′ l (ρ, σ) which is compatible with the gluing maps. However, we do not discuss this in this paper and we just state it as a conjecture.
If this is true, HF F * (Y ; γ) depends only on the homology classes [γ l ] ∈ H 1 (Y ; Z).

Relative Donaldson invariants.
In this subsection, we will see that the Fukaya-Floer homology groups allow us to extend the Donaldson invariants for closed 4-manifolds to 4manifolds with boundary. Let Y be a closed, oriented 3-manifold and We also denote Σ l ∪(γ l ×R ≥0 ) byΣ l for surfaces Σ l embedded in X 0 with boundary γ l . Assume that X 0 is simply connected and b + > 1. Moreover suppose that we have a U(2)-bundle P 0 on X 0 such that the restriction Q = P 0 | Y satisfies Hypothesis 2.1. Take a metricĝ 0 onX 0 such that the restriction ofĝ 0 to Y × R ≥0 is equal to g Y + dt 2 for some metric g Y on Y . Using families of twisted∂ operators overΣ l , we can define complex line bundles L ⊗2 l (ρ) over the moduli space MX 0 (ρ) of instantons on the extensionP 0 of P 0 toX 0 with limit ρ. As before, we can show that for ρ with dim MX 0 (ρ) < 8 there are sections s l (ρ) of the line bundles with properties similar to those in Proposition 2.8.
Let ρ 0 be the fixed flat connection used to define δ Y . For simplicity of notations, we suppose that where A(ρ 0 ) is a connection onP 0 with limit ρ 0 . Then we have Under these hypotheses, we define an element The following proposition follows from a combination of a discussion of the case when Y is a homology 3-sphere and techniques we have developed in this paper.
. This is true if an analogy of Conjecture 2.25 holds for the line bundles L ⊗2 l (ρ).

Gluing formula.
We consider a situation where a closed, oriented 4-manifold X is cut into two parts X 0 , X 1 along a closed 3-manifold Y . Let P be a U(2)-bundle over X such that the restriction Q = P | Y satisfies Hypothesis 2.1 and dim M P = 2d for some integer d with 0 ≤ d ≤ 3. We suppose that X 0 , X 1 are simply connected and b + of X 0 , X 1 are larger than of the closed manifold X in terms of the relative invariants Ψ X 0 , Ψ X 1 . To do this, we define a pairing For flat connections ρ, we have Hence CF j (Y ) and CF −j−3 (Ȳ ) are dual to each other. We define a pairing Here L is a subset of {1, . . . , d} with |L| = β and L c is the complement of L. It is easy to see the following.
Therefore we get the pairing on Fukaya-Floer homology groups.
Theorem 2.29. In the above situation, we have To prove the gluing formula, we need sections of L ⊗2 Σ l which are compatible with gluing maps as usual. Fix a Riemannian metric g Y on Y and choose Riemannian metrics g 0 , g 1 on X 0 , X 1 such that the restrictions to Y are equal to g Y . For each T > 0, we define a manifold The transversality conditions for the sections imply that MX 0 (ρ; L) and MX 1 (ρ; L c ) are finite sets. From an argument like that in the proof of (7), we can show there is a natural identification for large α. Counting the number of elements of both sides with signs, we obtain the gluing formula.

Floer homology for 2-torsion instanton invariants
In this section, we consider a variant Ψ u 1 X of Donaldson invariants for non-spin 4-manifolds X, which is a linear function This invariant is defined by a 2-torsion cohomology class u 1 of the moduli spaces. Originally this was defined for spin manifolds by Fintushel-Stern [9], and the author extended to non-spin 4-manifolds in [13]. In this section, we construct a variant of Floer homology group and prove a gluing formula for Ψ u 1 X . In [13], it was shown that Ψ u 1 X is non-trivial for X = CP 2 0 #CP 2 1 #CP 2 , where CP 2 0 and CP 2 1 are copies of CP 2 . More precisely Here H 0 , H 1 and E are the generators of H 2 (CP 2 0 ; Z), H 2 (CP 2 1 ; Z) and H 2 (CP 2 ; Z) respectively. We can consider X = CP 2 0 #CP 2 1 #CP 2 as a connected sum of X 0 = CP 2 0 and X 1 = CP 2 1 #CP 2 . Since the only flat connection over S 3 is the trivial one, we can deduce from (27) that the trivial flat connection has an important role in the gluing formula for Ψ u 1 X in contrast to the case of usual Donaldson invariants. Note also that the homology class −H 1 + E in (27) is a sum of homology classes −H 1 and E of X 0 and X 1 with self-intersection numbers odd.
On the other hand, for closed, simply connected, non-spin 4-manifolds X 0 , X 1 with b + positive, we can show the following vanishing theorem. For homology classes This follows from a standard dimension-counting argument. The formula (28) implies that the key of the non-vanishing result (27) is that the homology class −H 1 + E is split into two homology classes of X 0 , X 1 with self-intersection numbers odd.
In this section, we consider a situation where X has a decomposition X = X 0 ∪ Y X 1 for a homology 3-sphere Y and a closed, oriented surface Σ in X is split into two surfaces with self-intersection numbers odd along Y . We will give a gluing formula for Ψ u 1 X ([Σ]) in terms of differential-topological date about X 0 , X 1 . This situation is similar to that of Section 2 and we can apply the method developed in Section 2. The main difference is that a U(2)-bundle over a homology 3-sphere has the trivial flat connection, which is reducible, and we need to pay attention to the effect of the trivial flat connection on the gluing formula.
Let X be a closed, oriented, simply connected, non-spin 4-manifold with b + > 1. Take a U(2)-bundle P over X with w 2 (P ) = w 2 (X). For [Σ] ∈ H 2 (X; Z), we have the complex line bundleL We can show that if [Σ] · [Σ] ≡ 0 mod 2 then the center {±1} of SU(2) acts trivially on the line bundle. Hence we get a line bundle Let c be a spin-c structure of X. For each connections A on P , we have the twisted Dirac operator Here E is the rank two complex vector bundle associated with P and S ± are the spinor bundles.
If c 1 (det c) = −c 1 (P ), we have a "real part" of the Dirac operators: (See [1].) The family of real operators We also suppose that c 2 (P ) ≡ 0 mod 2. Then we can see that the center {±1} of SU(2) acts trivially onΛ, and we obtain a real line bundle to be w 1 (Λ). When b + (X) is even, the virtual dimension of the moduli space M P is odd. We can write dim M P = 2d + 1 for some integer d. Assume d ≥ 0. Then we define In general M P is not compact, however we can define the pairing as follows. Let s Σ i : B * Σ i → L Σ i be sections and V Σ i be the zero loci. We can prove the following lemma by a standard dimension counting argument.
When dim M P is 2d + 1, the intersection is independent of the metric on X and sections of L Σ i , and hence it is a differential-topological invariant of X.
Hence we need not to use the pull-back r * Σ 1 (s Σ 1 ). We can use any section of r * Σ 1 (L Σ 1 ) which is transverse to the zero section.
3.2. Line bundles. In this subsections, we introduce line bundles over B * Y ×R (ρ, σ) which are defined by families of Dirac operators over Y × R and twisted∂ operators over surfaces in Y × R. These are used to define a variant of Floer homology groups for 2-torsion instanton invariants.
Let Y be a homology 3-sphere and Q = Y × U(2) be the trivial U(2)-bundle over Y . Fix a connection a det on Q det = Y × U(1). As before, connections on Q and π * Q are compatible with a det , unless explicitly stated otherwise.
We assume the following. For irreducible flat connections ρ, σ, we define B * Y ×R (ρ, σ),B * Y ×R (ρ, σ) as before. We have just one gauge equivalence class of projectively flat connections which are not irreducible. It is represented by a connection θ 0 which induce the trivial connection on the adjoint bundle of Q. Fix a smooth map g 1 : Y → SU(2) with degree 1 and put θ a := g a Y ×R (ρ, σ) induced by families of the real part of Dirac operators. Since the action of {±(1, 1)} ⊂ Γ ρ × Γ σ on the line bundle is not trivial in general, the line bundle may not descend to B * Y ×R (ρ, σ). To avoid this problem, we will introduce real line bundles R(θ a , ρ) and R(σ, Then we have the Dirac operators We denote the numerical index of these operators by The wight of the action of Here It follows from the additivity of index and the index formula that and we can see that this isomorphism induces an isomorphism from Λ ab (ρ, σ) to Λ a ′ b ′ (ρ, σ). Therefore we obtain: Definition 3.6. For irreducible flat connections ρ, σ and a ∈ {0, 1}, we write Λ (a) (ρ, σ) for Λ aa (ρ, σ). Moreover we define u in general, we can however apply the technique used to define Λ ab (ρ, σ).
Let θ γ,0 be the restriction of θ 0 to γ. Choose a smooth map g γ : γ → U(2) such that the homotopy class [g γ ] is the generator of π 1 (U(2)) ∼ = Z. Then for a ∈ Z we put θ γ,a := g a γ (θ γ,0 ). Note that these connections are not compatible with the fixed connection a det on Q det . For a, b ∈ Z and irreducible flat connections ρ, σ, we take connections A(θ γ,a , ρ), A(σ, θ γ,b ) on π * Q such that (These connections are not compatible a det either.) Let C(θ γ,a , ρ) and C(σ, θ γ,b ) be the trivial complex line bundles overB * Y ×R (ρ, σ) with Γ ρ -action of weight Ind −+∂ A(θγ,a,ρ) and Γ σ -action From the additivity of index and the index formula, we have This lemma implies that for a, b ∈ Z with a ≡ b mod 2 we can define . We can also show that for a ≡ b ≡ a ′ ≡ b ′ mod 2 we have a natural isomorphism between L Γ,ab (ρ, σ) and L Γ,a ′ b ′ (ρ, σ). We get two bundles L The additivity of index implies that This is a contradiction.
This lemma means that we can apply the method in the previous section to showing that we can take admissible sections of the line bundles Λ (a) (ρ, σ), L Thus we obtain the following.
To prove (29), we need to describe the ends of M 0 Y ×R (ρ, σ) as in the previous section. In the cases (1), (2), (3), the dimension of the moduli space M 0 Y ×R (ρ, σ) is less than 4. Lemma 3.8 means that (29) follows from the same discussion in the previous section. Therefore we need to consider only the case (4).
. We can show this by using a partition of unity on the moduli space as Proposition 2.8. The following lemma follows from the usual dimension-counting argument.
]) with r > 1. Then we have either • r = 2 and This lemma means that the reducible flat connections θ n do not appear in the description of the end of M 0 Y ×R (ρ, σ) if [ρ] ∈ CF 8n+1 (Y ) for any n. Therefore (29) follows from a discussion like that in the previous section in this case.
From now on, we assume that [ρ] ∈ CF 8n+1 (Y ), [σ] ∈ CF 8n−4 (Y ). To describe the end of the cut-down moduli space M 0 Y ×R (ρ, σ; L β ), we put Here  (1)), M 0 Y ×R (ρ(1), σ) satisfy the transversality conditions in Lemma 3.9. Hence from a discussion like that in the proof of (7), we have for large T 1 . Perturbing the sections s Λ , s L outside the end (30) of M ′ , we get a 1-dimensional, smooth, compact manifold with boundary  Proof. Let X 1 , X 2 be two copies of Y × R. And we write X 1 (T ), we get a manifold X #(T ) . Let E be the rank-two complex vector bundle over Y associated with Q. Take a trivialization φ of E such that θ n is trivial under φ. We write E i for π * i E and φ i for π * i φ. Here π i are the projections from X i (T ) to Y . Forũ ∈ SU(2) we define a vector bundle E(ũ) by Gluing two instantons  (2). We have an action of {±1} on this bundle defined by the following diagram: Here A(θ γ,1 ,θn|γ) . Here A(θ γ,1 , θ n | γ ) is the connection on Γ = γ × R with limits θ γ,1 , θ n | γ . Let g 1 be the fixed map from Y to SU(2) with degree 1. The class [g 1 | γ ] represented by the restriction of g 1 to γ is trivial in π 1 (U(2)). Hence we have A(θ γ,1 ,θ γ,0 ) = −1 by Lemma 3.7. Therefore the restriction L Γ (ρ, σ). Remark 3.17. We assumed that Y is a homology 3-sphere, and it is sufficient to take a very small loop γ which is trivial in π 1 (Y ) when we calculate invariants using the gluing formula. However we have considered a general loop γ which may not be trivial in π 1 (Y ). One of the reason is that we would like to prepare techniques to extend the construction to more general 3-manifolds. Another reason is that the dependence of I * (Y ; γ) on γ is not clear and investigating the dependence may be an interesting problem.
We show the subtlety of the dependence of I * (Y ; γ) on γ. Let γ be a very small loop in Y whose homotopy class is trivial in π 1 (Y ). Then we can see that the complex line bundles L  Γ (ρ, σ) is defined to be the quotient of C(θ γ,a , ρ) ⊗L Γ (ρ, σ) ⊗ C(σ, θ γ,a ) by the action of Z 2 × Z 2 , and the action of Z 2 × Z 2 is not trivial in general. Therefore we do not get a canonical trivialization of L  3.4. Relative invariants. Let X be a closed, oriented, simply connected, non-spin 4-manifold with b + (X) positive and even. We consider a situation where we have a decomposition X = X 0 ∪ Y X 1 . Here Y is an oriented homology 3-sphere and X 0 , X 1 are compact, simply connected, non-spin 4-manifold with boundary Y ,Ȳ and b + (X 0 ), b + (X 1 ) > 1. We consider invariants Ψ u 1 X,P (z), where P is a U(2)-bundle over X with w 2 (P ) ≡ w 2 (X) and dim M P = 3 and z ∈ H 2 (X; Z) with z · z ≡ 0 mod 2. The purpose is to write Ψ u 1 P (z) in terms of data from X 0 , X 1 as in the case of the usual Donaldson invariants.
We fix a trivialization ϕ 0 of Q := P | Y and we write θ 0 for the trivial connection. Choose a smooth map g 1 from Y to SU(2) of degree 1. For n ∈ Z, we denote g n 1 (θ 0 ) by θ n as before.
This lemma means that the construction of the gluing formula for Ψ u 1 X (z) is the same as that in Subsection 2.8 if dim MX 0 (θ n ) = 0 for all n ∈ Z. Hence we suppose the following. Hypothesis 3.19. There is an integer n 0 with dim MX 0 (θ n 0 ) = 0.
We will define relative invariants for X 0 under this assumption. Put P 0 := P | X 0 and let γ ∼ = S 1 be a loop in Y and Σ 0 be an oriented surface embedded in X 0 with boundary γ. Using Dirac operators associated with a spin-c structure c 0 over X 0 with c 1 (det c 0 ) = −c 1 (P 0 ) and twisted∂ operators overΣ 0 , we have line bundles . For irreducible flat connections ρ and a ∈ {0, 1} we put . Here θ γ,1 is defined as the previous subsection. Then Γ ρ = Z 2 naturally acts on these bundles. Lemma 3.20. Let ϕ 0 be the fixed trivialization of Q and put ϕ a := g a 1 ϕ 0 . When Proof. The wight of the Γ ρ action onΛ (a) (ρ) is Ind + D A(θa) . Hence it is sufficient to show that the index is even if c 2 (P 0 ; ϕ a ) is even. Choose a compact, spin 4-manifold X ′ 0 with boundarȳ Y and we consider a manifold X ′ = X 0 ∪ Y X ′ 0 . Let P ′ be a U(2)-bundle over X ′ obtained by gluing P 0 and the trivial bundle P X ′ 0 over X ′ 0 through ϕ a . Let c ′ 0 be a spin-c structure over X ′ 0 induced by a spin structure. Then we have a spin-c structure c ′ on X ′ induced by c 0 , c ′ 0 and we have the Dirac operator Here A ′ is a connection induced by A(θ a ) and the trivial connection θ X ′ 0 on P X ′ 0 and E ′ is the complex vector bundle of rank two associated with P ′ . Since D θ X ′ 0 is the direct sum of two copies of a Dirac operator, the index Ind − D θ X ′ 0 is even. Thus we get Applying the index formula, we have is the index of a spin-c Dirac operator, it is an integer. Hence the first term is even. From c 1 (det c ′ ) = −c 1 (E ′ ), the second term is zero. Therefore we obtain Thus the first part of Lemma follows.
These numbers define a chain ψ u 1 8n 0 (Y ) as usual. As before, we have: −8n (Ȳ ), we define < θ n , θ −n >≡ 1 mod 2. Then we get a pairing We can easily show This lemma means that the pairing induces a pairing The main statement in this section is the following.
Theorem 3.26. Let X be a closed, oriented, simply connected 4-manifold with b + positive and even. Assume that X has a decomposition X = X 0 ∪ Y X 1 , where Y is a homology 3sphere and X 0 , X 1 are compact, non-spin, simply connected 4-manifolds with boundary Y ,Ȳ . Take a loop γ in Y and compact, oriented surfaces Σ 0 , Σ 1 in X 0 , X 1 with boundary γ. If the self-intersection numbers of We can generalize this formula to the case when the dimension of M X,P is more than 3. Assume that the dimension of M X,P is 2(d 0 + d 1 ) + 3 for some d 0 , d 1 ∈ Z ≥0 . For x 1 , . . . , x d 0 ∈ H 2 (X 0 ; Z), y 1 , . . . , y d 1 ∈ H 2 (X 1 ; Z) with x i · x i ≡ y j · y j ≡ 0 mod 2, we can define relative invariants . Here Σ i are surfaces in X i as in Theorem 3.26. Then we have Applying this formula to the case Y = S 3 , we immediately obtain the following.
Remark 3.28. In general, usual Donaldson invariants Ψ P (x 1 , . . . , x d ) do not lie in Z even if x i ∈ H 2 (X; Z). However if < w 2 (P ), x i >≡ 0 mod 2 for all i then the value Ψ P (x 1 , . . . , x d ) lies in Z. Therefore the above formula makes sense.
As mentioned before, if there are no integers n such that dim MX 0 (θ n ) = 0 then the proof of Theorem 3.26 is quite the same as that of Theorem 2.29. Hence we suppose Hypothesis 3.19. To prove Theorem 3.26, we fix a sequence T α → ∞ and consider manifolds X α = X 0 ∪ (Y × [0, T α ]) ∪ X 1 as usual. As in Subsection 2.3, we can take sections s α − 1)). Moreover suppose that ρ(i) are irreducible for i = 0, . . . , r − 1. Then we have . A standard dimension-counting argument shows that for any sequence Here SO (3) [A] is the image of SO(3) by a gluing map as in the proof of Lemma 3.11. Therefore we get Thus we must show This follows from the following lemma. Proof. As in the proof of Lemma 3.14, we can see that a necessary and sufficient condition for the restrictions of the line bundles to be non-trivial is that the indexes Ind +∂ A 0 , Ind + D A 0 are odd. The calculations in the proof of Lemma 3.20 show that Therefore we obtain the statements.

A non-existence result
Let X be a closed, oriented, simply connected 4-manifold and Q X be the intersection form of X on H 2 (X; Z). For positive integers k and [Σ 1 ], . . . , [Σ 2k ] ∈ H 2 (X; Z), we define In [4], Donaldson showed that when X is spin and b + = 1, Q X must satisfy the condition In this section, we will consider the intersection forms of compact 4-manifolds with boundaries homology 3-spheres. We prove that the condition also holds for compact, spin 4-manifold X with b + = 1 and with boundary a homology 3-sphere Y , if HF 1 (Y ; Z 2 ) and HF 2 (Y ; Z 2 ) are trivial. Here HF * (Y ; Z 2 ) are the usual Floer homology groups with coefficients Z 2 . This is originally due to Fukaya, Furuta and Ohta. (See [12].) Making use of techniques developed in this paper, we write down the proof of this theorem.

Relative invariants for spin 4-manifolds.
In the previous section, we defined relative invariants for non-spin 4-manifolds with boundary. Here we define relative 2-torsion instanton invariants for spin 4-manifolds, which are used to prove Theorem 4.1. To define the invariants, we modify the construction in the previous section.
We denote by θ the trivial connection on P | Y with respect to ϕ. The dimension of the moduli space MX (θ) of instantons onX with limit θ is For generators [ρ] ∈ CF β (Y ), we have moduli spaces MX (ρ) of dimension 10 − β.
Let Σ be a closed, oriented surface embedded in X. We put B * Σ,+ = B * Σ ∪ {[θ Σ ]}. It is known that the determinant line bundle L Σ → B * Σ extends to B * Σ,+ . (See [8].) We use the same notation L Σ for the extension. We can take a section s Σ : B * Σ,+ → L Σ such that s Σ does not vanish near [θ Σ ]. We always assume that sections of L Σ have this property. Let V Σ be the zero locus of s Σ and we define MX (ρ) ∩ V Σ to be A standard dimension-counting argument shows the following: (1) If [ρ] ∈ CF 2 (Y ) and dim MX(ρ) = 8 then r = 0, Z 0 = ∅.
These define a chain ψ u 1 X ∈ C    We can easily show this by considering the long exact sequence associated with a short exact sequence Then we have a linear map D (a) from C  We can see that we are in one of the following cases: (1) r = 0, Z 0 = ∅.
. We describe the ends of the moduli space relevant to the case (2). Let x 1 ∈ Σ l 1 ∩ Σ l 2 , x 2 ∈ Σ l 3 ∩ Σ l 4 for distinct l 1 , . . . , l 4 and take small neighborhoods U x 1 , U x 2 inX. There is an SO(3)-equivariant map (3) is homeomorphic to an open set in MX (θ). As in [4] we can see that where ℓ is a loop in SO(3) which represents the generator of π 1 (SO(3)) ∼ = Z 2 . For each x = (x 1 , x 2 ), we have a gluing map which is given by gluing the trivial connection onX and two copies of the fundamental instanton J over S 4 at x 1 , x 2 . The intersection of the image of this map and V Λ is one of the ends of the moduli space.

Y ). Then we have a gluing map
Gl Put If we perturb the section s Λ over a compact set in M ′ then the intersection M ′ ∩V Λ is a smooth compact manifold with boundary Here [ρ] runs over the generators of CF 2 (Y ) and [σ] runs over the generators of CF 1 (Y ). Therefore we get We can see that # Gl x (ℓ, T 0 ) ∩ V Λ ) ≡ 1 mod 2 for each x. This follows from arguments similar to those which deduce (31). The point is that J is an instanton on an SU(2)-bundle P ′ with c 2 (P ′ ) = 1 and hence the index Ind D J is odd. Therefore we get as required.

Introduction
In this paper, we construct a variant of instanton Floer homology groups and prove a gluing formula for a variant of Donaldson invariants introduced in [19].
As S. K. Donaldson showed in [4,5,6,7], the moduli spaces of instantons have crucial information about the topology of closed, oriented, smooth 4-manifolds X. Mainly there are two methods to draw out the information from the moduli spaces. First one is to describe the structures of the singular points and the ends of the moduli spaces [4,5]. The description gives strong restrictions to the realization of unimodular quadratic forms as the intersection form of a smooth 4-manifold. Second one is to integrate cohomology classes over moduli spaces [6,7]. This gives differential-topological invariants of 4-manifolds which distinguish different smooth structures on the same topological 4-manifold.
It used to be hard to compute Donaldson invariants in general. However instanton Floer theory gives us a way to compute Donaldson invariants Ψ X when X has a decomposition X = X 0 ∪ Y X 1 for some compact 4-manifolds X 0 and X 1 with boundary Y andȲ . Here Y is a closed, oriented 3-manifold andȲ is Y with the opposite orientation. A. Floer introduced instanton homology groups HF * (Y ) for homology 3-spheres Y in [11]. Floer's groups allow us to generalize Donaldson invariants for compact, oriented 4-manifolds X 0 whose boundaries are homology 3-spheres Y . The relative invariant Ψ X 0 is an element of HF * (Y ), which is defined by integrating cohomology classes of the moduli spaces of instantons overX 0 = X 0 ∪(Y ×R ≥0 ). Assume that a closed 4-manifold X has a decomposition X = X 0 ∪ Y X 1 for some homology 3sphere Y and two compact 4-manifolds X 0 , X 1 with boundary Y ,Ȳ and with b + (X 0 ), b + (X 1 ) > 1. Here b + stands for the dimension of a maximal positive subspace of the intersection form on H 2 (X). Then we have the relative invariants Ψ X 0 ∈ HF * (Y ), Ψ X 1 ∈ HF * (Ȳ ). There is a natural pairing < , >: HF * (Y ) ⊗ HF * (Ȳ ) −→ Q, and we have a relation Instanton Floer homology groups also enable us to generalize the results on the non-existence of closed 4-manifolds with b + = 0 and with non-standard intersection forms. Donaldson [8] showed that if HF 1 (Y ) = 0 then there is a restriction to the realization of unimodular quadratic forms as the intersection form of a 4-manifold X 0 with boundary Y and with b + = 0. The 2000 Mathematics Subject Classification. Primary 57R57, 57R58. 1 proof involves the description of the singular points and the ends of the moduli spaces of instantons overX 0 as in the case of closed 4-manifolds. We also refer to K. A. Froyshov's papers [13,14,15]. He gave obstructions for (rational) homology 3-spheres to bound a 4manifold, using Seiberg-Witten theory and "h invariant" defined by instanton Floer theory.
On the other hand, there are some variants of Donaldson invariants. R. Fintushel and R. Stern defined a variant of Donaldson invariants for closed, simply connected, spin 4-manifolds in [10]. They used a 2-torsion cohomology class u 1 ∈ H 1 (M P ; Z 2 ) of moduli spaces M P which was originally defined by Donaldson in [5]. In [1], S. Akbulut, T. Mrowka and Y. Ruan extended the construction of u 1 to non-spin 4-manifolds and they showed a universal constraint of Donaldson invariants for non-spin 4-manifolds. Using u 1 for non-spin 4-manifolds, the author [19] defined a variant Ψ u 1 X of Donaldson invariants for non-spin 4-manifolds, which can be regarded as an extension of Fintushel-Stern's invariant to non-spin 4-manifolds. A remarkable feature of these variants is that they are non-trivial for connected sums of the form X = Y #S 2 × S 2 in general.
In [17], it was announced (without proof) that K. Fukaya, M. Furuta and H. Ohta showed a non-existence result of compact, spin 4-manifolds with b + = 1 and with some intersection form when their boundaries are diffeomorphic to some homology 3-spheres. In [17], Furuta explained that a variant of instanton Floer homology groups and an extension of Ψ u 1 X to compact, spin 4-manifolds with boundary a homology 3-sphere are used in the proof. This is a nice generalization of a result in [5], however, explicit examples were not given.
The main purpose of this paper is to construct a variant of instanton Floer homology groups and to extend the relation (1) to Ψ u 1 X . As a corollary, we obtain a non-vanishing result for connected sums of some 4-manifolds. Moreover we will write down a proof of Fukaya-Furuta-Ohta's non-existence result using the variant of instanton Floer homology groups. In the proof, we make use of the structure of the ends of some moduli spaces as in [5]. There is a difference between our variant of instanton Floer homology and Fukaya-Furuta-Ohta's one. In our construction we use the determinant line bundles of families of∂-operators over γ × R.
Here γ is a loop in Y . This enables our gluing formula to be applied to not only the case when 4-manifolds are spin but also the case when 4-manifold are non-spin.
In [19], the author showed that the variant of Donaldson invariants is non-trivial for 2CP 2 #CP 2 . As we will explain in Section 3, this calculation is suggesting that we need to take into account the following aspects in the construction of our variant of instanton Floer homology groups: • The trivial flat connection on Y should play an important role in the gluing formula for Ψ u 1 X . • We need a similar construction to Fukaya's extension [16] of instanton Floer homology groups.
In the gluing formula for the usual Donaldson invariants, the trivial flat connection has no contribution. Hence the contribution of the trivial flat connection to the gluing formula for Ψ u 1 X is a new phenomenon. See Donaldson's book [8] and Froyshov's papers [14,15] for other treatments of the trivial flat connection. In Fukaya's construction, we use the determinant line bundles of∂-operators over γ × R, and we need sections of the determinant line bundles with certain properties. The properties have to do with non-compactness of the moduli spaces over Y × R and transversality. (See Proposition 2.8.) We call sections having the properties admissible sections. One of the main purposes of this paper is to give a construction of admissible sections of determinant line bundles and it will be done in Section 2.
Instanton Floer homology groups are defined for any homology 3-spheres. For other 3manifolds Y , there are some problems coming from reducible flat connections over 3-manifolds and the decomposition of homology classes of X along Y . In order to overcome the first difficulty, D. M. Austin and P. J. Braam [2] introduced an equivariant version of instanton Floer homology for 3-manifolds Y with b 1 (Y ) = 0 or H 1 (Y ; Z) torsion free. Kronheimer and Mrowka [18] gave another method to treat reducible solutions in the context of Seiberg-Witten theory. On the other hand, Fukaya's construction [16] gives an effective way to overcome the second difficulty when there are no reducible flat connections. However it does not seem that we have a complete method to construct suitable instanton Floer homology groups fitting to the situation where we treat both the problems at the same time. To construct a gluing formula for our invariant Ψ u 1 X , we must treat both the problems at the same time, even when the 3manifold Y is a homology 3-sphere. Although we treat only homology 3-spheres in this paper, we can regard our construction as a first step to give a method simultaneously applicable to both the problems.
The organization of this paper is as follows. In Section 2, we recall Fukaya's construction. We will basically follow [16] and [3], however, with some modification. In Section 3, making use of techniques developed in Section 2, we will introduce a variant I (a) * (Y ; γ) of instanton Floer homology groups for homology 3-spheres Y , loops γ in Y and a ∈ {0, 1}. We define relative invariants Ψ u 1 X 0 ∈ I (a) * (Y ; γ) for compact 4-manifolds X 0 with boundary Y . We also prove a formula for Ψ u 1 X similar to (1) (Theorem 3.26). In particular, we deduce a non-vanishing result for connected sums of 4-manifolds (Corollary 3.27). In Section 4, we will prove the result on the non-existence of spin 4-manifolds with boundaries some homology 3-spheres and with some intersection forms (Theorem 4.1).
Acknowledgments . The author is grateful to Mikio Furuta for his suggestions. He would also like to thank Yukio Kametani and Nobuhiro Nakamura for useful conversations.

Fukaya-Floer homology
Let X be a closed, oriented 4-manifold with b + > 1 and take a U(2)-bundle P over X and assume that the dimension of the moduli space of instantons on P is 2d for some integer d ≥ 0. We can associate cohomology classes µ([Σ]) of degree 2 on the moduli space to homology classes [Σ] ∈ H 2 (X; Z). Here Σ is a closed, oriented surface embedded in X which represent the homology class. Suppose that we have a decomposition X = X 0 ∪ Y X 1 and that the surfaces Σ l are split into two surfaces with boundary γ l ∼ = S 1 along Y . We consider how to recover the Donaldson invariants of X from relative invariants of X 0 and X 1 . Fukaya's construction [16] provide us suitable homology groups to define the relative invariants. In the construction, we need sections of determinant line bundles of families of twisted∂ operators on γ l × R which behave nicely on the ends of moduli spaces of instantons over Y × R and satisfy certain transversality conditions. We call such sections admissible. The main point of this section is the construction of admissible sections of the determinant line bundles (Proposition 2.8).
2.1. Outline of proof of gluing formula. In this subsection, we give a sketch of the construction of a gluing formula for Donaldson invariants.
Let X be a closed, oriented, simply connected smooth 4-manifolds with b + > 1 and odd, and P be a U(2)-bundle over X with w 2 (P ) non-trivial. We write P det for the U(1)-bundle over X induced by P , and fix a connection a det on P det . We take a Riemannian metric on g on X and suppose that the virtual dimension of M P is 2d with d ≥ 0. Here M P = M P (g) is the moduli space of instantons on P which induce the connection a det on P det . The moduli space M P is a smooth manifold for generic metrics g, and a choice of an orientation of the space H + g (X) of self-dual harmonic 2-forms on X orients M P . The moduli space M P gives us the Donaldson invariant Ψ X,P : H 2 (X; Z) ⊗d −→ Q. This is defined as follows.
Let [Σ] be a class in H 2 (X; Z). We denote byB * Σ the space of gauge equivalence classes of framed, irreducible connections on the restriction P | Σ of P to Σ which are compatible with a det | Σ . We have the determinant line bundlẽ Σ ) is the∂ operator twisted by the rank-two complex vector bundle E associated to P and a square root K Σ is trivial, we have the line bundle L ⊗2 Σ may not be a square of a genuine line bundle. Take a section s Σ of L ⊗2 Σ and denote the zero locus by V Σ . We define M P ∩ V Σ by Since the restriction of elements of M P to Σ may not be irreducible, this is not well-defined. But we can avoid this problem by replacing Σ with a small neighborhood ν(Σ) of Σ. (See [9].) Take d homology classes [Σ 1 ], . . . , [Σ d ] ∈ H 2 (X; Z). We can show that the intersection We can see that this is independent of the choices of metric and sections.
Suppose that X has a decomposition X = X 0 ∪ Y X 1 . Choose a Riemannian metric g Y on Y and let g 0 , g 1 be Riemannian metrics on X 0 , X 1 whose restrictions to Y are equal to g Y . Assume that the dimension of the moduli space M P is 2d with 0 ≤ d ≤ 3. Take homology We assume that γ l are diffeomorphic to S 1 for all l. We show how to compute Ψ X ([Σ 1 ], . . . , [Σ d ]) from data of X 0 , X 1 briefly under this situation. We use some facts about instantons which can be found in [8].
Take a sequence {T α } ∞ α=1 of positive real numbers which diverges to infinity. We have manifolds which converge to projectively flat connections ρ(0), ρ(r−1) at infinity. MX 0 (ρ(0)) is a moduli space of instantons overX 0 with limit ρ(0) and similarly for MX 1 (ρ(r − 1)). For is defined by translations. ) Since dim M P is less than 8, bubbling phenomena do not occur. As we will see below, we can take sections s α l of L ⊗2 then at least one of the components of the limit of s α Σ ([A α ]) vanishes. A dimension counting argument shows that r is 1 and that Here L is a subset of {1, . . . , d} and L c is its complement. We denote the number of elements of L by |L|. Then we also have are finite (if the sections over the moduli spaces satisfy some transversality conditions and behave nicely on the end of the moduli spaces). A standard theory of gluing of instantons shows that Here #MX 0 (ρ; L) · #MX 1 (ρ; L c ).
From this formula, formal sums The formal sums ψ X 0 , ψ X 1 depend on the metrics and sections. We will define a boundary map such that the composition ∂ • ∂ is identically zero, and show that ∂ψ X 0 = 0 and the class The formula (4) implies This is the gluing formula.

Fukaya's construction.
Let Y be a closed, oriented 3-manifold. Take a Riemannian metric g Y on Y and a U(2)-bundle Q over Y , and fix a connection a det on the U(1)-bundle Q det induced by Q. We consider connections on Q which induce the fixed connection a det on Q det . Let A Q be the space of connections on Q with fixed determinant and G Q be the space of automorphisms of Q of determinant 1. The Chern-Simons functional is an S 1 -valued functional on the quotient space B Q = A Q /G Q . The critical points are the gauge equivalence classes of projectively flat connections. If the Hessian of the Chern-Simons functional at a projectively flat connection is non-generate, we say that the projectively flat connection is non-degenerate. For simplicity, we will refer projectively flat connections over Y with fixed determinant as "flat connections". Throughout this section we assume the following hypothesis.
Hypothesis 2.1. All flat connections on Q are irreducible and non-degenerate.
We are always able to perturb the Chern-Simons functional such that any critical points are non-degenerate. See [11], [8].
Let R(Y ) = R(Y, Q) be the set of gauge equivalence classes of flat connections on Q. It follows from Hypothesis 2.1 that R(Y ) is a finite set. We define a Z 8 -grading function δ Y on R(Y ) as follows. Let π be the projection from Y × R to Y . Fix a flat connection ρ 0 on Q.
For each flat connection ρ on Q, choose a connection A 0 = A 0 (ρ, ρ 0 ) over Y × R, which is compatible with π * a det , such that Then we have an operator Here g Q is the bundle of trace free, skew adjoint, endomorphisms of the rank-two complex vector bundle E associated with Q. Under Hypothesis 2.1, this operator is a Fredholm operator and we have the numerical index ind D A 0 ∈ Z. We set We can show that this depends only on the gauge equivalence class of ρ (and ρ 0 ). For j ∈ Z we write CF j (Y ) for the Q-vector space spanned by Let d be an integer with 1 ≤ d ≤ 3 and γ l ∼ = S 1 be a loop in Y for l = 1, . . . , d. We write γ for {γ l } d l=1 . We define the Fukaya-Floer chain group CF F * (Y ; γ) by where γ L := γ l 1 · · · γ l β ∈ Sym Q < γ 1 , . . . , γ d > for L = {l 1 , . . . , l β }. We define a boundary operator Then we have a moduli space M Y ×R (ρ, σ) of instantons with limits ρ, σ and the dimension is 2(β 2 − β 1 ) + 1. We write M 0 Y ×R (ρ, σ) for the quotient M Y ×R (ρ, σ)/R, where the action of R is defined by translations. When L 1 ⊂ L 2 , we define . . , l β } = L 2 \L 1 and L ⊗2 l i are line bundles defined by families of twisted∂ operators over γ l i × R. Since the moduli spaces are non-compact in general, we must specify the meaning of the pairing, and it will be done later. When L 1 is not included in L 2 , we define < ∂( give the boundary operator ∂. We will show that ∂ • ∂ is identically zero. The Fukaya-Floer homology group HF F * (Y ; γ) is defined to be the homology group of (CF F * (Y ; γ), ∂).
We give the precise definition of (6). To do this, we introduce some spaces of connections on Y × R and γ l × R. For flat connections ρ, σ on Q, take a smooth connection A 0 = A 0 (ρ, σ) on π * Q as before. Let τ > 0 be a small positive number and we set The wighted Sobolev space L 2,τ 4 is defined as follows. Take a function W τ on Y × R such that W τ (y, t) > 0 for ∀ (y, t) ∈ Y × R, W τ (y, t) = e τ |t| for |t| > 1.
For a smooth, compact supported section f , we define the weighted L 2 4 -norm by The weighted Sobolev space L 2,τ 4 is the completion of the space of smooth, compact supported sections. Note that since ρ, σ are irreducible, every connection in A * Y ×R (ρ, σ) is irreducible. By Hypothesis 2.1, we need not to introduce the weighted Sobolev space for the construction of the moduli space. However we need the weighted Sobolev space to define the determinant line bundles as explained below.
We introduce a gauge group acting on the space of connections. We put σ) . For g ∈ G Y ×R , we have g(y, t) → 1 or g(y, t) → −1 as t → ±∞. ( See Proposition 4.7 in [8]. ) We set be the moduli space of instantons with limits ρ, σ. We can perturb the instanton equation such that the operators D A are surjective for all [A] ∈ M 0 Y ×R (ρ, σ) ( [11,8]). For simplicity, we always assume the following.

Hypothesis 2.2. The operators D A defined by (5) are surjective for all [A] ∈ M 0
Y ×R (ρ, σ). Under this hypothesis, the moduli space M 0 Y ×R (ρ, σ) is smooth of expected dimension. Next we introduce spaces of connections and gauge groups over Γ l = γ l ×R. Let A l = A l (ρ, σ) be the restriction of the fixed connection A 0 (ρ, σ) to Γ l . We set We denote the quotient space A Γ l (ρ, σ)/G 0 Γ l byB Γ l (ρ, σ). Note that the restrictions of ρ, σ to γ l may be reducible, and hence some connections in A Γ l (ρ, σ) are reducible.
We define the determinant line bundle overB Γ l (ρ, σ). We need a spin structure on Γ l . Since H 1 (Γ l ; Z 2 ) is isomorphic to Z 2 , there are two spin structures on each Γ l (up to isomorphism). We fix a spin structure on γ l which represent the trivial class in the 1-dimensional spin bordism group. This spin structure induces a spin structure on Γ l . We use this spin structure. (We will explain the reason why we take this spin structure in Remark 2.3 below.) The spin structure induces a square root K 1 2 Γ l of the canonical line bundle K Γ l . For connections A ∈ A Γ l (ρ, σ) we have the twisted∂ operators Here are the weighted Sobolev spaces with weight function W ′ τ > 0 such that W ′ τ (y, t) = e τ t for |t| > 1. We do not take the absolute value of t in the exponent this time. The operators are Fredholm operators for small τ > 0. Since we have the universal bundles we obtain complex line bundles Letr l be the map fromB * Y ×R (ρ, σ) toB Γ l (ρ, σ) defined by restricting connections to Γ l . Then we have a natural action of Z 2 × Z 2 on the pull-backr * lL ⊗2 Γ l and the action of the diagonal Z 2 = {±(1, 1)} is trivial. Hence we get the line bundle Remark 2.3. We explain the reason why we choose the spin structure on Γ l induced by the spin structure on γ l representing the trivial class of 1-dimensional spin bordism group Ω spin 1 = Z 2 . When we prove the gluing formula for Donaldson invariants, we consider Γ l as a neck of a closed surface in a closed 4-manifold. We need the restriction of a spin structure on the closed surface, which is used to define Donaldson invariants, to Γ l . This spin structure is induced by the spin structure on γ l which represent the trivial element in Ω spin 1 .
To define (6) we need sections s l = s l (ρ, σ) of L ⊗2 l (ρ, σ) which behave nicely on the ends of the moduli spaces and satisfy suitable transversality conditions. We briefly recall some basic definitions and facts which are relevant to the end of the moduli spaces. (See [8] for details.) For a real number T let c T be the translation We call a sequence T of real numbers T 1 < · · · < T r−1 a translation vector.
i ] over any compact sets of (Y × R)\Z i and |c * Here dµ Y ×R is the volume form on Y × R, Z i = [z 1 , . . . , z s i ] and δ z l are the delta functions.
). It follows from the additivity of the index of the operator D A that if the dimension of M 0 Y ×R (ρ, σ) is less than 8 then Z i are empty for all i. In Section 2 and 3, we only consider the case when dim M 0 Y ×R (ρ, σ) < 8 and hence Z i are always empty. In Section 4, we will analyze the end of moduli spaces under the situation where Z i are not empty.
The end of the moduli spaces are described by gluing maps. We consider the case when Z i are empty. Let U 1 , . . . , U r be precompact, open sets of M 0 Y ×R (ρ, ρ(1)), . . . , M 0 Y ×R (ρ(r − 1), σ). Then we have a gluing map for some T 0 > 0. The map Gl is a diffeomorphism onto its image.
Then for large α, [A α ] are in the image of the gluing map.
We also have a gluing map Using these definitions and facts, we state the properties of sections of the line bundles which are required to define (6).
. Definition 2.7. Let s l be sections of the line bundles L ⊗2 l (ρ, σ). Under the above situation, we say that Here · is the norm on L ⊗2 l (ρ, σ) induced by the L 2,(−τ,τ ) -norms on the spaces of sections of K The following proposition is the key in this paper.

) is transverse and compact. Hence the intersection is a finite set.
Here we introduce the following definition. Definition 2.9. If sections s l (ρ, σ) of L ⊗2 l (ρ, σ) have the properties in Proposition 2.8, we call them admissible.
The proof of Proposition 2.8 will be given in the following two subsections. In this subsection, we assume that we have admissible sections s l (ρ, σ) and define the boundary operator for the Fukaya-Floer homology groups.
are finite by the property (c) in Proposition 2.8. Hence we can count the number of points in the intersections. They numbers give the definition of (6). More precisely we need to attach a sign ±1 to each point. However we mention nothing about signs. (The main purpose of this paper is to construct variants of Floer homology groups for 2-torsion instanton invariants. They are defined over Z 2 and we do not need signs for the construction. ) We prove the following.
This is given by counting the number of the ends of 1-dimensional moduli spaces. Let We have a reduced moduli space . Hence there is a number i(l) ∈ {1, . . . , r} such that i(l) ]) = 0 for each l ∈ L 2 \L 1 . This means that l ∈ L 2 \L 1 lies in L(i(l)). Therefore we have , ρ(i); L(i)), the intersection are not empty. The transversality condition (b) in Proposition 2.8 implies that for each i. From the additivity of the index, we have 2(β 2 − β 1 ) Therefore we get r ≤ 2. We assumed that r > 1, so r = 2. Put L := L(1) L 1 . Using (b), (c) in Proposition 2.8, we can easily see that ] are sufficiently small, the transversality conditions in Proposition 2.8 imply that intersections are transverse and that and let V ′ l be the zero locus of s ′ l . Then for each T 1 > T 0 we have and the first intersection is transverse. The condition (a) in Proposition 2.8 means that s l (ρ, σ) and s ′ l are close to each other on the end of the moduli space. Hence for T 1 large enough, the intersection is transverse and consists of a single point which is close to ([A 1 ], T 1 , [A 2 ]). We consider a subset Then M ′ is compact and if we perturb sections s l (ρ, σ) outside of neighborhoods of the triples , M ′ becomes a smooth manifold of dimension 1. Moreover there is a natural identification By counting the number of ∂M ′ with signs, we get and this gives ∂ • ∂ = 0 as required.
Define sections s l (ρ, σ) by (9). Then we have: The proof will be given in Subsection 2.4. Here we prove that we can perturb the sections s l (ρ, σ) such that the sections become admissible, assuming Lemma 2.16. We show the next lemma to do this.
Hence we have Using this inequality, we show This is a contradiction since we assumed that dim M 0 Y ×R (ρ, σ) ≤ 2|L|. We have obtained (12). By the hypothesis of induction, s l (ρ(i 0 − 1), ρ(i 0 )) satisfy (b) in Proposition 2.8. Hence the inequality (12) means that ) by the definition of L(i 0 ). We have obtained a contradiction. Therefore In the first lemma, we consider the situation where the connections [A α ] can be written as gluing of three instantons. For example, such a situation occurs if [A α ] are given by and T 1 are independent of α, and T α 2 → ∞. Assume that [A α ] ∈ Gl λ α with |λ α | = 3. We can write ). for some [B α i ] ∈ U λ α i (i = 1, 2, 3) and S α j > T λ α (j = 1, 2). The first lemma is the following. Lemma 2.18. There is a subsequence {[A α ′ ]} α ′ which satisfies the following.
We can show similar statements to this lemma when we assume |λ α | = ℓ with ℓ ≥ 4. As we As before, we can deduce that . This is a contradiction. 2.5. Evaluation of difference of two gluing maps. In this subsection, we prove Lemma 2.19. We need to compare two gluing maps defined on different regions intersecting each other. To do this, we must explicitly see the construction of gluing maps. We give outline of the construction following [8] before we prove Lemma 2. 19.
First note that elements in M 0 Y ×R (ρ, σ) can be considered as elements in M Y ×R (ρ, σ) with center of mass 0. Here the center of mass of Gluing X 1 (2T ) and X 2 (2T ) through ϕ, we obtain a manifold X #(T ) = X 1 (2T ) ∪ ϕ X 2 (2T ). We have a natural identification between X #(T ) and Y × R such that (y, 3T /2) ∈ X 1 (2T ) ⊂ X #(T ) corresponds to (y, 0) ∈ Y × R. Let ρ = ρ(0), ρ(1), σ = ρ(2) be flat connections and choose [A 1 ] ∈ M 0 X 1 (ρ, ρ(1)), [A 2 ] ∈ M 0 X 2 (ρ(1), σ). Here A 1 , A 2 are instantons on X 1 , X 2 with center of mass 0. We write ρ(0), ρ(2) for ρ, σ respectively. Then we can write and a i are π * i g Q -valued 1-forms on That is, A ′ is an almost instanton. To obtain a genuine instanton, we consider the equation F + A ′ +a = 0 for a small π * g Q -valued 1-form a. The equation can be rewritten as (16) d When T is large enough, we can construct a right inverse P = P T of D A ′ by using P 1 , P 2 and some cut-off functions. We can see the operator norm P of P is bounded by 2( P 1 + P 2 ). If we substitute a = P φ into (16), we get the equation We can see that for large T > 0 this equation has a unique solution φ = φ T with φ T → 0 as T → ∞. Hence for large T we get an instanton of the form (17) A For simplicity of notations, we suppose that ∂ A i are surjective for i = 1, 2. Let Γ 1 , Γ 2 be copies of Γ and fix cut-off functions γ i = γ T i on Γ i with Here supp(γ i ) is the support of γ i , and Γ i (2T ), Γ #(T ) are defined as before. In the third equation, we consider γ i as functions on Γ #(T ) in the natural way. Take f i ∈ ker∂ A i for i = 1, 2 and put Then we can show that We may construct a right inverse Q T from right inverses Q i of∂ A i for large T . The difference Applying this construction to all connections in U 1 × U 2 , we get the map Gl.
When∂ A i are surjective, we choose maps Applying the above method to these operators, we obtain the gluing maps. Proof of Lemma 2.19 Assume that we have a sequence Then we have two gluing maps . For large α, we have two different expressions of [A α ]: -norms. From (19), (20) and (21), we have for some sections g α , g ′ α with g α , g ′ α → 0. Here we show that the difference T α − T ′ α goes to zero as α → ∞.  (15), (17) and (18) that Comparing (23) with (24), we have Similarly we have Hence we obtain the required result.

From (22), we get
Here · is the L 2,(−τ,τ ) norm. The first term on the right hand side is equal to

This is bounded by
Hence the first term goes to zero. The second term is equal to It follows from Lemma 2.22 that δ α goes to zero. Since f ∞ 1 is finite and {δ α } α is bounded, for any ε > 0 there exist T (ε) > 0 independent of α such that Thus we have Similarly Since and δ α converge to zero, we have Since ε is arbitrary, the norm of difference 1 ) goes to zero. Thus the limit of the second term is zero. Similarly the limits of the third and fourth terms are zero. Since g α , g ′ α → 0, we have lim 2 ) = 0 as required.
2.6. Well-definedness. A priori, Fukaya-Floer homology groups seem to depends on the choices of Riemannian metric on Y and sections of L ⊗2 l (ρ, σ). But we will prove that Fukaya-Floer homology does not depend on these choices up to canonical isomorphism. In the proof of well-definedness of usual Floer homology groups, we need the functorial property of Floer homology groups with respect to Riemannian bordisms. We shall generalize the functorial property to Fukaya-Floer homology groups and prove the well-definedness by using this property.
Here A(ρ, σ) is a connection onX with limits ρ 0 , ρ 1 . Then the dimension of MX(ρ, σ) is equal Then the matrix elements give the map ζ : We give outline of the proof that ζ has the properties stated in Lemma 2.24.
The first part follows from a similar discussion to that in the proof that ∂ • ∂ = 0. Take a generator [τ ] ∈ CF j−2β 1 −1 (Y 1 ). Then we have a cut-down moduli space MX (ρ, τ ; L 1 \L 0 ) with dimension 1. Counting the number of the end of this moduli space, we get To prove the second part, take other metricĜ ′ and sectionsŝ ′ l (ρ, σ) overX. Then we have another map ζ ′ : CF F * (Y 0 ; γ) → CF F * (Y 1 , ; γ ′ ). We will construct a chain homotopy Take a path {G s } 0≤s≤1 of metrics on X from G to G ′ , and put Then M (ρ, σ) is smooth for generic paths of metrics. We can define line bundles L ⊗2 l (ρ, σ) over M (ρ, σ) as usual. As in Subsection 2.3, for ρ, σ with dim M(ρ, σ) < 8, we can construct sections s l (ρ, σ) such that s(ρ, σ) are compatible with gluing maps, {s l (ρ, σ)} l satisfy the transversality conditions as in Proposition 2.8, and Here V l is the zero locus of s l (ρ, σ). We put These matrix elements give H : CF F j (Y 0 ; γ) → CF F j+1 (Y 1 ; γ ′ ) as usual. We can show the chain homotopy condition (25) by counting the number of the ends of the cut-down moduli spaces M (ρ, τ ; L 1 \L 0 ) for generators [τ ] ∈ CF j−2β 1 (Y 1 ). Thus we obtain the second part of Lemma 2.24.
The proof of the third part is essentially same as the proof of the gluing formula for Donaldson invariants which will be given in the next subsection, and we omit the proof here.
We give a remark on the dependence of HF F * (Y ; γ) on γ. It seems that HF F * (Y ; γ) depends only on the homology classes [γ l ] ∈ H 1 (Y ; Z). If γ l and γ ′ l are homologous, then we have oriented, compact, surfaces Σ l in Y × [0, 1] with boundary γ l γ ′ l . As above we can construct a linear map we can show that for ρ with dim MX 0 (ρ) < 8 there are sections s l (ρ) of the line bundles with properties similar to those in Proposition 2.8. Let ρ 0 be the fixed flat connection used to define δ Y . For simplicity of notations, we suppose that ind D A(ρ 0 ) ≡ 0 mod 8, where A(ρ 0 ) is a connection onP 0 with limit ρ 0 . Then we have ) mod 8. Under these hypotheses, we define an element ψ X 0 = ψX 0 ∈ CF F 0 (Y ; γ) as follows. Let β be an integer with 0 ≤ β ≤ d and take L ⊂ {1, . . . , d} with |L| = β and a generator [ρ] ∈ CF −2β (Y ). Then we get a number < ψ X 0 , [ρ] ⊗ γ L >:= #MX 0 (ρ; L). Here MX 0 (ρ; L) is the 0-dimensional cut-down moduli space by the sections {s l (ρ)} l∈L . We define ψ X 0 by The following proposition follows from a combination of a discussion of the case when Y is a homology 3-sphere and techniques we have developed in this paper.
Remark 2.27. It seems that Ψ X 0 (Σ 1 , . . . , Σ d ) depend only on the homology classes [Σ 1 ], . . . , [Σ d ] ∈ H 2 (X 0 , Y ; Z). This is true if an analogy of Conjecture 2.25 holds for the line bundles L ⊗2 l (ρ). 2.8. Gluing formula. We consider a situation where a closed, oriented 4-manifold X is cut into two parts X 0 , X 1 along a closed 3-manifold Y . Let P be a U(2)-bundle over X such that the restriction Q = P | Y satisfies Hypothesis 2.1 and dim M P = 2d for some integer d with 0 ≤ d ≤ 3. We suppose that X 0 , X 1 are simply connected and b + of X 0 , X 1 are larger than 1. Let Σ 1 , . . . , Σ d be embedded surfaces in X such that the intersections γ l := Y ∩ Σ l are diffeomorphic to S 1 . We denote X 0 ∩ Σ l , X 1 ∩ Σ l by Σ ′ l , Σ ′′ l . Then we have relative invariants . We will express the invariant Ψ X ([Σ 1 ], . . . , [Σ d ]) of the closed manifold X in terms of the relative invariants Ψ X 0 , Ψ X 1 . To do this, we define a pairing < , >: HF F j (Y ; γ) ⊗ HF F −j−3+2d (Ȳ ; γ) −→ Q as follows.
For flat connections ρ, we have Hence CF j (Y ) and CF −j−3 (Ȳ ) are dual to each other. We define a pairing Here L is a subset of {1, . . . , d} with |L| = β and L c is the complement of L. It is easy to see the following.
Therefore we get the pairing on Fukaya-Floer homology groups.
Theorem 2.29. In the above situation, we have To prove the gluing formula, we need sections of L ⊗2 Σ l which are compatible with gluing maps as usual. Fix a Riemannian metric g Y on Y and choose Riemannian metrics g 0 , g 1 on X 0 , X 1 such that the restrictions to Y are equal to g Y . For each T > 0, we define a manifold X #(T ) by X #(T ) = X 0 ∪ Y × [0, T ] ∪ X 1 . Then g 0 , g 1 naturally induce a Riemannian metric g #(T ) on X #(T ) . Fix a sequence T α → ∞ and we write X α for X #(T α ) . As in Subsection 2.3, we can construct sections s α l of L ⊗2 . . , d} with |L| = β. The transversality conditions for the sections imply that MX 0 (ρ; L) and MX 1 (ρ; L c ) are finite sets. From an argument like that in the proof of (7), we can show there is a natural identification for large α. Counting the number of elements of both sides with signs, we obtain the gluing formula.

Floer homology for 2-torsion instanton invariants
In this section, we consider a variant Ψ u 1 X of Donaldson invariants for non-spin 4-manifolds X, which is a linear function Ψ u 1 X : A ′ (X) −→ Z 2 . Here A ′ (X) is the subspace of ⊕ d≥0 H 2 (X; Z) ⊗d generated by the set This invariant is defined by a 2-torsion cohomology class u 1 of the moduli spaces. Originally this was defined for spin manifolds by Fintushel-Stern [10], and the author extended to nonspin 4-manifolds in [19]. In this section, we construct a variant of Floer homology group and prove a gluing formula for Ψ u 1 X . In [19], it was shown that Ψ u 1 X is non-trivial for X = CP 2 0 #CP 2 1 #CP 2 , where CP 2 0 and CP 2 1 are copies of CP 2 . More precisely Here H 0 , H 1 and E are the generators of H 2 (CP 2 0 ; Z), H 2 (CP 2 1 ; Z) and H 2 (CP 2 ; Z) respectively. We can consider X = CP 2 0 #CP 2 1 #CP 2 as a connected sum of X 0 = CP 2 0 and X 1 = CP 2 1 #CP 2 . Since the only flat connection over S 3 is the trivial one, we can deduce from (27) that the trivial flat connection has an important role in the gluing formula for Ψ u 1 X in contrast to the case of usual Donaldson invariants. Note also that the homology class −H 1 + E in (27) is a sum of homology classes −H 1 and E of X 0 and X 1 with self-intersection numbers odd.
On the other hand, for closed, simply connected, non-spin 4-manifolds X 0 , X 1 with b + positive, we can show the following vanishing theorem. For homology classes [Σ l ] ∈ H 2 (X 0 ; Z), ) ≡ 0 mod 2. This follows from a standard dimension-counting argument. The formula (28) implies that the key of the non-vanishing result (27) is that the homology class −H 1 + E is split into two homology classes of X 0 , X 1 with self-intersection numbers odd.
In this section, we consider a situation where X has a decomposition X = X 0 ∪ Y X 1 for a homology 3-sphere Y and a closed, oriented surface Σ in X is split into two surfaces with self-intersection numbers odd along Y . We will give a gluing formula for Ψ u 1 X ([Σ]) in terms of differential-topological date about X 0 , X 1 . This situation is similar to that of Section 2 and we can apply the method developed in Section 2. The main difference is that a U(2)-bundle over a homology 3-sphere has the trivial flat connection, which is reducible, and we need to pay attention to the effect of the trivial flat connection on the gluing formula.
Let X be a closed, oriented, simply connected, non-spin 4-manifold with b + > 1. Take a U(2)-bundle P over X with w 2 (P ) = w 2 (X). For [Σ] ∈ H 2 (X; Z), we have the complex line bundleL Let c be a spin-c structure of X. For each connections A on P , we have the twisted Dirac operator . Here E is the rank two complex vector bundle associated with P and S ± are the spinor bundles. If c 1 (det c) = −c 1 (P ), we have a "real part" of the Dirac operators: (See [1].) The family of real operators {( D A ) R } [A]∈B P gives a real line bundlẽ Λ R −→B * P . We also suppose that c 2 (P ) ≡ 0 mod 2. Then we can see that the center {±1} of SU(2) acts trivially onΛ, and we obtain a real line bundle When b + (X) is even, the virtual dimension of the moduli space M P is odd. We can write dim M P = 2d + 1 for some integer d. Assume d ≥ 0. Then we define In general M P is not compact, however we can define the pairing as follows. Let s Σ i : B * Σ i → L Σ i be sections and V Σ i be the zero loci. We can prove the following lemma by a standard dimension counting argument.
Lemma 3.1. Assume that the dimension of M P is 2d + r for some d ≥ 0 and 0 ≤ r ≤ 3. Then the intersection M P ∩ V Σ 1 ∩ · · · ∩ V Σ d is compact and smooth for generic sections s Σ i .
When dim M P is 2d + 1, the intersection is independent of the metric on X and sections of L Σ i , and hence it is a differential-topological invariant of X.
Here r Σ i is the map M P → B * Σ i defined by restricting connections to Σ i . However Lemma 3.1 shows that when the dimension of M P is 2d + 1 with d ≥ 0 M P ∩ V Σ 2 ∩ · · · ∩ V Σ d is a compact, smooth manifold of dimension 3. Hence we need not to use the pull-back r * Σ 1 (s Σ 1 ). We can use any section of r * Σ 1 (L Σ 1 ) which is transverse to the zero section.
3.2. Line bundles. In this subsections, we introduce line bundles over B * Y ×R (ρ, σ) which are defined by families of Dirac operators over Y × R and twisted∂ operators over surfaces in Y × R. These are used to define a variant of Floer homology groups for 2-torsion instanton invariants.
Let Y be a homology 3-sphere and Q = Y × U(2) be the trivial U(2)-bundle over Y . Fix a connection a det on Q det = Y × U(1). As before, connections on Q and π * Q are compatible with a det , unless explicitly stated otherwise.
Proof. Let P γ,b−a be a U(2) bundle over γ × S 1 with c 1 (P γ,b−a ) = b − a. From the additivity of index and the index formula, we have This lemma implies that for a, b ∈ Z with a ≡ b mod 2 we can define . We can also show that for a ≡ b ≡ a ′ ≡ b ′ mod 2 we have a natural isomorphism between L Γ,ab (ρ, σ) and L Γ,a ′ b ′ (ρ, σ). We get two bundles L 3.3. The complex. Using the line bundles introduced in the previous subsection, we construct homology groups which allow us to extend 2-torsion instanton invariants to 4-manifolds with boundary. The idea of the construction is fundamentally the same as Fukaya-Floer homology groups in Section 2. The main difference is the existence of a reducible flat connection.
To prove (29), we need to describe the ends of M 0 Y ×R (ρ, σ) as in the previous section. In the cases (1), (2), (3), the dimension of the moduli space M 0 Y ×R (ρ, σ) is less than 4. Lemma 3.8 means that (29) follows from the same discussion in the previous section. Therefore we need to consider only the case (4). Lemma 3.9. Hence from a discussion like that in the proof of (7), we have for large T 1 . Perturbing the sections s Λ , s L outside the end (30) of M ′ , we get a 1-dimensional, smooth, compact manifold with boundary Here  Proof. Let X 1 , X 2 be two copies of Y × R. And we write X 1 (T ), X 2 (T ) for Y × (−∞, 2T ), Y × (−2T, ∞). Gluing X 1 (T ), X 2 (T ) through the identification we get a manifold X #(T ) . Let E be the rank-two complex vector bundle over Y associated with Q. Take a trivialization φ of E such that θ n is trivial under φ. We write E i for π * i E and φ i for π * i φ. Here π i are the projections from X i (T ) to Y . Forũ ∈ SU(2) we define a vector bundle E(ũ) by  (2). We have an action of {±1} on this bundle defined by the following diagram: Here

The restriction L
(1) Γ,[A(ũ)] }ũ ∈SU (2) by this action. The necessary and sufficient condition for L (1) Γ (ρ, σ)| SO (3) [A] to be non-trivial is that m 1 + m 2 is odd. From the additivity of the index, we have m 1 + m 2 = Ind −,+∂ A(θ γ,1 ,θn|γ) . Here A(θ γ,1 , θ n | γ ) is the connection on Γ = γ × R with limits θ γ,1 , θ n | γ . Let g 1 be the fixed map from Y to SU(2) with degree 1. The class [g 1 | γ ] represented by the restriction of g 1 to γ is trivial in π 1 (U(2)). Hence we have m 1 + m 2 = Ind −,+∂ A(θ γ,1 ,θ γ,0 ) = −1 by Lemma 3.7. Therefore the restriction L Γ (ρ, σ). Remark 3.17. We assumed that Y is a homology 3-sphere. In this case, it is sufficient to consider a very small loop γ which represents the trivial class in π 1 (Y ) or one point which is a degenerate loop, when we calculate invariants using our gluing formula. However we have considered a general loop γ which may not be trivial in π 1 (Y ). One of the reason is that even if we take a degenerate loop, we can see that the components of the boundary operator are not trivial, and we need to consider the components as in the case where the homotopy class of the loop is non-trivial. Another reason is that we hope our construction is extended to more general 3-manifolds.
3.4. Relative invariants. Let X be a closed, oriented, simply connected, non-spin 4-manifold with b + (X) positive and even. We consider a situation where we have a decomposition X = X 0 ∪ Y X 1 . Here Y is an oriented homology 3-sphere and X 0 , X 1 are compact, simply connected, non-spin 4-manifold with boundary Y ,Ȳ and b + (X 0 ), b + (X 1 ) > 1. We consider invariants Ψ u 1 X,P (z), where P is a U(2)-bundle over X with w 2 (P ) ≡ w 2 (X) and dim M P = 3 and z ∈ H 2 (X; Z) with z · z ≡ 0 mod 2. The purpose is to write Ψ u 1 P (z) in terms of data from X 0 , X 1 as in the case of the usual Donaldson invariants.
Proof. The wight of the Γ ρ action onΛ (a) (ρ) is Ind + D A(θa) . Hence it is sufficient to show that the index is even if c 2 (P 0 ; ϕ a ) is even. Choose a compact, spin 4-manifold X ′ 0 with boundarȳ Y and we consider a manifold X ′ = X 0 ∪ Y X ′ 0 . Let P ′ be a U(2)-bundle over X ′ obtained by gluing P 0 and the trivial bundle P X ′ 0 over X ′ 0 through ϕ a . Let c ′ 0 be a spin-c structure over X ′ 0 induced by a spin structure. Then we have a spin-c structure c ′ on X ′ induced by c 0 , c ′ 0 and we have the Dirac operator Here A ′ is a connection induced by A(θ a ) and the trivial connection θ X ′ 0 on P X ′ 0 and E ′ is the complex vector bundle of rank two associated with P ′ . Since D θ X ′ 0 is the direct sum of two copies of a Dirac operator, the index Ind − D θ X ′ 0 is even. Thus we get Applying the index formula, we have Ind D A ′ = ch(E ′ )Â(X ′ )e  SinceÂ(X ′ )e 1 2 c 1 (det c ′ ) /[X ′ ] is the index of a spin-c Dirac operator, it is an integer. Hence the first term is even. From c 1 (det c ′ ) = −c 1 (E ′ ), the second term is zero. Therefore we obtain Ind + D A(θa) ≡ −c 2 (E ′ )/[X ′ ] ≡ c 2 (P 0 ; ϕ a ) mod 2.
Thus the first part of Lemma follows.
These numbers define a chain ψ u 1 X 0 ∈ C (a) 8n 0 (Y ) as usual. As before, we have: Remark 3.28. In general, usual Donaldson invariants Ψ P (x 1 , . . . , x d ) do not lie in Z even if x i ∈ H 2 (X; Z). However if < w 2 (P ), x i >≡ 0 mod 2 for all i then the value Ψ P (x 1 , . . . , x d ) lies in Z. Therefore the above formula makes sense.
Let Σ be a closed, oriented surface embedded in X. We put B * Σ,+ = B * Σ ∪ {[θ Σ ]}. It is known that the determinant line bundle L Σ → B * Σ extends to B * Σ,+ . (See [9].) We use the same notation L Σ for the extension. We can take a section s Σ : B * Σ,+ → L Σ such that s Σ does not vanish near [θ Σ ]. We always assume that sections of L Σ have this property. Let V Σ be the zero locus of s Σ and we define MX (ρ) ∩ V Σ to be (1) If [ρ] ∈ CF 2 (Y ) and dim MX(ρ) = 8 then r = 0, Z 0 = ∅.
If we perturb the section s Λ over a compact set in M ′ then the intersection M ′ ∩V Λ is a smooth compact manifold with boundary MX (σ; Σ 1 , . . . , Here [ρ] runs over the generators of CF 2 (Y ) and [σ] runs over the generators of CF 1 (Y ). Therefore we get We can see that # Gl x (ℓ, T 0 ) ∩ V Λ ) ≡ 1 mod 2 for each x. This follows from arguments similar to those which deduce (31). The point is that J is an instanton on an SU(2)-bundle P ′ with c 2 (P ′ ) = 1 and hence the index Ind D J is odd. Therefore we get