On the Bauer-Furuta and Seiberg-Witten invariants of families of 4 -manifolds
aa r X i v : . [ m a t h . DG ] J un ON THE BAUER-FURUTA AND SEIBERG-WITTENINVARIANTS OF FAMILIES OF -MANIFOLDS DAVID BARAGLIA, HOKUTO KONNO
Abstract.
We show how the families Seiberg-Witten invariants of a family ofsmooth 4-manifolds can be recovered from the families Bauer-Furuta invariantvia a cohomological formula. We use this formula to deduce several propertiesof the families Seiberg-Witten invariants. We give a formula for the Steenrodsquares of the families Seiberg-Witten invariants leading to a series of mod2 relations between these invariants and the Chern classes of the spin c indexbundle of the family. As a result we discover a new aspect of the ordinarySeiberg-Witten invariants of a 4-manifold X : they obstruct the existence ofcertain families of 4-manifolds with fibres diffeomorphic to X . As a concretegeometric application, we shall detect a non-smoothable family of K K -theoretic Seiberg-Witten invariants and givea formula expressing the Chern character of the K -theoretic Seiberg-Witteninvariants in terms of the cohomological Seiberg-Witten invariants. This leadsto new divisibility properties of the families Seiberg-Witten invariants. Introduction
In [3], Bauer and Furuta constructed a refinement of the Seiberg-Witten invari-ants of smooth 4-manifolds taking values in stable cohomotopy. The refined Bauer-Furuta invariant contains strictly more information than the ordinary Seiberg-Witten invariants. Moreover the existence of the refined invariant is useful even ifone’s primary interest is in the Seiberg-Witten invariants themselves. To under-stand this point, recall that the usual definition of the Seiberg-Witten invariantsinvolves the construction of a smooth moduli space and one must use perturbationsto achieve transversality. However the Bauer-Furuta stable cohomotopy refinementaffords us the luxury of working in the setting of algebraic topology so that issuesof transversality can be bypassed.The Seiberg-Witten invariants of smooth 4-manifolds have been extended toinvariants of families of smooth manifolds [23, 24, 15, 16, 21]. By a family ofsmooth 4-manifolds we mean a smooth locally trivial fibre bundle over a smoothbase manifold whose fibres are a fixed compact smooth 4-manifold. Additionallythe family is assumed to be equipped with a spin c -structure on the vertical tangentbundle. Naturally one may also ask for a families extension of the Bauer-Furutainvariant. The existence of such an extension can already been seen implicitlyin Bauer-Furuta ([3, Theorem 2.6]) and was further developed by Szymik in [26].Neither of these works establish how, if at all, one can recover the families Seiberg-Witten invariants from the families Bauer-Furuta invariant. In this paper we answerthis question in the affirmative, showing that the families Seiberg-Witten invariants Date : June 26, 2019. can be recovered from the families Bauer-Furuta invariant and we give an explicitcohomological formula relating the two (Theorem 3.6). In subsequent sections of thepaper we use our cohomological formula to extract a number of results concerningthe families Seiberg-Witten invariants. Specifically: • We compute the Steenrod powers of the families Seiberg-Witten invariants(Section 4). • We give a simple new proof of the wall crossing formula of Li and Liu [16]for for families Seiberg-Witten invariants (Section 5). • We introduce K -theoretic families Seiberg-Witten invariants and show howthey are related to the usual cohomological families invariants via theChern character. This leads to certain divisibility properties of the familiesSeiberg-Witten invariants (Section 6).1.1. Families Seiberg-Witten invariants.
To state our main results we need torecall the construction of the families Seiberg-Witten and Bauer-Furuta invariants.The setting is as follows: let π : E → B be a smooth locally trivial fibre bundleover a compact smooth base manifold B whose fibres are a fixed compact smooth4-manifold X . More precisely, fix a basepoint b ∈ B and a diffeomorphism X ∼ = E b = π − ( b ) of X with the fibre of E over b . Assume that the vertical tangentbundle E is equipped with a spin c -structure s E and let s X be the correspondingspin c -structure on X obtained by restriction to the fibre E b . The data ( π : E → B, s E ) (or ( E, s E ) for short) will be referred to as a spin c family of 4-manifolds withfibres diffeomorphic to ( X, s X ). In this introduction we will consider only the casethat b ( X ) = 0. The necessary modifications to accommodate the case b ( X ) > g = { g b } b ∈ B on E and a smoothlyvarying family of 2-forms η = { η b } b ∈ B , where η b is a g b -self-dual 2-form on thefibre E b . The families Seiberg-Witten invariants will in general depend on a choiceof chamber φ (see Definition 2.14). If b + ( X ) > dim ( B ) + 1 then there exists aunique chamber, otherwise the Seiberg-Witten invariants will generally depend on φ . Assume that the metric g and 2-form perturbation η are chosen so that ( g, η )lies in the chamber corresponding to φ .Let M = M ( E, s E , g, η ) denote the moduli space of gauge equivalence classesof solutions to the Seiberg-Witten equations on the fibres of E with respect to thefibrewise spin c -structure s E , the fibrewise metric g and fibrewise perturbation η .For generic ( g, η ) the moduli space M ( E, s E , g, η ) is a smooth manifold of dimension dim ( B )+ (2 d − b + − d = c ( s X ) − σ ( X )8 and b + = b + ( X ) is the dimension ofthe space of harmonic self-dual 2-forms on X . Let π M : M → B denote the naturalprojection to B . The gauge theoretical construction of M determines a complexline bundle L → M as follows. Recall that the gauge group in Seiberg-Wittentheory is G = M ap ( X, U (1)). In the families setting this single group is replacedby a bundle of groups over B and M is obtained as a fibrewise quotient. Let G bethe subgroup of G given by: G = (cid:26) g ∈ G (cid:12)(cid:12)(cid:12)(cid:12) g = e if , Z X f dvol X = 0 (cid:27) . Since b ( X ) = 0, we have that G / G ∼ = S . The definition of G also works inthe families setting since the smoothly varying family of metrics { g b } determinesa smoothly varying family of fibrewise volume forms { dvol E b } . Let f M be the N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 3 families moduli space obtained by taking the quotient by the reduced gauge group G . Then f M → M is a principal circle bundle and we let
L → M be the associatedline bundle.Let H + → B be the vector bundle on B whose fibre over b is the space H + ( E b , g b )of g b -harmonic self-dual 2-forms on E b . To keep the introduction simple we willassume here that H + is oriented. The general case is dealt with in the paper usinglocal systems. In the unparametrised setting, an orientation of the Seiberg-Wittenmoduli space corresponds to a choice of orientation on H + ( X ) (when b ( X ) = 0).In the families setting this translates to the statement that a relative orientation of π M : M → B is determined by a choice of orientation of the bundle H + . In thispaper we define the families Seiberg-Witten invariants of ( E, s E ) (with respect tothe chamber φ ) to be the collection of cohomology classes { SW m ( E, s E , φ ) } m ≥ ,where SW m ( E, s E , φ ) = ( π M ) ∗ ( c ( L ) m ) ∈ H m − (2 d − b + − ( B ; Z ) . In the case that H + is not orientable, the families Seiberg-Witten invariants areinstead valued in a local coefficient system. One needs to check that this definitiondoes not depend on the particular choice of metric and perturbation ( g, η ). Theproof is much the same as in the unparametrised setting.1.2. Families Bauer-Furuta invariants.
In [3] Bauer and Furuta constructeda stable cohomotopy refinement of the Seiberg-Witten invariant by taking a finitedimensional approximation of the Seiberg-Witten equations. This construction wasextended to families of 4-manifolds in [26]. We will recall how this finite dimensionalapproximation is constructed in Subsection 2.3. Suppose again that π : E → B is a spin c family of 4-manifolds with fibres diffeomorphic to X . For simplicity wecontinue to assume in this introduction that b ( X ) = 0. The case b ( X ) > B produces an S -equivariant mapof sphere bundles over B : f : S V,U → S V ′ ,U ′ . Here
V, V ′ are complex vector bundles over B of ranks a, a ′ and U, U ′ are real vectorbundles over B of ranks b, b ′ . S V,U denotes the unit sphere bundle of R ⊕ V ⊕ U , orequivalently the fibrewise one point compactification of V ⊕ U and S V ′ ,U ′ is definedsimilarly. The unit circle S acts on V, V ′ by fibrewise scalar multiplication andacts trivially on U, U ′ . The S actions on V ⊕ U and V ′ ⊕ U ′ extend smoothly tothe sphere bundles S V,U , S V ′ ,U ′ .As will be shown in Subsection 2.3, the vector bundles V, V ′ , U, U ′ satisfy: V − V ′ = D ∈ K ( B ) , U ′ − U = H + ∈ KO ( B ) , where D ∈ K ( B ) denotes the families index of the families spin c Dirac operator of( E, s E ) and H + is the vector bundle over B whose fibres are the space of harmonicself-dual 2-forms on the corresponding fibres of the family E → B . Therefore a − a ′ = d = rank C ( D ) = c ( s X ) − σ ( X )8 , b ′ − b = b + = rank R ( H + ) . Lastly, we will see that the finite dimensional approximation f can be constructedso as to have the property that U ′ ∼ = U ⊕ H + and that the restriction f | U of f to U is given by the inclusion U → U ′ . DAVID BARAGLIA, HOKUTO KONNO
For simplicity we assume in this introduction that the bundle H + is orientable.In this case U and U ′ can also be chosen to be orientable. The general case where H + is not necessarily orientable is dealt with in the body of the paper. By defini-tion, a chamber of f is a homotopy class of section φ : B → U ′ \ U , or equivalentlya homotopy class of section of H + \ { } . As explained in Subsection 2.3, there isa canonical bijection between chambers for the families Seiberg-Witten equationsof ( E, s | E ) and chambers of the finite dimensional approximation f . Thus to anychamber [ φ ] represented by a map φ : B → U ′ \ U , the families Seiberg-Witteninvariants are defined: SW m ( E, s E , [ φ ]) ∈ H m − (2 d − b + − ( B ; Z ). We prove a coho-mological formula for these invariants in terms of f and φ . To state this formulawe must introduce one additional construction.Let us define e Y = S V,U \ N V,U , where N V,U is an S -invariant tubular neighbour-hood of S ,U in S V,U . Then e Y is a smooth compact manifold with boundary ∂ e Y .We also have that S acts freely on e Y and the quotient Y = e Y /S is a compactmanifold with boundary ∂Y = ∂ e Y /S . Then we have isomorphisms of cohomologygroups: H ∗ S ( S V,U , S ,U ; Z ) ∼ = H ∗ S ( e Y , ∂ e Y ) ∼ = H ∗ ( Y, ∂Y ) , where the first isomorphism is excision and the second isomorphism follows since S acts freely on e Y . Using Poincar´e-Lefschetz duality we get a push-forward map( π Y ) ∗ : H ∗ ( Y, ∂Y ) → H ∗− (2 a + b − ( B ; Z ) , since the fibres of π Y have dimension 2 a + b − φ : B → U ′ \ U of the chamber [ φ ] induces a pushforwardmap φ ∗ : H ∗ S ( B ; Z ) → H ∗ +(2 a ′ + b ′ ) S ( S V ′ ,U ′ , S ,U ; Z )and the finite dimensional approximation f induces a pullback map f ∗ : H ∗ S ( S V ′ ,U ′ , S ,U ; Z ) → H ∗ S ( S V,U , S ,U ; Z ) . Now we are ready to state our first main result:
Theorem 1.1.
Let ( E, s E ) be a spin c family over B . Let f : S V,U → S V ′ ,U ′ be afinite dimensional approximation of the Seiberg-Witten monopole map, as describedabove. Let [ φ ] be a chamber of the families Seiberg-Witten equations represented bya section φ : B → U ′ \ U . Then: SW m ( E, s E , [ φ ]) = ( π Y ) ∗ ( x m ` f ∗ φ ∗ (1)) ∈ H m − (2 d − b + − ( B ; Z ) , where x is the standard generator of H S ( pt ; Z ) . Theorem 1.1 is a special case of the theorem that we prove in the paper (Theorem3.6). Our general result applies also to families of 4-manifolds with b ( X ) > H + .1.3. Applications.
The remainder of the paper is concerned with applications ofour main formula, Theorem 1.1, which we will now discuss.Assume once again that E → B is a spin c family of 4-manifolds with fibresdiffeomorphic to X and for simplicity we continue to assume that b ( X ) = 0.Recall that D ∈ K ( B ) denote the virtual index bundle of the family of spin c Diracoperators determined by E and that H + ∈ KO ( B ) denotes the bundle of harmonicself-dual 2-forms on the fibres of E → B . Let c ( D ) = 1 + c ( D ) + c ( D ) + · · · denote the total Chern class of D . For each j ≥
0, define the j -th Segre class N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 5 s j ( D ) ∈ H j ( B ; Z ) of D as follows. Letting s ( D ) = 1 + s ( D ) + s ( D ) + · · · denotethe total Segre class of D , then s ( D ) is defined by the equation c ( D ) s ( D ) = 1.Let φ denote a chamber for the families Seiberg-Witten equations of the family( E, s E ). Consider the mod 2 reductions of the families Seiberg-Witten invariants: SW Z m ( E, s E , φ ) ∈ H m − (2 d − b + − ( B ; Z ) . Then as a consequence of Theorem 1.1, we compute the Steenrod squares of themod 2 families Seiberg-Witten invariants:
Theorem 1.2.
The Steenrod squares of the mod families Seiberg-Witten invari-ants are given by: Sq j ( SW Z m ( E, s E , φ )) = j X l =0 j − l X k =0 (cid:18) d − − m + l + kl (cid:19) s k ( D ) w j − l − k ( H + ) SW Z m + l ( E, s E , φ ) ,Sq j +1 ( SW Z m ( E, s E , φ )) = j X l =0 j − l X k =0 (cid:18) d − − m + l + kl (cid:19) s k ( D ) w j − l − k +1 ( H + ) SW Z m + l ( E, s E , φ ) . for all m, j ≥ . These formulas imply some surprising mod 2 relations between the familiesSeiberg-Witten invariants and the characteristic classes of D , H + . For example, inthe case of Sq , we get: Sq ( SW Z m ( E, s E , φ )) = ( d + m ) SW Z m +1 ( E, s E , φ )+( c ( D )+ w ( H + )) SW Z m ( E, s E , φ ) . Suppose that b + = 2 p + 1 is odd and set m = d − p −
1. Then SW Z b m ( E, s E , φ ) ∈ H ( B ; Z ) ∼ = Z is the mod 2 reduction of the ordinary Seiberg-Witten invariant SW ( X, s X ) of the 4-manifold X with respect to the spin c -structure s | X = s E | X .In this case the above equation reduces to:( p + 1) SW Z m +1 ( E, s E , φ ) = ( c ( D ) + w ( H + )) SW ( X, s X ) ∈ H ( B ; Z ) . If p is even, this gives a formula for SW Z m +1 ( E, s E , φ ) in terms of SW ( X, s X ), c ( D ) and w ( H + ). On the other hand if p is odd (so b + = 3 ( mod c ( D ) + w ( H + )) SW ( X, s X ) = 0 ∈ H ( B ; Z ), which leads to the followinginteresting consequence: Corollary 1.3.
Let ( X, s X ) be a compact smooth spin c -manifold with b ( X ) = 0 and b + ( X ) = 3 (mod 4) . Then: • If there exists a spin c family E → B with fibre ( X, s X ) and c ( D ) = w ( H + ) (mod 2) , then SW ( X, s X ) is even. • If SW ( X, s X ) is odd, then c ( D ) = w ( H + ) (mod 2) for any spin c family E → B with fibre ( X, s X ) . In particular, we see that the mod 2 Seiberg-Witten invariants of X can be usedto obstruct the existence of certain spin c families with fibres diffeomorphic to X .We give an application of this result to K Theorem 1.4.
There exists a continuous family E of K surfaces over the twotorus T satisfying the following conditions: DAVID BARAGLIA, HOKUTO KONNO • The total space of E is smoothable as a manifold. • The restriction of E to any -dimensional submanifold of T is smoothableas a family. • However, E is not smoothable as a family.Remark . Here are three remarks on Theorem 1.4.(1) The precise meaning of smoothablity as a family will be given in Defini-tion 4.20.(2) For comparison, see Theorem 1.4 of [10], in which other non-smoothablefamilies have been detected by a different technique based on 10/8-typeinequalities.(3) The proofs that E satisfies the first and second conditions are based on [10]and [17] respectively. To prove that E satisfies the third condition, we shalluse Corollary 1.3.Theorem 1.4 immediately implies that the fundamental group of the homotopyfiber of the natural map BDif f ( K → BHomeo ( K
3) is non-trivial. Recall thatthis homotopy fiber is homotopy equivalent to the homotopy quotient
Homeo ( K (cid:12) Dif f ( K
3) := (
EDif f ( K × Homeo ( K /Dif f ( K . Therefore we have:
Corollary 1.6. π ( Homeo ( K (cid:12) Dif f ( K = 0 .Remark . Here are two remarks on Corollary 1.6.(1) By the long exact sequence of homotopy groups it follows that π ( Dif f ( K → π ( Homeo ( K π ( Dif f ( K → π ( Homeo ( K π (cid:0) Homeo ( K S × S ) (cid:12) Dif f ( K S × S ) (cid:1) = 0 . Our next application of Theorem 1.1 is to give a simple new proof of the wallcrossing formula for the families Seiebrg-Witten invariants. The families wall cross-ing formula was originally proven in [16] using parametrised Kuranishi models andobstruction bundles. Using the technique of finite dimensional approximations ofthe Seiberg-Witten equations allows us to bypass these technicalities. Here we willstate the result for spin c families E → B with fibres diffeomorphic to a 4-manifoldwith b ( X ) = 0, however our general result also covers the case that b ( X ) > Theorem 1.8.
Let ( E, s E ) be a spin c family with fibres diffeomorphic to a -manifold X with b ( X ) = 0 . Let D denote the virtual index bundle of the family ofspin c Dirac operators determined by E and set d = rank C ( D ) . Let φ, ψ be chambersof the families Seiberg-Witten equations for ( E, s E ) . Then: SW m ( E, s E , φ ) − SW m ( E, s E , ψ ) = ( if m < d − ,Obs ( φ, ψ ) ` s m − ( d − ( D ) if m ≥ d − , where Obs ( φ, ψ ) ∈ H b + ( X ) − ( B ; Z w ( H + ) ) is the primary difference class of φ, ψ (see § s j ( D ) is the j -th Segre class of D . N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 7
In Subsection 5.3, we show how to recover the wall crossing formula for theordinary Seiberg-Witten invariant of a 4-manifold X with b + ( X ) = 1 and b ( X )even as a special case of the families wall crossing formula. Note that even in thiscase the families formalism is still relevant: the parameter space of the family isthe Jacobian torus T = H ( X ; R / Z ).Our final application concerns certain divisibility conditions of the families Seiberg-Witten invariants that arise from K -theoretic considerations. Recall from Subsec-tion 1.1 that the families Seiberg-Witten invariants of a spin c family ( E, s E ) over B with fibres diffeomorphic to X are certain cohomology classes SW m ( E, s E , φ )valued in the cohomology of B :(1.1) SW m ( E, s E , φ ) = ( π M ) ∗ ( c ( L ) m ) ∈ H m − (2 d − b + − ( B ; Z ) , where for simplicity we are assuming in this introduction that b ( X ) = 0 and H + isorientable. Here M is the families Seiberg-Witten moduli space and π M : M → B is the natural map to B . In Section 6, we introduce a new set of invariants ofthe family ( E, s E ), the K -theoretic Seiberg-Witten invariants SW Km ( E, s E , φ ) ∈ K b + − ( B ). To define them, assume that H + admits a spin c -structure. Then onecan show that π M : M → B is relvatively spin c and imitate (1.1) in K -theory: SW Km ( E, s E , φ ) = ( π M ) K ∗ ( L m ) ∈ K b + − ( B ) , where ( π M ) K ∗ denotes the K -theoretic pushforward map induced by π M . The proofof Theorem 1.1 carries over to the K -theoretic setting, and thus we have: Theorem 1.9.
Let ( E, s E ) be a spin c family over B . Let f : S V,U → S V ′ ,U ′ be afinite dimensional approximation of the Seiberg-Witten monopole map, as describedabove. Let [ φ ] be a chamber of the families Seiberg-Witten equations represented bya section φ : B → U ′ \ U . Then: SW Km ( E, s E , φ ) = ( π Y ) K ∗ ( ξ m ` f ∗ φ K ∗ (1)) , where ξ ∈ K S ( pt ) = R [ S ] corresponds to the standard -dimensional representa-tion of S . Then by an application of Grothendieck-Riemann-Roch, we obtain a formularelating the Chern character of the K -theoretic Seiberg-Witten invariants to thecohomological Seiberg-Witten invariants. To state the result, we introduce the fol-lowing notation: for a complex vector bundle V over B , let T d S ( V ) ∈ H ∗ S ( B ; Q ) ∼ = H ∗ ( B ; Q ) ⊗ H ∗ S ( pt ; Q ) denote the S -equivariant Todd class of V , where S actson V by scalar multiplication. This can be expanded as a formal power series in x ,where x is the standard generator of H S ( pt ; Z ): T d S ( V ) = X j ≥ T d j ( V ) x j , for some characteristic classes T d j ( V ) ∈ H ev ( B ; Q ). Theorem 1.10.
Let κ ∈ H ( B ; Z ) be the first Chern class of the spin c line bundleassociated to H + . Then the K -theoretic and cohomological Seiberg-Witten invari-ants are related by: Ch ( SW Km ( E, s E , φ )) = e − κ/ ˆ A ( H + ) − X j ≥ T d j ( D ) X k ≥ m k k ! SW j + k ( E, s E , φ ) ∈ H ∗ ( B ; Q ) DAVID BARAGLIA, HOKUTO KONNO
This formula can be used to extract certain divisibility properties of the familiesSeiberg-Witten invariants. For example, we prove the following:
Corollary 1.11.
Let ( E, s E ) be as above and suppose that B = S r is an evendimensional sphere with r ≥ . Suppose that fibres of E are diffeomorphic to a -manifold X with b ( X ) = 0 and b + ( X ) = 2 p +1 odd. Suppose that n = r + d − p − ≥ . Then SW n ( E, s E , φ ) ∈ H r ( S r ; Z ) ∼ = Z is divisible by the denominators of a p − r,l , for l = 0 , , . . . n , where the rational numbers a p,l are defined to be thecoefficients of the Taylor expansion: log (1 − y ) p = ∞ X l =0 a p,l y p + l . Structure of paper.
A brief outline of the contents of this paper is as fol-lows. In Subsection 2.1 we introduce the notion an infinite dimensional familiesmonopole map, of which the families Seiberg-Witten monopole map is a specialcase. We define chambers and families Seiberg-Witten invariants for such maps.In Subsection 2.2 we introduce the notion of a finite dimensional monopole mapand define chambers and Seiberg-Witten invariants for them. In Subsection 2.3 weshow that any infinite dimensional families monopole map has a finite dimensionalapproximation and in Subsection 2.4 we prove that the families Seiberg-Witteninvariants are preserved under the process of taking a finite dimensional approxi-mation (provided the approximation is “sufficiently large”). In Section 3 we proveour main formula which expresses the families Seiberg-Witten invariants of a finitedimensional monopole map in terms of purely cohomological operations. The re-maining sections of the paper are concerned with applications of the cohomologicalformula. In Section 4 we compute the Steenrod powers of the families Seiberg-Witten invariants and in Subsection 4.2 we give an application of these results to K K -theoreticfamilies Seiberg-Witten invariants and compute their Chern character in terms ofthe cohomological Seiberg-Witten invariants. We also prove a wall crossing formulafor these invariants. Acknowledgments . From Nobuo Iida, the second author heard Mikio Furuta’sargument to relate the Seiberg-Witten moduli space to the corresponding modulispace of a finite dimensional approximation of the monopole map in the unparam-eterised setting. This helped the authors to consider its families version. Afterthe version 1 of this paper had appeared on arXiv, the second author uploadedthe paper [10] with Tsuyoshi Kato and Nobuhiro Nakamura on arXiv. The workon [10] and a comment by Nakamura on the version 1 of this paper prompted theauthors to conceive the idea of Theorem 1.4 and Corollary 1.6, which are addedafter [10] appeared on arXiv. The authors are grateful to Kato and Nakamura forthese. D. Baraglia was financially supported by the Australian Research CouncilDiscovery Early Career Researcher Award DE160100024 and Australian ResearchCouncil Discovery Project DP170101054. H. Konno was supported by JSPS KAK-ENHI Grant Number 16J05569 and the Program for Leading Graduate Schools,MEXT, Japan.
N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 9 The families monopole map and finite dimensional approximations
We now turn to the construction of the families monopole map and its finitedimensional approximations. While the primary motivation for this construction isthe Bauer-Furuta and Seiberg-Witten invariants for a family of smooth 4-manifolds,the construction holds in a more general setting which we outline below.2.1.
Families monopole map and families Seiberg-Witten invariants.
Inthis subsection we introduce a general notion of a families monopole map and itsassociated families Seiberg-Witten invariants. In the case of the families Seiberg-Witten monopole map of a spin c family of 4-manifolds, we recover the notion ofthe families Seiberg-Witten invariants. In the following subsections we will showhow to construct a finite dimensional approximation of the families monopole mapand recover the families Seiberg-Witten invariants from the approximation.The general setting for the families monopole map is as follows. Let B be acompact smooth manifold. Let V , W → B be infinite dimensional separable Hilbertspace bundles, each a direct sum of a real and a complex Hilbert space bundle ofinfinite dimension: V = V C ⊕ V R , W = W C ⊕ W R where V C , W C are complex infinite dimensional Hilbert bundles and V R , W R arereal infinite dimensional Hilbert bundles.Let S act on V C , W C by scalar multiplication and act trivially on V R , W R . Let f : V → W be an S -equivariant bundle map. We will say that f is a familiesmonopole map (or monopole map for short) if it satisfies the following conditions:(M1) f = l + c , where l : V → W is an S -equivariant fibrewise linear Fredholmmap and c : V → W is an S -equivariant family of compact operators.(M2) l and c are both smooth as maps between Hilbert manifolds (hence f itselfis smooth)(M3) f satisfies a boundedness property: the preimage under f of a disc bundlein W is contained in a disc bundle in V .(M4) By S -equivariance we can write l = l C ⊕ l R , where l C : V C → W C , l R : V R → W R . We assume that l R is injective.(M5) We assume that c ( v ) = 0 for all v ∈ V R . Example 2.1.
The following example is the main motivation for the above notionof a monopole map. Suppose we have a family π : E → B of 4-manifolds over B with fibres diffeomorphic to X . Here we discuss the simple case that b ( X ) = 0.The case b ( X ) > E is equipped with a spin c -structure s E . Choose a smoothlyvarying fibrewise metric g = { g b } b ∈ B on E . Fix a smoothly varying family of U (1)-connections A = { A b } b ∈ B for the determinant line of the spin c -structure (forinstance one could choose a globally defined connection on the total space of E anddefine A b as the restriction of this connection to the fibre over b ). Fix an integer k >
2. We define the following Hilbert bundles over B : V C = L k ( S + ) , V R = L k ( ∧ T ∗ X ) , W C = L k − ( S − ) , W R = L k − ( ∧ + T ∗ X ) ⊕ L k − ( R )
00 DAVID BARAGLIA, HOKUTO KONNO where S ± denote the spinor bundles, L k ( E ) denotes the Sobolev space of L k sec-tions of E and L k − ( R ) denotes the subspace of sections f ∈ L k − ( R ) satisfying R X f dvol X = 0.We define the families Seiberg-Witten monopole map f : V → W to be given by f ( ψ, a ) = ( D A + ia ψ, − iF + A + ia + iσ ( ψ ) + iF + A , d ∗ a )where D A + ia denotes the spin c Dirac operator associated to A + ia and σ ( ψ ) is thequadratic spinor term in the Seiberg-Witten equations. The second term of f isusually defined to be − iF + A + ia + iσ ( ψ ), but we have added a harmless term iF + A which merely shifts the level sets of f . The reason for including this extra term istwofold: first it makes f satisfy (M5). Secondly it makes it easier to describe thechamber structure of the families Seiberg-Witten invariants associated to f . Notethat f can be re-written as: f ( ψ, a ) = ( D A + ia ψ, d + a + iσ ( ψ ) , d ∗ a ) . Then f = l + c , where l ( ψ, a ) = ( D A ψ, d + a, d ∗ a ) , c ( ψ, a ) = ( i a · ψ, iσ ( ψ ) , . So l C ( ψ ) = D A ψ , l R ( a ) = ( d + a, d ∗ a ). It follows that l R is injective since we assumedthat b ( X ) = 0. So f satisfies (M4). Clearly c (0 , a ) = 0, so f also satisfies (M5).Conditions (M1), (M2) are clearly satisfied as well. Condition ( M
3) was provenin [3] for a single 4-manifold. The same estimates easily extend to the case of asmoothly varying family over a compact base. Hence the families Seiberg-Wittenmonopole map satisfies conditions (M1)-(M5).
Definition 2.2.
Let f : V → W be a monopole map. A chamber for f is ahomotopy class of section η : B → W R \ l R ( V R ). We let CH ( f ) denote the set ofchambers for f (note that CH ( f ) could be empty).Let f be a monopole map. Recall that l R is Fredholm and injective by (M4).Hence W , R = Ker ( l ∗ R ) is a finite rank subbundle of W R which can be identifiedwith coker ( l R ). Clearly W R \ l R ( V R ) admits a fibrewise deformation retraction to W , R \ { } , where { } denotes the zero section. Thus a chamber for f is equivalentto a homotopy class of non-vanishing section of W , R . By rescaling, we have thatfor any fixed δ >
0, any chamber can be represented by a section η : B → W R whose norm satisfies || η || < δ (pointwise with respect to B ).In order to prove our main result we need to make two further assumptions on f :(M6) Let δ > f can be represented by a section η : B → W R such that || η || < δ pointwise with respect to B and such that η is transverse to f . Furthermore any two such η representing the samechamber can be joined by a path η ( t ) such that || η ( t ) || < δ for all t and η ( t ) : [0 , × B → W R \ l R ( V R ) is transverse to f ◦ pr V : [0 , × V → W ,where pr V : [0 , × V → V is projection to the second factor.(M7) There exists a smooth R -bilinear map q : V × B V → W such that c ( v ) = q ( v, v ). Example 2.3.
In the case that f is the Seiberg-Witten monopole map of a familyof 4-manifolds with b ( X ) = 0, then (M6)-(M7) are satisfied. (M6) follows fromthe existence of regular perturbations of the families Seiberg-Witten equations (this N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 11 is well known in the case of a single 4-manifold and easily adapts to the case offamilies). (M7) is easy to verify by looking at the Seiberg-Witten equations.
Example 2.4.
We now explain how the construction of the families Seiberg-Wittenmonopole map extends to the case of a spin c -family of 4-manifolds π : E → B whosefibres are diffeomorphic to a 4-manifold X with b ( X ) >
0, equipped with a section x : B → E . Let H ( R ) → B denote the vector bundle whose fibre H b ( R ) over b ∈ B is the space of harmonic 1-forms on the fibre E b = π − ( b ). Then on the totalspace of H ( R ) we define the following Hilbert bundles: V C = L k ( S + ) , V R = L k ( ∧ T ∗ X ) , W C = L k − ( S − ) , W R = L k − ( ∧ + T ∗ X ) ⊕ L k − ( R ) where L k − ( R ) is defined as in Example 2.1 and L k ( ∧ T ∗ X ) is the subbundle of L k ( ∧ T ∗ X ) consisting of 1-forms L -orthogonal to the finite dimensional subbundleof harmonic 1-forms. Define f : V C ⊕ V R → W C ⊕ W R over θ ∈ H ( R ) by: f ( θ, ( ψ, a )) = ( D A + ia + iθ ( ψ ) , d + a + iσ ( ψ ) , d ∗ a )where A , σ ( ψ ) are as in Example 2.1.The map f is a combination of the Seiberg-Witten equations and the gaugefixing term d ∗ a = 0. To obtain the (families) Seiberg-Witten moduli space we stillneed to factor out by harmonic gauge transformations. This is done as follows.Let H ( Z ) → B denote the locally constant bundle of groups whose fibre H b ( Z )over b ∈ B is the space of harmonic 1-forms on E b with integral periods. Toeach ω ∈ H ( Z ) we can define a unique harmonic map f ω : E b → U (1) such that f − ω df ω = 2 πiω and f ω ( x ( b )) = 1, namely f ω is given by f ω ( y ) = exp πi Z yx ( b ) ω ! . We let H ( Z ) act fibrewise on H ( R ) by: ω · θ = θ + 4 πω. Let J = H ( R ) / H ( Z ) be the quotient with respect to this action. Then J is atorus bundle over B (in fact, J is isomorphic to the bundle of Jacobians associatedto E → B , where the Jacobian J b of the fibre E b is defined to be the space ofisomorphism classes of flat topologically trivial unitary line bundles on E b ).The action of H ( Z ) on H ( R ) just defined lifts to fibrewise linear actions on thetotal spaces of V = V C ⊕ V R and W = W C ⊕ W R as follows: ω · ( θ, ( ψ, a )) = ( θ + 4 πω, ( f − ω ψ, a )) , ( ψ, a ) ∈ V θ ω · ( θ, ( φ, ν, g )) = ( θ + 4 πω, ( f − ω φ, ν, g )) , ( φ, ν, g ) ∈ W θ . This is precisely the action of the group of harmonic gauge transformations (whichare normalised to equal the identity at x ( b )). Note that H ( Z ) acts freely on H ( R )with quotient space J and so the vector bundles V , W descend to vector bundles V / H ( Z ) , W / H ( Z ) on J . Moreover f is H ( Z ) × S -equivariant so descends to an S -equivariant map f : V / H ( Z ) → W / H ( Z )of Hilbert bundles over J . This is our desired families Seiberg-Witten monopolemap in the case that b ( X ) >
0. One can verify that f satisfies (M1)-(M7) in muchthe same way as was done in the b ( X ) = 0 case in Example 2.1. Lemma 2.5.
Let f satisfy (M1)-(M7). Then for any v ∈ V , the linear map V → W given by u q ( v, u ) is compact.Proof. Note that since c ( v ) = q ( v, v ), it follows that:(2.1) q ( v, u ) = 12 ( c ( u + v ) − c ( u ) − c ( v )) . Fix v ∈ V b and let { u i } be a bounded sequence of elements in V b . Then bycompactness of c , we can take a subsequence such that { c ( u i ) } converges. Takinga further subsequence, we can also assume that { c ( v + u i ) } converges. Then byEquation (2.1), the sequence q ( v, u i ) also converges. (cid:3) Let f be a monopole map satisfying (M1)-(M7). Let η : B → W R be a repre-sentative of a chamber such that η is transverse to f . By (M6) such an η existsfor any chamber and we can take || η || to be as small as desired. By transversality f M η = f − ( η ) is a smooth manifold equipped with a free S -action and an S -invariant projection map π f M η : f M η → B . We claim that f M η is finite-dimensionaland in fact dim ( f M η ) = dim ( B ) + ind ( l ). To see this, note that the linearisationof f ( v ) = η around v (in the fibre directions) is given by df v ( ˙ v ) = l ( ˙ v ) + 2 q ( v, ˙ v ).By Lemma 2.5, 2 q ( v, · ) is a compact operator. It follows that the total derivativeof f : V → W at any point is Fredholm with index equal to ind ( l ). Then since f is transverse to η , dim ( f M η ) = dim ( f − ( η ( B ))) = dim ( B ) + ind ( l ). Moreover,since f M η is finite-dimensional and contained in a bounded subset of V (by (M3)),it follows that f M η is compact.Since S acts freely on f M η , the quotient M η = f M η /S is a compact smoothmanifold of dimension dim ( B ) + ind ( l ) −
1. Moreover the map π f M η : f M η → B is S -invariant and so descends to a map π M η : M η → B . Let L → M η denote thecomplex line bundle associated to the principal S -bundle f M η → M η . Lemma 2.6.
Let
Ind ( l R ) ∈ KO ( B ) denote the real K -theory class of the familyof real Fredholm operators l R : V R → W R . Then there is a natural isomorphism ψ : det ( T M η ⊕ π ∗M η ( T B )) ∼ = π ∗M η ( det ( Ind ( l R )) .Proof. Using the S -action we have that det ( T f M η ) descends to f M η and one easilysees that det ( T f M η ) /S = det ( T M η ). Therefore it suffices to show that thereis an S -equivariant isomorphism det ( T f M η ⊕ π ∗ f M η ( T B )) ∼ = π ∗ f M η ( det ( Ind ( l R ))).However we have seen that the linearisation of f around any point of f M η differsfrom l by a compact operator. In particular we can write down an S -equivarianthomotopy through Fredholm operators from the family { df v } v ∈ f M to the family { l } v ∈ f M . From this it follows that det ( T f M η ) is S -equivariantly isomorphic to thepullback of det ( Ind ( l ) ⊕ det ( T B )), where
Ind ( l ) ∈ KO ( B ) is the families index of l . But l = l R + l C , so we get a decomposition Ind ( l ) = Ind ( l C ) + Ind ( l R ). Moreover l C is a complex Fredholm operator, so det ( Ind ( l C )) is trivial and det ( Ind ( l )) = det ( Ind ( l R )). (cid:3) Let w = w ( Ind ( l R )) ∈ H ( B ; Z ) be the first Stiefel-Whitney class of Ind ( l R ).Then from the above lemma, w is the relative orientation class of π M η : M η → B .Let Z w denote the local system with coefficient group Z determined by w . N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 13
Definition 2.7.
Let m ≥ η ] ∈ CH ( f ) be achamber of f represented by a section η : B → W R \ l R ( V R ) which is transverse to f . The m -th families Seiberg-Witten invariant of f with respect to η is defined as SW m ( f, η ) = ( π M η ) ∗ ( c ( L ) m ) ∈ H m − ( ind ( l ) − ( B ; Z w )where w = det ( Ind ( l R )) ∈ H ( B ; Z ). Remark . If Ind ( l R ) is orientable, then we can regard SW m ( f, η ) as a cohomologyclass with integral coefficients. However this depends on the choice of orientationof Ind ( l R ). Reversing orientation on Ind ( l R ) will change the sign of SW m ( f, η ). Lemma 2.9.
Let f : V → W satisfy (M1)-(M7). Then SW m ( f, η ) depends onlyon the chamber [ η ] ∈ CH ( f ) in which η lies and not on the particular choice of η .Proof. Let η , η : B → W R \ l R ( V R ) represent the same chamber and assume η , η are both transverse to f . Let M > sup b ∈ B {|| η ( b ) || , || η ( b ) ||} . Then by (M6) wecan find a generic path η ( t ) joining η to η and such that || η ( t, b ) || < M for all( t, b ) ∈ [0 , × B . Then f M = ` t ∈ [0 , f − ( η ( t )) is a smooth manifold with free S -action and with boundary ] M η ` ] M η . Moreover f M is compact, by (M3). Let M = f M /S . Then M is a smooth compact manifold with boundary M η ` M η .Moreover the line bundles on M η , M η extend to a common line bundle L → M (namely, the line bundle associated to f M → M ). It follows that SW m ( f, η ) = ( π M η ) ∗ ( c ( L ) m | M η ) = ( π M η ) ∗ ( c ( L ) m | M η ) = SW m ( f, η ) . (cid:3) Remark . In the case that f is the families Seiberg-Witten monopole map ofa spin c family of 4-manifolds ( E, s E ) with b ( X ) = 0, we have that SW m ( f, η ) = SW m ( E, s E , [ η ]) is exactly the m -th families Seiberg-Witten invariant of the family( E, s E ) with respect to the chamber [ η ], as defined in the introduction.If E → B is a spin c family of 4-manifolds with b ( X ) ≥ x : B → E , we have constructed a monopole map defined on the total spaceof the Jacobian bundle J → B (Example 2.4). Associated to this monopole mapare families Seiberg-Witten invariants SW m ( f, η ) ∈ H m − (2 d − b + − ( J ; Z w ), where w = w ( H + ). We interpret these as the families Seiberg-Witten invariants of thequadruple ( E, s E , x, [ η ]) and set SW m ( E, s E , x, [ η ]) = SW m ( f, η ).When b ( X ) = 0 we clearly have that SW m ( E, s E , x, [ η ]) agrees with our previousdefinition of SW m ( E, s E , [ η ]) and in particular does not depend on the choice ofsection x (and can be defined even if no such section exists).If b ( X ) >
0, then in general SW ( E, s E , x, [ η ]) may depend on the choice of sec-tion x , but only up to homotopy. To see homotopy invariance, let x : [0 , × B → E be a homotopy of section from x to x . Then we can view x as a section ofthe pullback [0 , × E of E over [0 , × B and obtain a families Seiberg-Witteninvariant SW m ([0 , × E, s E , x, [ η ]) ∈ H m − (2 d − b + − ([0 , × J ; Z w ). Restric-tion of SW m ([0 , × E, s E , x, [ η ]) to t = 0 , SW m ( E, s E , x , [ η ]) and SW m ( E, s E , x , [ η ]) respectively, but the homotopy equivalence [0 , × J ∼ = J im-plies that SW m ( E, s E , x , η ) = SW m ( E, s E , x , [ η ]).In order to compare the Seiberg-Witten invariants of f with those of a finitedimensional approximation we would like to restate the definition of the invariants SW m ( f, η ) in a slightly different way. Fix a real line bundle U → B and considertriples ( M, π M , ψ M ) consisting of: • a smooth compact manifold M with a free S -action • an S -invariant smooth map π M : M → B • an S -equivariant isomorphism of real line bundles ψ M : det ( T M ⊕ π ∗ M ( T B )) ∼ = π ∗ M ( U ).Then M/S is a smooth compact manifold and π M descends to π M/S : M/S → B .Let L M → M be the complex line bundle associated to the principal S -bundle M → M/S and define the m -th Seiberg-Witten invariant of ( M, π M , ψ M ) to be: SW ( M, π, ψ ) = ( π M/S ) ∗ ( c ( L M ) m ) ∈ H m + dim ( B ) − dim ( M ) − ( B ; Z w )where w = w ( U ) and where we use ψ M to identify the relative orientation classof π M/S : M/S → B with π ∗ ( U ).Now let f : V → W be a families monopole map satisfying (M1)-(M7) and let[ η ] ∈ CH ( f ) be a chamber represented by a map η : B → W R \ l R ( V R ) such that η is transverse to f . Set U = det ( Ind ( l R )). Then we obtain a triple ( M, π M , ψ M ),where M = f M η = f − ( η ) equipped with the natural S -action, π M is the naturalprojection to B and ψ M is the isomorphism in Lemma 2.6. Clearly the familiesSeiberg-Witten invariants of f , SW m ( f, η ) coincide with the Seiberg-Witten invari-ants of the triple ( M, π, ψ ).We say that two triples ( M , π M , ψ M ), ( M , π M , ψ M ) are cobordant if thereexists a smooth compact manifold M with boundary the disjoint union of M and M such that the S -actions on M , M extends to a free S -action on M , the maps π M , π M are the restrictions to ∂M of a S -invariant map π M : M → B and thereis an S -equivariant isomorphism ψ M : det( T M ⊕ π ∗ ( T B )) → π ∗ M ( U ) such that ψ | M = ψ M , ψ | M = − ψ M (where we use the outward normal first convention toidentify det ( T M ) | M j with det ( T M j ) for j = 1 , M, π, ψ ) are cobordism invariant. Thus to obtain the families Seiberg-Witteninvariants SW m ( f, η ) of f from a finite dimensional approximation, it suffices toobtain the cobordism class of associated triple ( M, π, ψ ). This will be our goal inthe following subsections.2.2.
Finite dimensional monopole maps and their Seiberg-Witten invari-ants.
In this subsection we introduce the notion of a finite dimensional familiesmonopole map and define the Seiberg-Witten invariants of such maps. In the fol-lowing subsections we will see how to obtain a finite dimensional monopole map asan approximation of an infinite dimensional one and show that the Seiberg-Witteninvariants are preserved under this process.Let B be a compact smooth manifold. Suppose that V, V ′ → B are complexvector bundles of ranks a, a ′ and U, U ′ → B are real vector bundles of ranks b, b ′ .We make V, V ′ into S -equivariant vector bundles by letting S act by scalar multi-plication on the fibres. Similarly we make U, U ′ into S -equivariant vector bundlesby giving them the trivial action.For any vector bundle W , let S W be the unit sphere bundle in R ⊕ W . Thus S W is obtained from W by one-point compactifying each fibre of W . Let B W ⊂ S W denote the section at infinity. If V is a complex vector bundle and U a real vectorbundle, then the S -action on V ⊕ U extends to S V ⊕ U preserving the section at N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 15 infinity. We often write S V,U instead of S V ⊕ U to remind ourselves that the S -actionon S V,U depends on the pair (
V, U ).Suppose that we have an S -equivariant map f : S V,U → S V ′ ,U ′ covering theidentity on B and which sends infinity to infinity. Our interest in such maps is thatthey arise as finite dimensional approximations of the Seiberg-Witten monopolemap of spin c families of 4-manifolds. More precisely, suppose that X is a compact,oriented, smooth 4-manifold with b ( X ) = 0 and suppose that E → B is a spin c family with fibres diffeomorphic to X . To such data we constructed the familiesSeiberg-Witten monopole map (Example 2.1). As will be shown in Subsection 2.3,we can take a finite dimensional approximation of the Seiberg-Witten monopolemap to obtain an S -equivariant map f : S V,U → S V ′ ,U ′ where V, V ′ , U, U ′ satisfy: V − V ′ = D ∈ K ( B ) , U ′ − U = H + ∈ KO ( B ) , where D ∈ K ( B ) denotes the families index of the families spin c Dirac operator of( E, s E ) and H + is the vector bundle over B whose fibres are the space of harmonicself-dual 2-forms on the corresponding fibres of the family E → B . Remark . A similar construction holds in the case that b ( X ) >
0. Considera spin c family ( E, s E ) over B which admits a section x : B → E . To such datawe constructed a families Seiberg-Witten monopole map (Example 2.4). Taking afinite dimensional approximation of this map one again obtains an equivariant map f : S V ⊕ U → S V ′ ⊕ U ′ , except that V, V ′ , U, U ′ are now vector bundles over the space J , where π J : J → B is the Jacobian bundle of the family E → B (see Example2.4). Further, we have that V, V ′ , U, U ′ satisfy: V − V ′ = D J ∈ K ( J ) , U ′ − U = π ∗ J ( H + ) ∈ KO ( J ) , where H + is defined as in the b ( X ) = 0 case and D J is defined as follows. Let D A be the spin c Dirac operator associated to the reference connection A . Eachpoint in J defines a flat line bundle on X . Coupling this line bundle to D A we geta family of spin c Dirac operators parametrised by J . Then D J is defined as thefamilies index of this family.Consider again an S -equivariant map f : S V,U → S V ′ ,U ′ , where V, V ′ , U, U ′ are vector bundles on B and f covers the identity. This will be the setup usedhenceforth. By analogy with with the case of the Seiberg-Witten monopole map,we let D denote the virtual bundle D = V − V ′ ∈ K ( B ) and similarly let H + denote H + = U ′ − U ∈ KO ( B ). Remark . As we will see in Subsection 2.3, if f arises as a finite dimensionalapproximation of the Seiberg-Witten monopole map, then U can be identified witha subbundle of U ′ and f | U : U → U ′ with the inclusion ι : U → U ′ .In light of Remark 2.12, we will assume henceforth that: Assumption 1 : U can be identified with a subbundle of U ′ and that f | U : U → U ′ is the inclusion ι : U → U ′ .Under Assumption 1 we can assume that U ′ splits into a direct sum U ′ = U ⊕ H + and that H + is a genuine vector bundle rather than a virtual vector bundle. Letus also set: d = rank C ( D ) = a − a ′ , b + = rank R ( H + ) = b ′ − b. In the case that f is the finite dimensional approximation of the Seiberg-Wittenmonopole map of a spin c family ( E X , s E ), we of course have d = c ( s X ) − σ ( X )8 , b + = b + ( X ) , where s X = s E | X is the spin c structure on X induced by restriction of s E to a fibre X of E . Definition 2.13.
Let
V, V ′ be complex vector bundles and U, U ′ be real vectorbundles over a compact smooth base manifold B . An S -equivariant map f : S V,U → S V ′ ,U ′ preserving sections at infinity and satisfying Assumption 1 will becalled a finite dimensional (families) monopole map .Let X, Y → B be fibre bundles over B equipped with sections x ∞ : B → X , y ∞ : B → Y . Then we can form the fibrewise smash product X ∧ B Y by taking thefibre product X × B Y and for each b ∈ B collapsing x ∞ ( b ) × Y b ∪ X b × y ∞ ( b ) to apoint. If X, Y have S -actions and x ∞ , y ∞ are S -invariant then X ∧ B Y inheritsan S -action. In particular if V , V are complex vector bundles and U , U realvector bundles, then one finds(2.2) S V ,U ∧ B S V ,U = S V ⊕ V ,U ⊕ U . In what follows, we will be interested primarily in the stable equivariant homo-topy class of f . By the stable equivariant homotopy class of f , we mean that weconsider the homotopy class of f up to stabilisation, where by stabilisation of f we mean taking the fibrewise smash product of f with id : S C m , R n → S C m , R n forany m, n ≥
0. Homotopies are taken to be homotopies through finite dimensionalmonopole maps, that is, through equivariant maps f t covering the identity on B ,preserving the section at infinity and such that f t | U : U → U ′ is a linear inclusionfor each t . By (2.2), stabilisation by C and R change V, V ′ , U, U ′ as follows: V V ⊕ C , V ′ V ′ ⊕ C (stabilisation by C ) ,U U ⊕ R , U ′ U ′ ⊕ R (stabilisation by R ) . Of course, such stabilisations do not change the K -theory classes D = V − V ′ and H + = U ′ − U .More generally, we can consider stabilisation by an arbitrary complex vectorbundle A or a real vector bundle B : V V ⊕ A, V ′ V ′ ⊕ A (stabilisation by A ) ,U U ⊕ B, U ′ U ′ ⊕ B (stabilisation by B ) . This corresponds to taking the fibrewise smash product of f with id : S A, → S A, or with id : S ,B → S ,B . Note that for any such vector bundles A, B we can findcomplementary vector bundles ˆ A, ˆ B such that A ⊕ ˆ A = C m , B ⊕ ˆ B = R n . Hencethe more general notion of stabilisation by vector bundles retains the underlyingstable homotopy class of f .After stabilising by a suitable choice of vector bundles, we can assume that V ′ ∼ = C a ′ × B and U ′ ∼ = R b ′ × B are trivial vector bundles over B . In this case f can be viewed as a stable equivariant homotopy class f : S V,U → ( C a ′ ⊕ R b ′ ) + from the total space of S V,U into the sphere ( C a ′ ⊕ R b ′ ) + . In this way f definesa stable equivariant cohomotopy class, which we may refer to as the Bauer-Furutainvariant of f , following [3]. On the other hand, we will see that it is sometimes N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 17 more convenient to stabilise such that V ∼ = C a × B and U ∼ = R b × B are trivial. Wewill make use of both types of stabilisations of f .Let f : S V,U → S V ′ ,U ′ be a finite dimensional monopole map. Thus we canassume that U ′ = U ⊕ H + and that f | U : U → U ′ is the inclusion map ι : U → U ′ . Definition 2.14. A chamber for f is a homotopy class of section φ : B → U ′ − U .Equivalently, this is the same as a homotopy class of section B → S ( H + ), where S ( H + ) is the unit sphere bundle of H + . Denote by CH ( f ) the set of chambers. Remark . In general, the set of chambers for f may be empty. Indeed thishappens precisely when H + does not admit a non-vanishing section. An obviousnecessary condition for this is that b + >
0. A sufficient condition for the existenceof a chamber is b + > dim ( B ). Using the fact that S ( H + ) → B is a sphere bundlewith fibres of dimension b + −
1, we see that if b + > dim ( B ) then there exists aunique chamber. Lemma 2.16.
Let
X, Y be smooth finite dimensional vector bundles over B . As-sume that X, Y are equipped with S -actions and that the projections π X : X → B , π Y : Y → B are S -invariant. Let f : X → Y be a smooth S -equivariant mapcovering the identity on B and suppose that η : B → Y is an S -invariant smoothmap which is transverse to f . Then M = f − ( η ) is a smooth manifold with an S -action. Let π M : M → B be the projection to B . Then there is a naturallydefined S -equivariant isomorphism ψ M : det ( T M ⊕ π ∗ M ( T B )) ∼ = π ∗ ( U ) . where U = det ( Y ⊕ X ) .Proof. T X | M can be decomposed in two ways: (1) using the vertical and horizontalsubspaces with respect to π X : X → B and (2) using the tangent and normalbundles of M . Equating these gives a canonical isomorphism: T M ⊕ N M ∼ = π ∗ M ( V ′ ⊕ T B ) , where N M is the normal bundle of M in X . But since M = f − ( η ) and η istransverse to f , we get a canonical isomorphism N M ∼ = f ∗ ( π ∗ Y ( W ′ )) | M ∼ = π ∗ M ( W ′ ).Hence T M ⊕ π ∗ M ( W ′ ) ∼ = π ∗ M ( V ′ ⊕ T B ) , which taking determinants gives det ( T M ⊕ π ∗ M ( T B )) ∼ = π ∗ M ( U ), where U = det ( Y ⊕ X ). (cid:3) Now we turn to the construction of Seiberg-Witten invariants of a finite dimen-sional monopole map f : S V,U → S V ′ ,U ′ . Let B V,U denote the section at infinityof S V,U and similarly define B V ′ ,U ′ . Our assumption that f preserves sections atinfinity means that we can regard f as a map of pairs: f : ( S V ⊕ U , B V ⊕ U ) → ( S V ′ ⊕ U ′ , B V ′ ⊕ U ′ )covering the identity on B . This point of view will quite useful in subsequentcalculations.Let [ φ ] ∈ CH ( f ) be a chamber and let φ : B → U ′ be a representative section(so in particular the image of φ is disjoint from U ). By [6, Theorem 7.1] thereexists an equivariant homotopy of f to a map f ′ such that f ′ is S -transversalto φ ( B ) (for the definition of G -transversality for a compact Lie group G , see [4]or [6]. The definitions of G -transversality given in these papers was shown to be equivalent in [7]). Note that f ′ can be chosen arbitrarily close to f in the C -topology. Thus we can assume that f ′− ( φ ( B )) is disjoint from U , so S acts freelyon f ′− ( φ ( B )). In such a case one easily verifies that the definition of S -equivarianttransversality reduces to transversality in the ordinary sense. So f ′ meets φ ( B )transversally and f M φ = f ′− ( φ ( B )) is a compact smooth submanifold of V ⊕ U of dimension 2 d − b + + dim ( B ). Moreover S acts freely on f M φ , so the quotient M φ = f M φ /S is a compact smooth manifold of dimension 2 d − b + − dim ( B ).The quotient map f M φ → M φ is a principal S -bundle. Let L = f M φ × S C be theassociated complex line bundle over M φ and let π M φ : M φ → B denote the naturalprojection map to B . By Lemma 2.16, we have a naturally defined isomorphism ψ : det ( T M φ ⊕ π ∗M φ ( T B )) ∼ = π ∗M φ ( det ( H + )). Definition 2.17.
Let [ φ ] ∈ CH ( f ) be represented by φ . Let m ≥ m -th Seiberg-Witten invariant of ( f, φ ) is the cohomologyclass SW m ( f, φ ) = ( π M φ ) ∗ ( c ( L ) m ) ∈ H m − (2 d − b + − ( B ; Z w )where w = w ( H + ) ∈ H ( B ; Z ) and Z w is the associated local system withcoefficient group Z .We have defined SW m ( f, φ ) as an invariant of the pair ( f, φ ). Later we willsee that SW m ( f, φ ) only depends on the stable homotopy class of f and on thechamber to which φ belongs. However, before we do this we should check that SW m ( f, φ ) does not depend on the choice of homotopy f ′ of f chosen so that f ′ is S -transversal to φ ( B ). In fact this will follow easily once we show that SW m ( f, φ )can be expressed in purely cohomological terms and so we defer the proof untilSubsection 3.2. Proposition 2.18.
The invariants SW m ( f, φ ) depend only on the stable homotopyclass [ f ] of f and the chamber [ φ ] . This will also follow easily once we have obtained a purely cohomological defini-tion of SW m ( f, φ ), so we also defer the proof of this until Subsection 3.2. We willalso prove a wall crossing formula, expressing the dependence of SW m ( f, φ ) on thechoice of chamber.2.3. Finite dimensional approximation.
Let f : V → W be a monopole mapsatisfying (M1)-(M7). In this subsection we will recall how a finite dimensionalapproximation of f is constructed, as in [3].Recall that l R is injective and we have set W , R = Ker ( l ∗ R ), a finite rank subbun-dle of W R . Next, consider l C : V C → W C . From [1, Proposition A5], there exists afinite rank complex subbundle W , C ⊂ W C such that W , C surjects to the cokernelof l C for any b ∈ B . Set W = W , C ⊕ W , R ⊂ W . Then W is a finite rank subbundle with the property that W surjects to thecokernel of l for any b ∈ B . Let D ( W ) = { w ∈ W | || w || ≤ } be the unit disc bundle in W . By (M3), there exists an R > f − ( D ( W )) ⊆ { v ∈ V | || v || < R } . N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 19
In particular this implies:(2.3) if || v || ≥ R, then || f ( v ) || > . Now let C ⊂ W be the closure of c ( D R ( V )). By compactness of c , the set C iscompact. Let C ′ be the image of C under the projection W → W /W ∼ = W ⊥ .Then C ′ is also compact. Fix a trivialisation W ⊥ ∼ = H × B for some Hilbertspace H (note that since W has finite rank, W ⊥ has infinite rank, so can betrivialised. Moreover the real and complex parts of W ⊥ both have infinite rank.Trivialising them separately, we see that the trivialisation W ⊥ ∼ = H × B can bechosen S -equivariantly for some action of S on H ). Under this trivialisation C ′ is a compact subset of H × B , hence, for any fixed ǫ >
0, can be covered by finitelymany open sets of the form { x ∈ H | || x − h i || < ǫ } × U i for some h , . . . , h n ∈ H and open subsets U , . . . , U n of B . Let H be the spanof h , . . . , h n , ih , . . . , ih n (where ih j means h j acted upon by i ∈ S ). Then H is a finite rank S -equivariant subspace of H . Let W be the preimage of H × B under the trivialisation W ⊥ ∼ = H × B . Then W is a finite rank subbundle of W ,orthogonal to W . Set W ′ = W ⊕ W . Then W ′ is a finite rank S -equivariant subbundle of W . By construction W ′ satisfies the following two properties: • W ′ surjects to the cokernel of l for any b ∈ B . • Every element of C is a distance less than ǫ from W ′ .Let p : W → W ′ denote the orthogonal projection to W ′ and p ⊥ = 1 − p theorthogonal projection to W ′′ = ( W ′ ) ⊥ . The second property of W ′ listed aboveimplies the following important estimate:(2.4) if || v || ≤ R, then || p ⊥ c ( v ) || < ǫ. We emphasise here that the choice of W ′ is dependent on the choice of ǫ . We willsay that a finite rank subbundle W ′ ⊂ W which contains W and which satisfies(2.4) is a subbundle of class ǫ (this notion depends on the choice of R > W , but it is the ǫ dependency that we wish to emphasise, hence thechoice of terminology).Given a subbundle W ′ ⊂ W of class ǫ , let V ′ = l − ( W ′ ) ⊆ V . Then V ′ is a finiterank subbundle of V (because of the fact that W ′ surjects to the cokernel of l foreach b ∈ B ). We also let V ′′ = ( V ′ ) ⊥ be the orthogonal complement. Then wehave V = V ′ ⊕ V ′′ W = W ′ ⊕ W ′′ and l decomposes into l = l + l + l , where l = l | V ′ : V ′ → W ′ , l = p ⊥ ◦ l | V ′′ : V ′′ → W ′′ , l = p ◦ l | V ′′ : V ′′ → W ′ .Now we are almost ready to give the construction of a finite dimensional ap-proximation of f . Let H be a Hilbert space. By definition the 1-point completedHilbert sphere is defined as S H = S ( R ⊕ H ), the unit sphere in R ⊕ H . Similarly fora Hilbert bundle V → B we define S V to be the unit sphere bundle of R ⊕ V . Thus S V is obtained from V by adding a point at infinity to every fibre. The boundedproperty (M3) ensures that f : V → W extends continuously to a map f : S V → S W .Consider the restriction f | S V ′ : S V ′ → S W . Lemma 2.19.
Assume that ǫ < . Then the image of f | S V ′ is disjoint from theimage of S ( W ′′ ) in S W .Proof. Since f sends the point at infinity in S V ′ to the point at infinity in S W , wejust have to show that the image of f | V ′ is disjoint from S ( W ′′ ) ⊂ W . Suppose onthe contrary that there exists some v ∈ V ′ with f ( v ) ∈ S ( W ′′ ). Thus pf ( v ) = 0and || p ⊥ f ( v ) || = || f ( v ) || = 1. By (2.3), we see that || v || < R and thus by (2.4) itfollows that || p ⊥ c ( v ) || ≤ ǫ . But if v ∈ V ′ , then p ⊥ l ( v ) = 0 and thus1 = || p ⊥ f ( v ) || = || p ⊥ c ( v ) || ≤ ǫ < , a contradiction. (cid:3) Recall from [3] that there is a deformation retraction ρ : S W \ S ( W ′′ ) → S W ′ . We will recall the construction of ρ in a moment. Then by Lemma 2.19, we obtaina map ˆ f = ρ ◦ f | S V ′ : S V ′ → S W ′ . We take ˆ f as our desired finite dimensional approximation of f .We recall the definition of ρ . First of all we need to identify W as a subset of S W = S ( R ⊕ W ). This is done by the stereographic map W ∋ w
11 + || w || (cid:0) || w || − , w (cid:1) and the point at infinity is ∞ = (1 , W = W ′ ⊕ W ′′ ,we have S W = S ( R ⊕ W ′ ⊕ W ′′ ) = { ( a, b, c ) ∈ R ⊕ W ′ ⊕ W ′′ | a + || b || + || c || = 1 } . Under the stereographic map, S ( W ′′ ) corresponds to ( a, b, c ) ∈ S W such that || c || 6 =1 or equivalently, such that a + || b || = 0. Then we may define ρ by ρ ( a, b, c ) = 1 p a + || b || ( a, b, . One can check that ρ ( ∞ ) = ∞ and if w ∈ W , then(2.5) ρ ( w ) = , if pw = 0 and || p ⊥ w || < , ∞ , if pw = 0 and || p ⊥ w || > ,λ ( w ) pw, otherwise , where λ ( w ) is a positive real number depending smoothly on w ∈ W \ S ( W ′′ ) (theprecise form of λ ( w ) is not important. All we will need to know about λ ( w ) is thatit is a positive real number). Note that ρ is a retraction to S W ′ , meaning that ρ ( w ) = w for any w ∈ S W ′ . Proposition 2.20.
The map ˆ f : S V ′ → S W ′ is a finite dimensional monopole map.Proof. We just have to check that ˆ f preserves the section at infinity and that itsatisfies Assumption 1. But f and ρ both clearly preserve the sections at infinityhence so does ˆ f . For Assumption 1, note that if v ∈ ( V ′ ) S , then c ( v ) = 0 by (M5),so f ( v ) = l ( v ) ∈ W ′ . But if w ∈ W ′ then ρ ( w ) = w because ρ is a retraction to S W ′ . Hence ˆ f ( v ) = ρ ( l ( v )) = l ( v ). So the restriction of ˆ f to ( V ′ ) S is just therestriction of l R to ( V ′ ) S and is therefore a linear injection. (cid:3) N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 21
In order to identify the Seiberg-Witten invariants of f and its finite dimensionalapproximation ˆ f , we must first be able to identify the chambers. Let ( W ′ ) S ,( V ′ ) S denote the S -invariant parts of W ′ , V ′ . Then in the notation of Subsection2.2 we have correspondences ( W ′ ) S ↔ U ′ , ( V ′ ) S ↔ U and W , R ↔ H + . Inparticular a chamber for ˆ f : S V ′ → S W ′ is a homotopy class of section ˆ η : B → ( W ′ ) S \ l (( V ′ ) S ).The following proposition is easily checked using the definitions of V ′ , W ′ . Proposition 2.21.
Let η : B → W R \ l R ( V R ) represent a chamber for f . Then p ◦ η : B → W ′ is valued in ( W ′ ) S \ l (( V ′ ) S ) and hence represents a chamber for ˆ f . The map η p ◦ η induces a bijection between chambers of f and ˆ f . Lemma 2.22.
Let ˆ η : B → ( W ′ ) S \ l (( V ′ ) S ) be a chamber for ˆ f . Then ˆ f − (ˆ η ) contains no fixed points.Proof. This is trivial, since the image of ˆ f | ( V ′ ) S is precisely l (( V ′ ) S ). (cid:3) Lemma 2.23.
Any chamber of ˆ f may be represented by a section ˆ η : B → ( W ′ ) S \ l (( V ′ ) S ) with || ˆ η || < δ . Moreover there exists an equivariant homotopy of ˆ f to amap ˆ f ′ : S V ′ → S W ′ such that ˆ f ′ is transverse to ˆ η . The map ˆ f ′ can additionallybe chosen arbitrarily close to ˆ f in the C -topology.Proof. This follows by the same type of argument used in Subsection 2.2. (cid:3)
Equality of Seiberg-Witten invariants.
Let f : V → W be a monopolemap satisfying (M1)-(M7). Let R > W be chosen as in Subsection 2.3. Let ǫ > W ′ ⊂ W be a finite rank subbundle of class ǫ . Let W ′′ , V ′ , V ′′ bedefined as in Subsection 2.3 and recall that l = l + l + l , where l : V ′ → W ′ , l : V ′′ → W ′′ , l : V ′′ → W ′ .Assume that ǫ < f : S V ′ → S W ′ is defined and is a finite dimensional monopole map.Let [ η ] ∈ CH ( f ) be a chamber represented by η : B → W R \ l R ( V R ) and assumethat η is transverse to f . Let ˆ η = pη : B → ( W ′ ) S \ l (( V ′ ) S ). Then ˆ η rep-resents the corresponding chamber of ˆ f . Under these conditions we have definedthe families Seiberg-Witten invariants of ( f, η ) and also the families Seiberg-Witteninvariants of ( ˆ f , ˆ η ). In this subsection we will establish their equality. Theorem 2.24.
Let R and W be fixed as above. For all sufficiently small ǫ > , if W ′ is a finite rank subbundle of class ǫ and ˆ f : S V ′ → S W ′ the correponding finitedimensional approximation, then SW m ( f, η ) = SW m ( ˆ f , ˆ η ) for all m ≥ . The rest of this subsection is concerned with the proof of Theorem 2.24. Theoverall strategy of the proof is as follows. Recall that in Subsection 2.1 we definedthe Seiberg-Witten invariants of a triple (
M, π M , ψ M ). We also defined the notionof a corbordism of triple and showed that the Seiberg-Witten invariants of a tripleare cobordism independent. Associated to the pair ( f, η ) is a triple ( M, π M , ψ M )where M = f − ( η ), π M is the natural projection and ψ M is the isomorphism given by Lemma 2.6. The Seiberg-Witten invariants of ( f, η ) are the Seiberg-Witteninvariants of the triple ( M, π M , ψ M ).Similarly, let ˆ f ′ and ˆ η be as in Lemma 2.23. Then ˆ M = f M ˆ η = ( ˆ f ′ ) − (ˆ η )is a compact smooth manifold with a free S -action and an S -invariant map to B . By Lemma 2.16 we obtain an S -equivariant isomorphism ψ ˆ M : det ( T f M ˆ η ⊕ π ∗M η ( T B )) ∼ = π ∗M η ( det ( Ind ( l R ))) and thus a triple ( ˆ M , π ˆ M , ψ ˆ M ). Then the Seiberg-Witten invariants of ( ˆ f , ˆ η ) are, essentially by definition, the Seiberg-Witten invari-ants of the triple ( ˆ M , π ˆ M , ψ ˆ M ).Thus to prove Theorem 2.24, it will be enough to show that the triples ( M, π M , ψ M )and ( ˆ M , π ˆ M , ψ ˆ M ) are cobordant, provided ǫ is sufficiently small. We will constructsuch a cobordism through a sequence of steps, but first we need some preliminaryresults. Lemma 2.25.
For any ǫ > and any W ′ of class ǫ we have: • l is invertible and || l − || ≤ K for some constant K which is independentof W ′ and ǫ . • || l || ≤ T for some constant T which is independent of W ′ and ǫ .Proof. We first show l is an isomorphism. It suffices to show l is injective andsurjective.For injectivity, suppose l ( v ′′ ) = 0, where v ′′ ∈ V ′′ . Then l ( v ′′ ) = l ( v ′′ ) + l ( v ′′ ) = l ( v ′′ ) ∈ W ′ . But V ′ = l − ( W ′ ), so l ( v ′′ ) = l ( v ′ ) for some v ′ ∈ V ′ . Thus l ( v ′′ − v ′ ) = 0 and v ′′ − v ′ ∈ Ker ( l ). But V ′ = l − ( W ′ ) implies that Ker ( l ) ⊆ V ′ ,and it follows that v ′′ = 0.For surjectivity, let w ′′ ∈ W ′′ . Then since W ′ surjects to coker ( l ), there exists w ′ ∈ W ′ such that w ′ and − w ′′ map to the same element of the cokernel. Hence w ′ + w ′′ = l ( v ) for some v ∈ V . Writing v as v = v ′ + v ′′ where v ′ ∈ V ′ , v ′′ ∈ V ′′ ,we see that w ′ + w ′′ = [ l ( v ′ ) + l ( v ′′ )] + l ( v ′′ ). Equating W ′′ components gives l ( v ′′ ) = w ′′ .Next, let us define V = l − ( W ). Then V is a constant rank subbundle of V ′ and hence V ′ = V ⊕ V where V is the orthogonal complement of V in V ′ .Similarly define W as the orthogonal complement of W in W ′ . Then we haveorthogonal decompositions: V = V ⊕ V ⊕ V ′′ W = W ⊕ W ⊕ W ′′ . Let p , p , p denote the orthogonal projections from W to W , W , W ′′ respectively.In particular, p = p + p and p ⊥ = p . Recall that l : V ′′ → W ′′ is defined as l = p ◦ l | V ′′ . Similarly define ˜ l : ( V ⊕ V ′′ ) → ( W ⊕ W ′′ ) by ˜ l = ( p + p ) ◦ l | V ⊕ V ′′ .Note that ( V ⊕ V ′′ ) = V ⊥ , ( W ⊕ W ′′ ) = W ⊥ and that ˜ l depends only on thechoice of W (and not on the choice of W ′ or ǫ ). Arguing in the same way thatwe did for l , we see that ˜ l is invertible. Let w ∈ W ′ . Let v = ˜ l − ( w ) so that˜ l ( v ) = w . Now decompose v into v = v + v ′′ , v ∈ V , v ′′ ∈ V ′′ . Then w = ( p + p ) l ( v ) = ( p + p ) l ( v ) + ( p + p ) l ( v ′′ ) . Extracting W ′′ components gives w = p l ( v ) + p l ( v ′′ ) . N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 23
But v ∈ V ⊆ V ′ = l − ( W ′ ), so l ( v ) ∈ W ′ and p l ( v ) = 0. Thus w = p l ( v ′′ ) = l ( v ′′ ) and v ′′ = l − ( w ). Thus || l − ( w ) || = || v ′′ || ≤ || v || = || ˜ l − ( w ) || ≤ K || w || where K = sup b ∈ B || ˜ l − || does not depend on W ′ or ǫ .Lastly, we note that if v ′′ ∈ V ′′ then l ( v ′′ ) = pl ( v ′′ ) and hence || l ( v ′′ ) || = || pl ( v ′′ ) || ≤ || l ( v ′′ ) || ≤ T || v ′′ || , where T = sup b ∈ B || l || does not depend on W ′ or ǫ . (cid:3) Lemma 2.26.
Let t ∈ [0 , be a real number and v ∈ V ′ with || v || = R . Then: • pf ( v ) = 0 , hence ρf ( v ) ∈ S W ′ does not equal ∞ and we can regard ρf ( v ) as an element of W ′ . • || (1 − t ) pf ( v ) + tρ ( f ( v )) || > .Proof. Let v ∈ V ′ with || v || = R and suppose on the contrary that pf ( v ) = 0.By (2.3), we have || f ( v ) || ≥ || p ⊥ c ( v ) || ≤ ǫ <
1. Also p ⊥ f ( v ) = p ⊥ c ( v ), since v ∈ V ′ . So1 ≤ || f ( v ) || = || p ⊥ f ( v ) || = || p ⊥ c ( v ) || <
1a contradiction.Let v ∈ V ′ with || v || = R . By the above, we can assume that ρf ( v ) = upf ( v ),where u = λ ( f ( v )) is a positive real number and pf ( v ) = 0. Thus(1 − t ) pf ( v ) + tρ ( f ( v )) = (1 − t + tu ) pf ( v )is non-zero, unless 1 − t + tu = 0. But this would imply that (1 − t ) = − tu . Clearly(1 − t ) ≥ − tu ≤ u is positive. So the only way we can get equality isif 1 − t = 0 = − tu . But 1 − t = 0 implies t = 1 and − tu = 0 implies t = 0, so thisis impossible. (cid:3) Fix once and for all a real number ǫ > ǫ < inf (cid:26) , R K , K ( T + 3 QR ) (cid:27) where Q = inf b ∈ B || q || (recall that c ( v ) = q ( v, v )). Fix a corresponding choice offinite rank subbundles W ′ , V ′ , where W ′ is of class ǫ and let ˆ f : S V ′ → S W ′ be thefinite dimensional approximation. As previously explained, we can assume that η is chosen with || η || < δ for any given δ >
0. We will assume that δ is chosen with δ < ǫ , but the precise value of δ will not be fixed until later.The cobordism from M = f − ( η ) to ˆ M = ( ˆ f ′ ) − (ˆ η ) will be carried out in asequence of cobordisms M = M ∼ M ∼ M = ˆ M . For this purpose we introducethe following notation: let f : V → W be given by f = f , let η = η : B → W R \ l R ( V R ) and recall that we may assume that || η || < δ and that η transverseto f . Write η = η ′ + η ′′ , where η ′ = pη , η ′′ = p ⊥ η . Lemma 2.27.
For all v, w ∈ V , we have || p ⊥ q ( v, w ) || ≤ ǫ R || v |||| w || . Proof.
Equation (2.1) together with (2.4) implies that: || p ⊥ q ( v, w ) || ≤ ǫ for all v, w ∈ V with || v || , || w || = 2 R . But since q is R -bilinear, a simple rescalingargument gives the result. (cid:3) Lemma 2.28.
Fix a point b ∈ B . Let v ′ ∈ V b satisfy || v ′ || ≤ R/ . Define a map φ : V ′′ b → V ′′ b by φ ( v ′′ ) = l − ( η ′′ ) − l − ( p ⊥ c ( v ′ + v ′′ )) . Then: (1) φ sends D R ( V ′′ b ) to itself. (2) φ : D R ( V ′′ b ) → D R ( V ′′ b ) is a contraction. Hence there exists a unique θ ( v ′ ) ∈ D R ( V ′′ b ) such that θ ( v ′ ) = φ ( θ ( v ′ )) . (3) We have that || θ ( v ′ ) || < inf { R, T +3 RQ ) } . (4) Letting b and v ′ vary, θ defines a smooth map θ : D R ( V ′ ) → D R ( V ′′ ) . (5) The map θ is S -equivariant.Proof. Let x, y ∈ D R ( V ′′ b ). Then || φ ( x ) − φ ( y ) || = || l − ( p ⊥ c ( v ′ + x ) − p ⊥ c ( v ′ + y )) ||≤ K || p ⊥ c ( v ′ + x ) − p ⊥ c ( v ′ + y ) || . But p ⊥ c ( v ′ + x ) − p ⊥ c ( v ′ + y ) = p ⊥ q ( v ′ + x, v ′ + x ) − p ⊥ q ( v ′ + y, v ′ + y )= 2 p ⊥ q ( v ′ , x − y ) + p ⊥ q ( x + y, x − y )= p ⊥ q (2 v ′ + x + y, x − y ) . Then using Lemma 2.27 we find:(2.6) || φ ( x ) − φ ( y ) || ≤ K || p ⊥ c ( v ′ + x ) − p ⊥ c ( v ′ + y ) ||≤ K || p ⊥ q (2 v ′ + x + y, x − y ) ||≤ ǫK R || v ′ + x + y |||| x − y ||≤ ǫK R (2 || v ′ || + || x || + || y || ) || x − y ||≤ ǫK R || x − y || < || x − y || where the last line follows from ǫ < R K . Then it will follow that φ acts as acontraction on D R ( V ′′ b ), provided we can show that φ sends D R ( V ′′ b ) to itself. Let x ∈ D R ( V ′′ b ). Then || φ ( x ) || ≤ || φ ( x ) − φ (0) || + || φ (0) ||≤ || x || + || φ (0) ||≤ R + || φ (0) || . But φ (0) = l − ( η ′′ ), so(2.7) || φ (0) || = || l − ( η ′′ ) || ≤ K || η ′′ || ≤ K || η || ≤ Kδ ≤ Kǫ.
Then since ǫ < R K we get that || φ (0) || ≤ R/ || φ ( x ) || ≤ R/ R/ < R .This proves that φ sends D R ( V ′′ b ) to itself and is a contraction. Moreover, setting x = θ ( v ′ ), we have || θ ( v ′ ) || ≤ || φ ( θ ( v ′ )) || < R. N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 25
To obtain the estimate || θ ( v ′ ) || < T +3 RQ ) , we note that the second to last line of(2.6) together with (2.7) gives: || θ ( v ′ ) || = || φ ( θ ( v ′ )) || ≤ || φ ( θ ( v ′ )) − φ (0) || + || φ (0) ||≤ ǫK R || θ ( v ′ ) || + Kǫ ≤ ǫK Kǫ ≤ Kǫ ≤ T + 3 RQ )where we also used that || θ ( v ′ ) || < R . The smoothness of θ : D R ( V ′ ) → D R ( V ′′ )follows easily from the implicit function theorem (note that θ extends smoothlyover the boundary of D R ( V ′ ) because θ ( v ′ ) is defined for || v ′ || ≤ R/ S -equivariance of θ , let z ∈ S . Recall that φ ( v ′′ ) = l − ( η ′′ ) − l − ( p ⊥ c ( v ′ + v ′′ )). Since this depends on v ′ and we wish to examine the dependence,we will write φ v ′ instead of φ , so φ v ′ ( v ′′ ) = l − ( η ′′ ) − l − ( p ⊥ c ( v ′ + v ′′ )) . Now using that c is S -equivariant, it follows that φ z · v ′ ( z · v ′′ ) = z · φ v ′ ( v ′′ ). It followsthat both θ ( z · v ′ ) and z · θ ( v ′ ) are fixed points of φ z · v ′ , hence θ ( z · v ′ ) = z · θ ( v ′ ) byuniqueness of the fixed point. (cid:3) We now set η = η ′ = pη and let f : V ′ → W ′ be defined by f ( v ′ ) = l ( v ′ ) + l ( θ ( v ′ )) + pc ( v ′ + θ ( v ′ )) . Lemma 2.29.
Let M = f − ( η ) ⊆ V and let M = { v ′ ∈ V ′ | f ( v ′ ) = η , || v ′ || < R } ⊆ V ′ . Then: • We have that M ⊆ D R ( V ) and M ⊆ D R ( V ′ ) . • The map V → V ′ , v v ′ = pv restricts to a bijection p | M : M → M . • The map D R ( V ′ ) → V , v ′ v = v ′ + θ ( v ′ ) restricts to a bijection M → M inverse to p | M . • The map p | M : M → M is an S -equivariant diffeomorphism. • η is transverse to f .Proof. Suppose that f ( v ) = η . Then since || η || ≤ δ < ǫ <
1, it follows from(2.3) that || v || < R . Hence M ⊆ D R ( V ). Also M ⊆ D R ( V ′ ) by its definition.Consider again the equation f ( v ) = η . Write v = v ′ + v ′′ with v ′ = pv ∈ V ′ , v ′′ = p ⊥ v ∈ V ′′ . Then as f = f = l + c , the W ′ and W ′′ components of f ( v ) = η are: l ( v ′ ) + l ( v ′′ ) + pc ( v ′ + v ′′ ) = η ′ l ( v ′′ ) + p ⊥ c ( v ′ + v ′′ ) = η ′′ . A solution of this pair of equations must have || v || < R , hence || v ′ || < R , || v ′′ || < R .Since l is invertible, the second equation can be rewitten as: v ′′ = l − ( η ′′ ) − l − p ⊥ c ( v ′ + v ′′ ) . Then since || v ′ || < R , Lemma 2.28 implies that this equation has a unique solution v ′′ = θ ( v ′ ). Substituting, we get that M = { v ∈ V | f ( v ) = η } = { v ′ ∈ V ′ | || v ′ || < R, l ( v ′ )+ l ( θ ( v ′ ))+ pc ( v ′ + θ ( v ′ )) = η ′ } , where the bijection is given by v v ′ = pv . However we recognise the set onthe right as M . So p : V → V ′ restricts to a bijection p | M : M → M . Theinverse map is v ′ v = v ′ + θ ( v ′ ). Since both of these maps are smooth and S -equivariant, we have that p | M is an S -equivariant diffeomorphism. The factthat f is transverse to η just follows from the fact that f is transverse to η . (cid:3) Note that M can be promoted to a triple ( M , π , ψ ) in the sense defined inSubsection 2.1, where π is the natural projection to B and ψ is the isomorphismof Lemma 2.6. Similarly M comes equipped with a natural map π : M → B andthe diffeomorphism M ∼ = M of Lemma 2.29 sends π to π . Then we can extend( M , π ) to a triple ( M , π , ψ ) where ψ is obtained from Lemma 2.16. Comparingthe proofs of Lemma 2.6 and 2.16 one easily sees that the diffeomorphism M → M that we have constructed sends ψ to ψ . Hence the triples ( M , π , ψ ) , ( M , π , ψ )are cobordant (indeed they are isomorphic).For t ∈ [0 , f t : D R ( V ′ ) → W ′ by f t ( v ) = l ( v ) + (1 − t ) l ( θ ( v )) + pc ( v + (1 − t ) θ ( v )) . Note that f t extends smoothly over the boundary of D R ( V ′ ) because θ does. Then f t defines an equivariant homotopy from f to f , where f : D R ( V ′ ) → W ′ isdefined by f ( v ) = l ( v ) + pc ( v ) = p ( l ( v ) + c ( v )) = p ( f ( v )) . Lemma 2.30.
Let v ′ ∈ D R ( V ′ ) . Then ρ ( f ( v )) = ∞ .Proof. Suppose on the contrary that v ∈ D R ( V ′ ) and ρ ( f ( v )) = ∞ . By (2.5) thismeans that pf ( v ) = 0 and || p ⊥ f ( v ) || >
1. But v ∈ V ′ means that p ⊥ f ( v ) = p ⊥ c ( v ).Also || v || ≤ R implies that || p ⊥ c ( v ) || ≤ ǫ , by (2.4). Thus1 < || p ⊥ f ( v ) || = || p ⊥ c ( v ) || ≤ ǫ < , a contradiction. (cid:3) By this lemma, if v ∈ D R ( V ′ ), then we can regard ρ ( f ( v )) as an element of W ′ and in fact ρ ( f ( v )) = λ ( f ( v )) pf ( v ) by (2.5). For t ∈ [0 , h t : D R ( V ′ ) → W ′ by h t ( v ) = (1 − t ) f ( v ) + tf ( v )where f : D R ( V ′ ) → W ′ is defined by: f ( v ) = ρ ( f ( v )) . In other words f = ˆ f | D R ( V ′ ) , where ˆ f is the finite dimensional approximation of f . Clearly h t is an equivariant homotopy from f to f . Lemma 2.31.
Let S R ( V ′ ) = { v ∈ V ′ | || v || = R } . Then for any t ∈ [0 , and v ∈ S R ( V ′ ) we have that || f t ( v ) || > ǫ and || h t ( v ) || > . N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 27
Proof.
The inequality || h t ( v ) || > v ∈ V ′ with || v || = R and || f t ( v ) || ≤ ǫ . By (2.3) we have that || f ( v ) || > || p ⊥ c ( v ) || < ǫ . So1 < || f ( v ) || ≤ || f ( v ) − f t ( v ) || + || f t ( v ) ||≤ ǫ + || f ( v ) − f t ( v ) ||≤ ǫ + || pc ( v + (1 − t ) θ ( v )) − pc ( v ) || + || l ( θ ( v )) || + || p ⊥ c ( v ) ||≤ ǫ + || c ( v + (1 − t ) θ ( v )) − c ( v ) || + T || θ ( v ) ||≤ ǫ + T || θ ( v ) || + || q ( v + (1 − t ) θ ( v ) , v + (1 − t ) θ ( v )) − q ( v, v ) ||≤ ǫ + T || θ ( v ) || + || q ( v, θ ( v )) || + || q ( θ ( v ) , θ ( v )) ||≤ ǫ + ( T + 2 Q || v || + Q || θ ( v ) || ) || θ ( v ) ||≤ ǫ + ( T + 3 QR ) || θ ( v ) ||≤ ǫ + 14 < ǫ < /
4. This is a contradiction,hence no such v exists. (cid:3) Lemma 2.32.
Let S + V ′ denote the complement in S V ′ of { v ∈ V ′ | || v || < R } . If v ∈ S + V ′ then ˆ f ( v ) = 0 .Proof. Suppose that v ∈ S + V ′ and ˆ f ( v ) = 0. Recall that ˆ f ( v ) = ρ ( f ( v )). Sinceˆ f ( ∞ ) = ∞ , we can assume v ∈ V ′ and since v ∈ S + V ′ we can further assumethat || v || ≥ R . From (2.5) we see that ρ ( f ( v )) = 0 if and only if pf ( v ) = 0 and || p ⊥ f ( v ) || <
1, hence || f ( v ) || <
1. But from (2.3), we have that || f ( v ) || ≥
1, so nosuch v can exist. (cid:3) Let us define ν = inf v ∈ V ′ , || v || = R || h t ( v ) || . By Lemma 2.31 and compactness of { v ∈ V ′ | || v || = R } (since V ′ is finite dimen-sional), we have that ν >
0. Similarly by Lemma 2.32 and compactness of S + V ′ there exists some ν ′ > f sends S + V ′ to S W ′ \ D ν ′ ( W ′ ). Now at last wefix a choice of δ > δ < inf { ǫ/ , ν/ , ν ′ } . Lemma 2.33.
Let t ∈ [0 , and v ∈ S R ( V ′ ) . Then || f t ( v ) − η || > δ and || g t ( v ) − η || > δ .Proof. From Lemma 2.31 we have that || f t ( v ) || > ǫ , hence || f t ( v ) − η || ≥ || f t ( v ) || − || η || > ǫ − δ > ǫ/ > δ. Similarly, || h t ( v ) || ≥ ν by the definition of ν and hence || h t ( v ) − η || ≥ || h t ( v ) || − || η || ≥ ν − δ > ν/ > δ. (cid:3) Let B ∞ ⊆ S W ′ denote the section at infinity. There exists a fibrewise defor-mation retraction S W ′ \ D δ ( W ′ ) → B ∞ , moreover we can choose the retractinghomotopy to be S -equivariant and smooth. It follows that if ϕ : D R ( V ′ ) → W ′ isany smooth S -equivariant map such that ϕ | S R ( V ′ ) takes values in W ′ \ D δ ( W ′ ),then ϕ admits a smooth S -equivariant extension ˜ ϕ : S V ′ → S W ′ with the property that if v ∈ S V ′ \ D R ( V ′ ), then ˜ ϕ ( v ) ∈ S W ′ \ D δ ( W ′ ). Therefore if η : B → W ′ is anysection with || η || < δ , it follows that ˜ ϕ − ( η ) = ϕ − ( η ). We apply this constructionto f , f , f and also the homotopies f t , h t to obtain ˜ f , ˜ f , ˜ f and homotopies ˜ f t from ˜ f to ˜ f and ˜ h t from ˜ f to ˜ f .Recall that η is transverse to f and that M = f − ( η ) is equivariantly diffeo-morphic to the original moduli space M . From the above remarks we have that˜ f is also transverse to η and that M = ˜ f − ( η ). Lemma 2.34.
There exists an S -equivariant map ˜ f ′ : S V ′ → S W ′ equivariantlyhomotopic to ˜ f such that ˜ f ′ is transverse to η . Moreover ˜ f ′ can be chosen tobe arbitrarily close to ˜ f and such that the preimage M = ( ˜ f ′ ) − ( η ) contains nofixed points of the S -action. In addition, M is equivariantly cobordant to M (andhence also to M ).Proof. Let τ : [0 , → [0 ,
1] be a smooth function such that τ ( t ) = 0 for 0 ≤ t ≤ / τ ( t ) = 1 for 2 / ≤ t ≤
1. Then the homotopies ˜ f τ ( t ) and ˜ h τ ( t ) can be joinedtogether smoothly to give a smooth homotopy k t from ˜ f to ˜ f with the propertythat if k t ( v ) ∈ D δ ( W ′ ), then || v || < R . We can equivariantly deform ˜ f to ˜ f ′ and equivariantly deform k t to a homotopy k ′ t from ˜ f to ˜ f ′ such that ˜ f , ˜ f ′ aretransverse to η and k ′ : [0 , × S V ′ → S W ′ is transverse to the constant path η : [0 , × B → W ′ . If we take our deformations to be sufficiently close tothe original maps, then M = ( ˜ f ′ ) − ( η ) and ( k ′ ) − ( η ) will be fixed point free.Moreover, ( k ′ ) − ( η ) defines an equivariant cobordism from M to M . (cid:3) Lemma 2.35.
There exists an equivariant homotopy of ˆ f to a map ˆ f ′ such that η is transverse to ˆ f ′ . Also ˆ f ′ can be chosen arbitrarily close to ˆ f and such that thereis an equivariant diffeomorphism M = ( ˆ f ′ ) − ( η ) .Proof. Let f ′ = ˜ f ′ | D R ( V ′ ) . Then f ′ is a deformation of f = ˆ f | D R ( V ′ ) . Let H :[0 , × D R ( V ′ ) → S W ′ be the homotopy from ˆ f D R ( V ′ ) to f ′ . Recall that theimage of ˜ f | S R ( V ′ ) = ˆ f | S R ( V ′ ) is disjoint from D δ ( W ′ ). We can therefore choose thedeformation ˜ f ′ and homotopy H such that the image of H | [0 , × S R ( V ′ ) is disjointfrom D δ ( W ′ ). It follows that there exists some ǫ ′ > H ( t, v ) / ∈ D δ ( W ′ )for all t ∈ [0 ,
1] and all v ∈ V ′ with R − ǫ ′ ≤ || v || ≤ R . Now define K : [0 , × S V ′ → S W ′ as follows. As in the previous lemma, let τ : [0 , → [0 ,
1] be a smooth functionsuch that τ ( t ) = 0 for 0 ≤ t ≤ / τ ( t ) = 1 for 2 / ≤ t ≤
1. Recall that wedefined S + V ′ = S V ′ \ { v ∈ V ′ | || v || < R } . Then we define K ( t, v ) = ˆ f ( v ) if v ∈ S + V ′ H ( tτ ( R −|| v || ǫ ′ ) , v ) if R − ǫ ′ ≤ || v || ≤ RH ( t, v ) if || v || ≤ R − ǫ ′ . Then K is a smooth equivariant homotopy from ˆ f to ˆ f ′ , where we set ˆ f ′ ( v ) = K (1 , v ). Since we can choose ˜ f ′ can be chosen arbitrarily close to ˜ f and such thatthe homotopy H also remains arbitrarily close to ˜ f at all times, it follows that K ( t, v ) can be made arbitrarily close to ˆ f for all times, in particular ˆ f ′ can bemade arbitrarily close to ˆ f . Next we claim that ˆ f ′ is transverse to η . Recall that || η || < δ . From Lemma 2.32, we see that if v ∈ S + V ′ then ˆ f ( v ) = ˆ f ′ ( v ) / ∈ D δ ( W ′ ), sothere are no solutions to ˆ f ′ ( v ) = η with v ∈ S + V ′ . Similarly, since H (1 , v ) / ∈ D δ ( W ′ ) N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 29 for all v ∈ V ′ with R − ǫ ′ ≤ || v || ≤ R , we see that there are no solutions to ˆ f ′ ( v ) = η with R − ǫ ′ ≤ || v || ≤ R as well. Next we note that ˆ f ′ | D R − ǫ ′ ( V ′ ) = f ′ | D R − ǫ ′ ( V ′ ) andthat there are no solutions to f ′ ( v ) = η with R − ǫ ′ ≤ || v || ≤ R , since H (1 , v ) = f ′ .Thus ˆ f ′ is transverse to η because f ′ is and ( ˆ f ′ ) − ( η ) = ( f ′ ) − ( η ) = M . (cid:3) Completion of proof of Theorem 2.24 : we have seen that the Seiberg-Witten invari-ants of the original monopole map are the Seiberg-Witten invariants of the triple( M , π , ψ ) and similarly the Seiberg-Witten invariants of ˆ f are those of the triple( M , π , ψ ), where M = ˆ M = ( ˆ f ′ ) − ( η ), π is the natural projection to B and ψ is the isomorphism given as in Lemma 2.16. We have previously defined thetriple ( M , π , ψ ) and shown an isomorphism ( M , π , ψ ) ∼ = ( M , π , ψ ). Lem-mas 2.34 and 2.35 show that there is a cobordism of pairs ( M , π ) ∼ ( M , π ).It just remains to check that that ψ , ψ extend over this cobordism. Recall that ψ is the isomorphism obtained by applying Lemma 2.16 to f and similarly ψ isthe isomorphism obtained by applying Lemma 2.16 to f ′ . Applying Lemma 2.16to the homotopy k ′ t | D R ( V ′ ) (where k ′ t is as in Lemma 2.34) joining f to f ′ , we getthe desired extension of ψ , ψ over the cobordism. So at last we have shown that( M , π , ψ ) and ( M , π , ψ ) are cobordant and thus have the same Seiberg-Witteninvariants.3. Cohomological formulation of the families Seiberg-Witteninvariants
In this section we will show how the families Seiberg-Witten invariants of a finitedimensional monopole map can be recovered from purely cohomological operations.This reformulation of the Seiberg-Witten invariants makes it easy to establish var-ious properties of them, such as the wall crossing formula or the computation oftheir Steenrod powers.3.1.
Equivariant cohomology computations.
Let B be a compact smoothmanifold as before. Let V → B be a complex vector bundle of rank a and U → B a real vector bundle of rank b . We make V into an S -equivariant vector bundlewhere S acts by scalar multiplication on the fibres of V . We make U into an S -equivariant vector bundle with the trivial S -action. Let S V,U denote the unitsphere bundle S V,U = S ( R ⊕ V ⊕ U ). Let S U denote S ,U , let q : S U → B be theprojection to B . Let P ( V ) be the projective bundle associated to V and note that q ∗ ( P ( V )) = P ( q ∗ V ).We will work with the S -equivariant cohomology of various spaces. Let C beequipped with the standard S -action by scalar multiplication. This defines anequivariant line bundle over a point and thus a class x = c ( C ) ∈ H S ( pt ; Z ).Then H ∗ S ( pt ; Z ) is isomorphic to Z [ x ], the ring of polynomials in x with integercoefficients. Equip B with the trivial S -action. Since we will be working withspaces that fibre (equivariantly) over B , the cohomology groups of interest willbe modules over the equivariant cohomology of B . We introduce the followingnotation: H ∗ = H ∗ S ( B ; Z ) = H ∗ ⊗ H ∗ ( B ; Z ) = H ∗ ( B ; Z )[ x ] . We will also need to work with local systems and with coefficient groups other than Z . Let A be a local system of abelian groups on B (equipped with trivial S -action) and define: H ∗ ( A ) = H ∗ S ( B ; A ) = H ∗ ⊗ H ∗ ( B ; A ) = H ∗ ( B ; A )[ x ] . We are mostly interested in local systems which arise as follows. Suppose w ∈ H ( B, Z ). Then w corresponds to a principal Z covering B w → B and we let Z w denote the local system B w × Z Z , where Z acts on Z as multiplication by ± S V,U = S ( R ⊕ V ⊕ U ). There is a global section B → S V,U given by(1 , , B V,U denote the image of this section.In the computations that follow we will first assume that U is oriented. The re-sults and proofs can easily be adapted to the case that U is non-orientable, providedwe use local coefficients. Proposition 3.1.
Suppose that U is orientable. Then H ∗ S ( S V,U , B
V,U ; Z ) is a freerank module over H ∗ with generator τ V,U ∈ H a + bS ( S V,U , B
V,U ; Z ) .Proof. This is the Thom isomorphism in equivariant cohomology. We briefly recallthe proof. Since G = S acts on V ⊕ U , we obtain a vector bundle ( V ⊕ U ) G over B × BG given by ( V ⊕ U ) × G EG → B × G EG = B × BG . Now we just applythe usual Thom isomorphism to ( V ⊕ U ) G . In particular we obtain an equivariantThom class τ V,U . This point of view also makes it clear that the choice of generator τ V,U corresponds to a choice of orientation of U . (cid:3) In the case that U is not orientable we instead have an equivariant Thom class τ V,U ∈ H a + bS ( S V,U , B
V,U ; Z w ( U ) ) and the Thom isomorphism continues to hold aslong as we use local coefficients.Recall that the equivariant Euler class e V,U ∈ H a + b ( Z w ( U ) ) of V ⊕ U is thepullback of the equivariant Thom class τ V,U under the zero section B → ( V ⊕ U )(viewed as a map of pairs ( B, ∅ ) → ( S V,U , B
V,U )). Clearly this factors as e V,U = e V e U , where e U is just the ordinary Euler class of U and e V is the equivariant Eulerclass of V . Using the splitting principle, one finds that e V is given by: e V = x a + c ( V ) x a − + c ( V ) x a − + · · · + c a ( V ) . Let b ≥ U be a real oriented orthogonal vector bundle. Suppose thatwe are given an oriented sub-bundle i W : W → U of positive codimension. Assumethat there exists a section φ : B → U which is disjoint from W . Equivalently,assume that the orthogonal complement W ⊥ of W in U admits a non-vanishingsection. Recall from [2] that pushforward maps can be constructed in equivariantcohomology. In particular we have that φ induces a pushforward map φ ∗ : H jS ( B ; Z ) → H j +2 a + bS ( S V,U , S W ; Z ) . The image of φ ∗ (1) under the natural map H a + bS ( S V,U , S W ; Z ) → H a + bS ( S V,U , B
V,U ; Z )is precisely the equivariant Thom class τ V,U . Thus φ ∗ (1) is a lift of τ V,U and hencewe sometimes denote it as φ ∗ (1) = e τ φV,U . Proposition 3.2.
Let U be a real oriented vector bundle of rank b ≥ and supposethat W ⊂ U is a proper subbundle of rank w which is also oriented. Suppose that U admits a section φ : B → U whose image is disjoint from W . Let V be a complexvector bundle of rank a . Then H ∗ S ( S V,U , S W ; Z ) is a free H ∗ -module of rank ,generated by e τ φV,U and δτ ,W , where e τ φV,U = φ ∗ (1) and δτ ,W is the image of τ ,W under the coboundary map δ : H wS ( S W , B W ; Z ) → H w +1 S ( S V,U , S W ; Z ) of the triple ( S V,U , S W , B W ) . N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 31
Proof.
Consider the long exact sequence in equivariant cohomology of the triple( S V,U , S W , B V,U ): · · · → H iS ( S V,U , S W ; Z ) i ∗ W −→ H iS ( S V,U , B
V,U ; Z ) ι ∗ −→ H iS ( S W , B V,U ; Z ) δ −→ · · · by our discussion above we have that i ∗ W ( e τ φV,U ) = τ V,U . It follows by the Thomisomorphism that i ∗ W is surjective and hence ι ∗ = 0. Therefore the above longexact sequence splits into short exact sequences:0 → H i − S ( S W , B V,U ; Z ) δ −→ H iS ( S V,U , S W ; Z ) i ∗ W −→ H iS ( S V,U , B
V,U ; Z ) → . But H iS ( S V,U , B
V,U ; Z ) is a free H ∗ -module generated by τ V,U . So the choice oflift e τ φV,U of τ V,U determines a splitting of the above short exact sequences. (cid:3)
Using local coefficients we can easily extend Proposition 3.2 to the case that U and W are not necessarily oriented.Next we will be interested in the relative equivariant cohomology of the pair( S V,U , S U ). In this case it is useful to replace S U with an equivariant tubularneighbourhood. Recall that we let q : S U → B denote the projection to B . Thenone sees that the normal bundle to S U in S V,U is given by q ∗ V , equipped withthe action of S by fibrewise scalar multiplication. Let e U be the unit open discbundle in q ∗ ( V ), which we can identify with an equivariant tubular neighbourhoodof S U . Let e Y = S V,U \ e U . We observe that e Y is a compact manifold with boundary ∂ e Y = S ( q ∗ V ), the unit sphere bundle of q ∗ V . We further observe that S actsfreely on e Y . Let Y = e Y /S be the quotient, which is a compact manifold withboundary ∂Y = ( ∂ e Y ) /S = S ( q ∗ V ) /S = P ( q ∗ V ) = q ∗ ( P ( V )). Note that e Y → Y is a principal circle bundle whose Chern class is just the image of x ∈ H in H ( Y ; Z ) under the natural map H ∗ → H ∗ S ( e Y ; Z ) ∼ = H ∗ ( Y ; Z ). Note further thatthe restriction of x to ∂Y is precisely the Chern class of the bundle O q ∗ V (1) → P ( q ∗ ( V )) (the dual of the tautological line bundle over P ( q ∗ ( V ))). Proposition 3.3.
Let U be an orientable real vector bundle. Then: • There is a naturally defined isomorphism H ∗ S ( S V,U , S U ; Z ) ∼ = H ∗ ( Y, ∂Y ; Z ) of H ∗ -modules. • H ∗ S ( S V,U , S U ; Z ) is a free module over H ∗ ( B ; Z ) with basis { ( δτ ,U ) x i } a − i =0 ,where δ : H i − S ( S V,U , S U ; Z ) → H ∗ S ( S V,U , S U ; Z ) is the coboundary map inthe long exact sequence of the triple ( S V,U , S U , B V,U ) .Proof. Using excision, we have isomorphisms: H ∗ S ( S V,U , S U ; Z ) ∼ = H ∗ S ( S V,U , ˜ U ; Z ) ∼ = H ∗ S ( e Y , ∂ e Y ; Z ) ∼ = H ∗ ( Y, ∂Y ; Z ) . Next, we observe that there is an isomorphism H ∗ S ( S V,U , S U ; Z ) ∼ = H ∗ S ( D ( V ⊕ U ) , S ( V ⊕ U ) ` D ( U ); Z ) . The long exact sequence of the triple ( D ( V ⊕ U ) , S ( V ⊕ U ) ` D ( U ) , S ( V ⊕ U )) hasthe form: · · · δ −→ H ∗ S ( D ( V ⊕ U ) , S ( V ⊕ U ) ` D ( U ); Z ) → H ∗ S ( D ( V ⊕ U ) , S ( V ⊕ U ); Z ) →→ H ∗ S ( S ( V ⊕ U ) ` D ( U ) , S ( V ⊕ U ); Z ) δ −→ · · · But H ∗ S ( S ( V ⊕ U ) ` D ( U ) , S ( V ⊕ U ); Z ) ∼ = H ∗ S ( S U , B U ; Z ) ∼ = H ∗ τ ,U and themap H ∗ τ V,U ∼ = H ∗ S ( D ( V ⊕ U ) , S ( V ⊕ U ); Z ) → H ∗ S ( S U , B U ; Z ) ∼ = H ∗ S ( D ( V ); Z ) ∼ = H ∗ , which corresponds to pullback under the zero section of V , is given by τ V,U e V τ ,U . In particular, this map is injective and the image is H ∗ e V τ ,U . Hence thecoboundary map induces isomorphisms δ : H i − / h e V i → H i + bS ( D ( V ) , S ( V ); Z ) , y y · δτ ,U . Recall that the equivariant Euler class e V has the form e V = x a + c ( V ) x a − + · · · + c a ( V )and it follows easily that H ∗ / h e V i is a free H ∗ ( B ; Z ) module generated by { x i } a − i =0 .Hence H ∗ S ( D ( V ) , S ( V ); Z ) is a free H ∗ ( B ; Z ) module generated by { δτ ,U x i } a − i =0 . (cid:3) As with other results in this subsection, Proposition 3.3 extends to the case that U is not orientable using local coefficients.Let π Y : Y → B denote the projection map from Y to B . One sees that Y hasthe structure of a locally trivial fibre bundle over B whose fibres are manifolds withboundary. An orientation of U determines a relative orientation of Y → B , that is,an orientation of T Y ⊕ π ∗ Y ( T B ). From this we obtain a pushforward map( π Y ) ∗ : H i ( Y, ∂Y ; Z ) → H i − l ( B ; Z ) , where l = 2 a + b − H i ( Y, ∂Y ; Z ) ( π Y ) ∗ / / ∼ = (cid:15) (cid:15) H i − l ( B ; Z ) ∼ = (cid:15) (cid:15) H dim ( B )+ l − i ( Y ; π ∗ Y Z B ) ( π Y ) ∗ / / H dim ( B )+ l − i ( B ; Z B )where the vertical arrows are given by Poincar´e-Lefschetz duality and Z B denotesthe orientation local system on B . Lemma 3.4.
Let Y be a compact manifold with boundary and let A be a local systemof abelian groups on Y . Let ι : ∂Y → Y denote the inclusion of the boundary.Let Z Y denote the orientation local system on Y . Then δ = ι ∗ , where δ is thecoboundary map δ : H i − ( ∂Y ; ι ∗ A ) → H i ( Y, ∂Y ; A ) in the long exact sequence ofthe pair ( Y, ∂Y ) and ι ∗ : H i − ( ∂Y ; ι ∗ A ) → H i ( Y, ∂Y ; A ) is the pushforward mapdefined by the commutative diagram: H i − ( ∂Y ; ι ∗ A ) ι ∗ / / ∼ = (cid:15) (cid:15) H i ( Y, ∂Y ; A ) ∼ = (cid:15) (cid:15) H dim ( Y ) − i ( ∂Y ; ι ∗ ( A ⊗ Z Y )) ι ∗ / / H dim ( Y ) − i ( Y ; A ⊗ Z Y ) where the vertical arrows are given by Poincar´e-Lefschetz duality. N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 33
Proof.
For a pair of spaces (
U, V ) and local system A , let C j ( U, V ; A ) and C j ( U, V ; A )denote the singular chain/cochain complexes. Let ω ∈ C j ( ∂Y ; ι ∗ A ) represent a class[ ω ] ∈ H j ( ∂Y ; ι ∗ A ). Choose an extension e ω ∈ C j ( Y ; A ) of ω to Y . Then δ [ ω ] = [ δ e ω ].Next let [ Y ] ∈ H n ( Y ; ∂Y ; Z Y ) denote the fundamental class of Y , where n = dim ( Y ). Taking cap product with [ Y ] gives maps:[ Y ] ∩ : H j ( Y ; A ) → H n − j ( Y, ∂X ; A ⊗ Z Y ) , [ Y ] ∩ : H j ( Y, ∂Y ; A ) → H n − j ( Y ; A ⊗ Z Y ) , which are the isomorphisms defining Poincar´e-Lefschetz duality. Moreover, thefundamental classes of Y and ∂Y are related by ∂ [ Y ] = [ ∂Y ] , where ∂ : H n ( Y, ∂Y ; Z Y ) → H n − ( ∂Y ; ι ∗ Z Y ) is the boundary map in the long exactsequence of the pair ( Y, ∂Y ). Let [ e Y ] , [ f ∂Y ] be representatives of [ Y ] , [ ∂Y ] at thechain level. Then we have: ∂ [ e Y ] = [ f ∂Y ] + exact term . The pushforward map ι ∗ is characterised at the chain level by[ e Y ] ∩ ι ∗ ω = ι ∗ ([ f ∂Y ] ∩ ω ) + exact term . However, we have: ι ∗ ([ f ∂Y ] ∩ ω ) = ι ∗ ([ f ∂Y ] ∩ ι ∗ e ω ) (since ι ∗ e ω = ω )= ι ∗ [ f ∂Y ] ∩ e ω = ∂ [ e Y ] ∩ e ω + exact term (since ∂ [ Y ] = [ ∂Y ])= [ e Y ] ∩ δ e ω + exact term (using the Leibniz rule for cap products) . So at the level of (co)homology classes [ Y ] ∩ ι ∗ ω = [ Y ] ∩ δ [ ω ] and thus ι ∗ ω = δω asclaimed. (cid:3) Proposition 3.5.
Suppose U is orientable and fix an orientation. The fibre inte-gration map ( π Y ) ∗ : H i ( Y, ∂Y ; Z ) → H i − l ( B ; Z ) is a morphism of H ∗ ( B ; Z ) -modules and is given by: ( π Y ) ∗ ( δτ ,U x j ) = ( if j < a − ,s j − ( a − ( V ) if j ≥ a − , where s j ( V ) ∈ H j ( B ; Z ) is the j -th Segre class of V [9, § , which is characterisedby the property that if we let s ( V ) = s ( V ) + s ( V ) + s ( V ) + · · · ∈ H ev ( B ; Z ) denote the total Segre class of V and similarly c ( V ) = 1 + c ( V ) + c ( V ) + · · · thetotal Chern class, then c ( V ) s ( V ) = 1 . In the case that U is not orientable, the analogous result holds using local coeffi-cients. Proof.
Clearly ( π Y ) ∗ is a morphisms of H ∗ ( B ; Z )-modules. Let ι : ∂Y → Y denotethe inclusion and let π ∂Y : ∂Y → B be the projection to B . Note that π ∂Y = π Y ◦ ι .It follows that ( π ∂Y ) ∗ = ( π Y ) ∗ ◦ ι ∗ and from Lemma 3.4, we have that ι ∗ = δ , where δ : H i − ( ∂Y ; Z ) → H i ( Y, ∂Y ; Z ) is the coboundary map of the long exact sequenceof the pair ( Y, ∂Y ). Therefore( π Y ) ∗ ( δτ ,U x j ) = ( π Y ) ∗ ◦ δ ( τ ,U x j ) = ( π ∂Y ) ∗ ( τ ,U x j ) . Next, we recall that ∂Y = q ∗ ( P ( V )), where q is the projection q : S U → B . Notethat τ ,U is precisely the Thom class of the bundle U → B and it follows that( π ∂Y ) ∗ ( τ ,U x j ) = ( π P ( V ) ) ∗ ( x j ) , where π P ( V ) is the projection π P ( V ) : P ( V ) → B . If j < a − x j is less than the fibre dimension of P ( V ), so we obviously have ( π P ( V ) ) ∗ ( x j ) = 0 inthis case. So it remains to prove the identity(3.1) ( π P ( V ) ) ∗ ( x j +( a − ) = s j ( V ) , for all j ≥ j . The case j = 0 is obvious since s ( V ) = 1 and x a − restricts to a generator of H a − ( CP a − ; Z ) on each fibre of P ( V ). Now recallthat on P ( V ) the class x satisfies the equation x a + c ( V ) x a − + · · · + c a ( V ) = 0(this is the Grothendieck approach to defining the Chern classes of a complex vectorbundle V ). Now let j ≥ j ′ ≤ j .Then (letting s k ( V ) = 0 if k <
0) we have:( π P ( V ) ) ∗ ( x ( j +1)+( a − ) = ( π P ( V ) ) ∗ ( x j + a )= ( π P ( V ) ) ∗ ( x j ( − c ( V ) x a − − c ( V ) a − x − · · · − c a ( V ))= − ( c ( V ) s j ( V ) − c ( V ) s j − ( V ) − · · · − c a ( V ) s j +1 − a )= s j +1 ( V ) , where we used the relation s j +1 ( V ) c ( V ) + s j ( V ) c ( V ) + · · · + s j +1 − a ( V ) c a ( V ) = 0 for all j ≥ , which follows by taking the degree 2 j + 2 part of the identity c ( V ) s ( V ) = 1. (cid:3) Cohomological formula for the families Seiberg-Witten invariants.
Let f : S V,U → S V ′ ,U ′ be a finite dimensional monopole map. Then as in Subsection2.2 we may assume that U ′ = U ⊕ H + and that f | U is the inclusion U → U ′ . Forconvenience, we will assume throughout this section that U, U ′ are oriented. Asusual, the results easily extend to the general case provided we use local coefficients.As before we assume f preserves sections at infinity: f : ( S V,U , B
V,U ) → ( S V ′ ,U ′ , B V ′ ,U ′ ) . Let [ φ ] ∈ CH ( f ) be a chamber and let φ : B → U ′ − U be a representative section.Since φ avoids U , we obtain a pushforward map φ ∗ : H jS ( B ; Z ) → H j +2 a ′ + b ′ S ( S V ′ ,U ′ , S U ; Z )and we also have a pullback map f ∗ : H jS ( S V ′ ,U ′ , S U ; Z ) → H jS ( S V,U , S U ; Z ) ∼ = H j ( Y, ∂Y ; Z ) N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 35
Theorem 3.6.
For each m ≥ we have SW m ( f, φ ) = ( π Y ) ∗ ( x m ` f ∗ φ ∗ (1)) . The proof will be given later in this subsection. But first we make some obser-vations about this formula: • This formula shows that SW m ( f, φ ) depends only of f and φ and not on achoice of perturbation f ′ of f meeting φ transversally. • This formula also shows that SW m ( f, φ ) depends only on the homotopyclasses of f and φ , in particular SW m ( f, φ ) depends only on φ only throughthe chamber [ φ ] in which φ lies. • We will show in Proposition 3.8 that SW m ( f, φ ) depends only on the stablehomotopy class of f . More precisely SW m ( f, φ ) is invariant under stabili-sation of f by real or complex vector bundles.Let e τ φV ′ ,U ′ = φ ∗ (1) ∈ H a ′ + b ′ ( S V ′ ,U ′ , S U ; Z ). Observe that the image of e τ φV ′ ,U ′ under the map H a ′ + b ′ ( S V ′ ,U ′ , S U ; Z ) → H a ′ + b ′ ( S V ′ ,U ′ , B V ′ ,U ′ ; Z ) is τ V ′ ,U ′ , theequivariant Thom class. This is simply because φ is a section of V ′ ⊕ U ′ , so theinduced pushforward map φ ∗ : H jS ( B ; Z ) → H j +2 a ′ + b ′ S ( S V ′ ,U ′ , B V ′ ,U ′ ; Z ) sends 1to the equivariant Thom class.Next, we note that f ∗ ( e τ φV ′ ,U ′ ) ∈ H a ′ + b ′ S ( S V,U , S U ; Z ) can be written uniquely as(3.2) f ∗ ( e τ φV ′ ,U ′ ) = a − X j =0 µ j ( f, φ ) δτ ,U x ( a − − j , where µ j ( f, φ ) ∈ H j − (2 d − b + − ( B ; Z ). Note that since H k ( B ; Z ) = 0 for k >dim ( B ), we have that µ j = 0 whenever 2 j − (2 d − b + − > dim ( B ), that is, µ j = 0whenever 2 j ≥ d − ( b + − dim ( B )).Let f ′ be a deformation of f such that f ′ meets φ transversally. Then f M φ = f ′− ( φ ( B )) is a compact embedded submanifold of S V,U without boundary. Recallthat we define e Y to be the complement in S V,U of an equivariant tubular neigh-bourhood of S U and that e Y is a compact manifold with boundary. Choosing theneighbourhood sufficiently small, we may assume that f M φ is an embedded sub-manifold of e Y and that f M φ ∩ ∂ e Y = ∅ . Noting that S acts freely on f M φ , we let M φ = f M φ /S , which is an embedded submanifold of Y = e Y /S . Let L → M φ bethe line bundle associated to the principal circle bundle f M φ → M φ . Then clearly c ( L ) = x | M φ . Note that the dimension of M φ is given by dim ( M φ ) = (2 a + b ) − (2 a ′ + b ′ ) − d − b + − . Recall that a choice of orientation of U defines a relative orientation of S V,U → B ,hence also a relative orientation of π Y : Y → B . Moreover a choice of orientationon U ′ defines an orientation of the normal bundle of M φ ⊆ Y . Let π M φ : M φ → B be the restriction of π Y to M φ . We see that an orientation of U ⊕ U ′ (equivalently,an orientation of H + ) determines a relative orientation of π M φ : M φ → B andhence a push-forward map ( π M φ ) ∗ : H j ( B ; Z ) → H j − (2 d − b + − ( B ; Z ). Recall thatfor each m ≥
0, we have defined the Seiberg-Witten invariants of ( f, φ ) to be givenby: SW m ( f, φ ) = ( π M φ ) ∗ ( c ( L ) m ) ∈ H m − (2 d − b + − ( B ; Z ) . Just as we saw for the µ j ( f, φ ), since H k ( B ; Z ) = 0 for k > dim ( B ), we have that SW m ( f, φ ) = 0 whenever 2 m ≥ d − ( b + − dim ( B )). Proposition 3.7.
The SW m ( f, φ ) are related to the classes µ j ( f, φ ) ∈ H j − (2 d − b + − ( B ; Z ) by: (3.3) X m ≥ SW m ( f, φ ) t m = a − X j =0 µ j ( f, φ ) t j X n ≥ s n ( V ) t n where t is a formal variable and s n ( V ) is the n − th Segre class of V . This rela-tion can be inverted to express the µ j ( f, φ ) invariants in terms of the SW m ( f, φ ) invariants: (3.4) a − X j =0 µ j ( f, φ ) t j = X m ≥ SW m ( f, φ ) t m X n ≥ c n ( V ) t n , where c n ( V ) is the n -th Chern class of V .Proof. By Theorem 3.6 we have SW m ( f, φ ) = ( π Y ) ∗ ( x m f ∗ φ ∗ (1)) . Recall that we set e τ φV ′ ,U ′ = φ ∗ (1). Then combined with Equation (3.2) and usingProposition 3.5, we get: SW m ( f, φ ) = ( π Y ) ∗ a − X j =0 µ a − − j ( f, φ ) δτ ,U x j + m = a − X j =0 µ a − − j ( f, φ )( π Y ) ∗ ( δτ ,U x j + m )= a − X j =0 µ a − − j ( f, φ ) s j + m − ( a − ( V )= X j + n = m µ j ( f, φ ) s n ( V ) , which proves Equation (3.3). Equation (3.4) follows from Equation (3.3) and theidentity ( c ( V ) + tc ( V ) + · · · )( s ( V ) + ts ( V ) + ts ( V ) + · · · ) = 1 . (cid:3) Proof of Theorem 3.6.
Let Z B denote the orientation local system on B . Fixchoices of orientations on U and U ′ . This gives an isomorphism of the orien-tation local system of Y with ( π Y ) ∗ Z B and an isomorphism of the orientationlocal system of M φ with ( π M φ ) ∗ Z B . The submanifold M φ ⊂ Y has a funda-mental class [ M φ ] ∈ H d − b + − ( Y ; ( π Y ) ∗ Z B ). Let η M φ denote the Poincar´e dualclass η M φ ∈ H dim ( X ) − (2 d − b + − ( Y, ∂Y ; Z ). Let ι : M φ → Y denote the inclusion.We obtain a pushforward map ι ∗ : H j ( M φ ; Z ) → H j +(2 d − b + − ( X, ∂X ; Z ). From N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 37 general properties of the pushforward map we have that if θ ∈ H j ( Y ; Z ), then ι ∗ ( ι ∗ θ ) = θ ` η M φ . Therefore(3.5) SW m ( f, φ ) = ( π M φ ) ∗ ( c ( L ) m ) = ( π Y ) ∗ ◦ ι ∗ ( ι ∗ ( x m )) = ( π Y ) ∗ ( x m ` η M φ ) . To complete the proof, we just need to show that η M φ = f ∗ ( φ ∗ (1)). We begin byobserving that the normal bundle N f M φ to f M φ in e Y is equivariantly identified viathe derivative of f ′ with the normal bundle of B φ = φ ( B ) ⊆ S V ′ ,U ′ . But since φ is a section, the normal bundle of B φ can be identified with the restriction of thevertical tangent bundle of S V ′ ,U ′ → B to B φ , which is exactly V ′ ⊕ U ′ . Thus N f M φ = f ′∗ ( V ′ ⊕ U ′ ) ∼ = f ∗ ( V ′ ⊕ U ′ ) . Since S acts freely on M φ with quotient M φ , we have that N f M φ descends to avector bundle N M φ on M φ , which is precisely the normal bundle of M φ in Y . Let U denote a tubular neighbourhood of M φ in Y , which we identify with the openunit disc bundle in N M φ . The inclusion ι : M φ → Y factors as: M φ ζ −→ U j U −→ Y, where ζ is the zero section of N M φ and j U is the inclusion map, which is an openimmersion. Passing to cohomology, we obtain a factorisation of ι ∗ as: H j ( M φ ; Z ) ζ ∗ −→ H j +(2 d − b + − c ( U ; Z ) ( j U ) ∗ −→ H j +(2 d − b + − ( Y, ∂Y ; Z ) , where H j +(2 d − b + − c ( U ; Z ) denotes compactly supported cohomology of U of degree j + (2 d − b + − ζ ∗ : H j ( M φ ; Z ) → H j +(2 d − b + − c ( U ; Z ) is theThom isomorphism for N M φ ∼ = U . Next we note that η M φ = ι ∗ (1) = ( j U ) ∗ ◦ ζ ∗ (1) = ( j U ) ∗ ( τ N M φ ) , where τ N M φ = ζ ∗ (1) ∈ H a ′ + b ′ c ( U ; Z ) ∼ = H a ′ + b ′ ( U + , ∞ ; Z ) is the Thom class of N M φ . Instead of computing ( j U ) ∗ ( τ N M φ ) directly we will work S -equivariantly on e Y . The inclusion e ι : f M → e Y factors as: f M e ζ −→ e U f j U −→ e Y , where e U is the open disc bundle in N g M φ .Note that H a ′ + b ′ S ( e U + , ∞ ; Z ) is isomorphic to H a ′ + b ′ ( U + , ∞ ; Z ). To see this,let O be an open neighbourhood of ∞ in U + which strongly deformation retractsto ∞ and let e O be the pre-image of O in e U + . Note that S acts freely on e O \ ∞ .Then we have a sequence of isomorphisms: H ∗ S ( e U + , ∞ ; Z ) ∼ = H ∗ S ( e U + , e O ; Z ) (by deformation retraction) ∼ = H ∗ S ( e U , e O \ ∞ ; Z ) (excision) ∼ = H ∗ ( U, O \ ∞ ; Z ) ( S acts freely) ∼ = H ∗ ( U + , O ; Z ) (excision) ∼ = H ∗ ( U + , ∞ ; Z ) (by deformation retraction) . Henceforth we use this sequence of isomorphisms to identity H a ′ + b ′ S ( e U + , ∞ ; Z )with H a ′ + b ′ ( U + , ∞ ; Z ). Let τ N g M φ ∈ H a ′ + b ′ S ( e U + , ∞ ; Z ) denote the equivari-ant Thom class of N g M φ . Then τ N g M φ agrees with τ N M φ under the isomorphism H a ′ + b ′ S ( e U + , ∞ ; Z ) ∼ = H a ′ + b ′ ( U + , ∞ ; Z ) constructed above. Let ( f j U ) ∗ : H ∗ S ( e U + , ∞ ; Z ) → H ∗ S ( e Y , ∂ e Y ; Z ) be the pushforward with respect to f j U . Then we have a commutativediagram: H ∗ S ( e U + , ∞ ; Z ) ∼ = (cid:15) (cid:15) ( f j U ) ∗ / / H ∗ S ( e Y , ∂ e Y ; Z ) H ∗ ( U + , ∞ ; Z ) ∼ = (cid:15) (cid:15) H ∗ ( Y, ∂Y ; Z ) ∼ = O O H ∗ c ( U ; Z ) ( j U ) ∗ ♠♠♠♠♠♠♠♠♠♠♠♠♠ Under the isomorphism H ∗ c ( U ; Z ) ∼ = H ∗ ( U + , ∞ ; Z ), the pushforward map ( j U ) ∗ : H ∗ ( U + , ∞ ; Z ) → H ∗ ( Y, ∂Y ; Z ) may be constructed as follows: let q : Y → U + bethe map given by q ( x ) = x if x ∈ U and q ( x ) = ∞ otherwise. Noting that q sends ∂Y to ∞ we have that q gives a map of pairs q : ( Y, ∂Y ) → ( U + , ∞ ) and we have( j U ) ∗ = q ∗ . Similarly define e q : e Y → e U + by e q ( x ) = x if x ∈ e U and e q ( x ) = ∞ otherwise. Tracing through the sequence of isomorphisms defining ( f j U ) ∗ , one mayverify that we likewise have ( f j U ) ∗ = e q ∗ .Our goal now will be to compute e q ∗ ( τ N g M φ ) = ( f j U ) ∗ ( τ N g M φ ) ∈ H a ′ + b ′ S ( e Y , ∂ e Y ; Z ),which of course coincides with ( j U ) ∗ ( τ N M ) = η M under the isomorphism H a ′ + b ′ S ( e Y , ∂ e Y ; Z ) ∼ = H a ′ + b ′ ( Y, ∂Y ; Z ). Recall that f ′ is transverse to B φ , andthe points of B φ are fixed points of the S -action and disjoint from S U . Since f ′ is transverse to B φ , we can find an S -equivariant open tubular neighbourhood W φ ⊆ S U ′ ,V ′ of B φ such that each point of W φ is a regular value of f ′ . Note alsothat W φ can be identified with the open unit disc bundle in the normal bundleof B φ in S U ′ ,V ′ , which is isomorphic U ′ ⊕ V ′ . Then (possibly after shrinking W φ and e U ), we can assume that e U = f ′− ( W φ ) and that f ′ | e U → W φ is a surjectivesubmersion. By choosing W φ small enough, we may assume that W φ is disjointfrom S U . Let j W φ : W φ → S U ′ ,V ′ denote the inclusion map. Consider the followingcommutative diagram of S -equivariant maps: e Y f ′ | e Y / / S U ′ ,V ′ e U f j U O O f ′ | e U / / W φj Wφ O O Let τ φU ′ ,V ′ ∈ H a ′ + b ′ S ( W + φ , ∞ ; Z ) be the equivariant Thom class. Then τ N g M φ =( f ′ | e U ) ∗ ( τ φU ′ ,V ′ ). So under the isomorphism H a ′ + b ′ ( Y, ∂Y ; Z ) ∼ = H a ′ + b ′ S ( e Y , ∂ e Y ; Z ), η M φ is given by:(3.6) η M φ = e q ∗ ( f ′ | e U ) ∗ ( τ φU ′ ,V ′ ) . N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 39
Let q W φ : S U ′ ,V ′ → W + φ be the map given by q W φ ( x ) = x if x ∈ W φ and q W φ ( x ) = ∞ otherwise. Then we have a commutative diagram of pairs:( e Y , ∂ e Y ) f ′ | e Y / / e q (cid:15) (cid:15) ( S U ′ ,V ′ , S U ) q Wφ (cid:15) (cid:15) ( e U + , ∞ ) f ′ | e U / / ( W + φ , ∞ )From commutativity of the diagram and Equation (3.6), we get: η M φ = ( f ′ | e Y ) ∗ q ∗ W φ ( τ φU ′ ,V ′ ) . But note that q ∗ W ( τ φU ′ ,V ′ ) is exactly the class e τ φV ′ ,U ′ = φ ∗ (1). Thus η M φ = f ′∗ ( φ ∗ (1)) = f ∗ ( φ ∗ (1)) . Substituting into Equation (3.5), we get the desired equality: SW m ( f, φ ) = ( π Y ) ∗ ( x m ` η M φ ) = ( π Y ) ∗ ( x m ` f ∗ φ ∗ (1)) . (cid:3) Proposition 3.8.
The Seiberg-Witten invariants SW m ( f, φ ) are independent ofstabilisation of f by real or complex vector bundles.Proof. We will focus on the case of stabilisation by a complex vector bundle, thereal case being considerably simpler. Let f : S V,U → S V ′ ,U ′ and a chamber φ : B → ( U ′ \ U ) be given. Let A be a complex vector bundle of rank α , set V A = V ⊕ A , V ′ = V ′ ⊕ A and let f A : V A ⊕ U → V ′ A ⊕ U denote the suspension of f . Observethat S V ′ A ,U ′ is the fibrewise smash product S V ′ A ,U ′ = S V ′ ,U ′ ∧ B S A, (equivariantly)and hence we have an external cup product ` : H iS ( S V ′ ,U ′ , S U ; Z ) × H jS ( S A, , B A, ; Z ) → H i + jS ( S V ′ A ,U ′ , S U ; Z ) . Using this cup product we see that e τ φV ′ A ,U ′ = e τ φV ′ ,U ′ ` τ A, and hence(3.7) f ∗ A ( e τ φV ′ A ,U ′ ) = f ∗ ( e τ φV ′ ,U ′ ) ` τ A, . Recall from Proposition 3.3 that H ∗ S ( S V,U , S U ; Z ) ∼ = H ∗ ( Y, ∂Y ; Z ) is a free moduleover H ∗ ( B ; Z ) with basis { ( δτ ,U ) x i } a − i =0 . Moreover, the proof of Proposition 3.3shows that as H ∗ S ( B ; Z )-modules we have an isomorphism H ∗ S ( S V,U , S U ; Z ) ∼ = (cid:18) Z [ x ] h e V i (cid:19) δτ ,U where e V ∈ H aS ( B ; Z ) is the equivariant Euler class of V . Similarly we have anisomorphism H ∗ S ( S V A ,U , S U ; Z ) ∼ = (cid:18) Z [ x ] h e V e A i (cid:19) δτ ,U where e A ∈ H aS ( B ; Z ) is the equivariant Euler class of A . Noting that S V A ,U = S V,U ∧ S A, we see that the external cup product with τ A, defines a morphism of H ∗ S ( B ; Z )-modules: ` τ A, : H ∗ S ( S V,U , S U ; Z ) → H ∗ + αS ( S V A ,U , S U ; Z ) . Tracing through the isomorphisms used in the proof of Proposition 3.3, we see that ` τ A, corresponds to the morphism (cid:18) Z [ x ] h e V i (cid:19) δτ ,U → (cid:18) Z [ x ] h e V e A i (cid:19) δτ ,U , u u · e A given by multiplication by e A . Let η = f ∗ ( e τ φV ′ ,U ′ ) ∈ H a ′ + b ′ S ( S V,U , S U ; Z ) so that SW m ( f, φ ) = ( π Y ) ∗ ( x m ` η ), where Y is defined as in Subsection 3.1. Similarly de-fine η A = f ∗ A ( e τ φV ′ A ,U ′ ) ∈ H a ′ + b ′ S ( S V A ,U , S U ; Z ) so that SW m ( f A , φ ) = ( π Y A ) ∗ ( x m ` η A ), where Y A is defined in the same way as Y , but with V A replacing V . Thenfrom Equation (3.7), we have that: SW m ( f A , φ ) = ( π Y A ) ∗ ( x m ` η ` e A ) . Therefore, to show that SW m ( f A , φ ) = SW m ( f, φ ) it suffices to show that:(3.8) ( π Y A ) ∗ ( u ` e A ) = ( π Y ) ∗ ( u )for any u ∈ H ∗ S ( S V,U , S U ; Z ). But since H ∗ S ( S V,U , S U ; Z ) ∼ = (cid:16) Z [ x ] h e V i (cid:17) δτ ,U , it suf-fices to check Equation (3.8) for u = x i δτ ,U , i = 0 , , . . . , a −
1. From Proposition3.5, we see that ( π Y ) ∗ ( x i δτ ,U ) is zero for 0 ≤ i ≤ a − i = a −
1. Onthe other hand, since the equivariant Euler class of A has the form e A = x α + x α − c ( A ) + · · · + c α ( A ) , we see again from Proposition 3.5 that( π Y A ) ∗ ( x i e A δτ ,U ) = ( π Y A ) ∗ ( x i + α + x i + α − c ( A ) + · · · + x i c α ( A ))is zero for 0 ≤ i ≤ a − i = a −
1. This completes the proof. (cid:3) Steenrod operations on the families Seiberg-Witten invariants
The Steenrod operations are stable cohomology operations. The stability of theseoperations makes them convenient to use in the study of families Seiberg-Witteninvariants, which are extracted from a stable homotopy class. For simplicity, letus consider the Steenrod squares. The Steenrod reduced power operations for oddprimes can also be considered, however the algebra turns out to be considerablymore difficult and so we will not purse the problem of calculating them here.4.1.
Steenrod squares computation.
Our calculations will be carried out using S -equivariant cohomology with Z -coefficients. The Borel model for equivariantcohomology allows us to extend the Steenrod operations to the equivariant setting.We have that H ∗ S ( pt ; Z ) ∼ = Z [ x ], where x has degree 2. It follows easily that Sq i ( x m ) = 0 whenever i is odd and Sq j ( x m ) = (cid:0) mj (cid:1) x m + j .Let f : ( S V,U , B
V,U ) → ( S V ′ ,U ′ , B V ′ ,U ′ ) be a finite dimensional monopole map.It turns out that for the purposes of computing Steenrod operations it is mostconvenient to stabilise f such that the bundles V, U are trivial, say V = C a , U = R b .Recall that U ′ = U ⊕ H + , hence in this case we have U ′ = R b ⊕ H + . In particular,this gives an equality of Stiefel-Whitney classes: w j ( U ′ ) = w j ( H + ) , for all j ≥ . Next recall that V − V ′ = D , hence V ′ = V − D = C a − D . From this we obtain: N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 41
Lemma 4.1.
Let c S ,j ( V ′ ) ∈ H j denote the j -th equivariant Chern class of V ′ .Then (4.1) c S ,j ( V ′ ) = j X l =0 x j − l (cid:18) a ′ − lj − l (cid:19) s l ( D ) . Remark . Note that since V ′ = C a − D , it follows that the Chern classes of V ′ are the Segre classes of the virtual bundle D . So s l ( D ) = 0 whenever l > a ′ , since V ′ has rank a ′ . Hence the non-zero terms in Equation (4.1) only involve binomialcoefficients whose upper index is non-negative. Proof.
We use the splitting principal to compute c S ,j ( V ′ ). Suppose that (non-equivariantly) the Chern roots of V ′ are y , . . . , y a ′ . Then the equivariant Chernroots of V ′ are ( x + y ) , . . . , ( x + y a ′ ) and hence c S ,j ( V ′ ) = X i < ···
Let η ∈ H ∗ ( P ( V ); Z ) be defined as in Equation (4.2). Then for all j ≥ , we have: Sq j ( η ) = j X l =0 c S ,l ( V ′ ) w j − l ( H + ) η,Sq j +1 ( η ) = j X l =0 c S ,l ( V ′ ) w j − l +1 ( H + ) η. Proof.
Recall that if τ E is the Thom class of a vector bundle E , then Sq j ( τ E ) = w j ( E ) τ E . Let t be a formal variable and set: Sq t = Sq + tSq + · · · w t ( E ) = w ( E ) + tw ( E ) + · · · so that Sq t ( τ E ) = w t ( E ) τ E . Applying this to our setting (and using the fact thatSteenrod operations commute with the coboundary map in the long exact sequenceof a pair), we obtain: Sq t ( δτ ,U ) = w t ( U ) δτ ,U , Sq t ( τ V ′ ,U ′ ) = w S ,t ( V ′ ⊕ U ′ ) τ V ′ ,U ′ where w S ,t = w S , + tw S , + · · · and w S ,j is the j -th equivariant Stiefel-Whitneyclass.Recall that e τ φV ′ ,U ′ = φ ∗ (1). To understand the action of Sq t on e τ φV ′ ,U ′ we factor φ ∗ as follows. Let B φ = φ ( B ) denote the image of B in S V ′ ,U ′ . Let W φ denote anequivariant tubular neighbourhood of B φ , which we can identify with the normalbundle of B φ ⊆ S V ′ ,U ′ . Let q : ( S V ′ ,U ′ , S U ) → ( W + φ , ∞ ) be the collapsing mapwhich acts as the identity on W φ and collapses the complement of W φ to ∞ . LetΦ : H ∗ S ( B ; Z ) → H ∗ +2 a ′ + b ′ S ( W + φ , ∞ ; Z ) be the Thom isomorphism. Then φ ∗ = q ∗ ◦ Φ, essentially by definition of the pushforward map of a submanifold. It followsthat e τ φV ′ ,U ′ = φ ∗ (1) = q ∗ (Φ(1)) = q ∗ ( τ W φ ), where τ W φ is the equivariant Thomclass of the normal bundle of B φ . But since φ is a section, the normal bundle of B φ can be identified with the restriction of the vertical tangent bundle of S V ′ ,U ′ → B to B φ , which is exactly V ′ ⊕ U ′ . Thus Sq t ( τ W φ ) = w S ,t ( V ′ ⊕ U ′ ) τ W φ and using e τ φV ′ ,U ′ = q ∗ ( τ W φ ), we get: Sq t ( e τ φV ′ ,U ′ ) = w S ,t ( V ′ ⊕ U ′ ) e τ φV ′ ,U ′ . Now by the definition of η , we have η · δτ ,U = f ∗ ( e τ φV ′ ,U ′ ) and applying Sq t we have: Sq t ( η ) w t ( U ) · δτ ,U = Sq t ( η ) Sq t ( δτ ,U )= Sq t ( η · δτ ,U )= Sq t ( f ∗ ( e τ φV ′ ,U ′ ))= f ∗ ( Sq t ( e τ φV ′ ,U ′ ))= w S ,t ( V ′ ⊕ U ′ ) f ∗ ( e τ φV ′ ,U ′ )= w S ,t ( V ′ ⊕ U ′ ) η · δτ ,U . N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 43
Recalling that U is taken to be trivial, we have w t ( U ) = 1 and hence Sq t ( η ) = w S ,t ( V ′ ⊕ U ′ ) η. In particular for any i ≥
0, we have: Sq i ( η ) = w S ,i ( V ′ ⊕ U ′ ) η. The proof of the lemma is completed by noting that: w S , j ( V ′ ⊕ U ′ ) = j X l =0 c S ,l ( V ′ ) w j − l ( H + ) ,w S , j +1 ( V ′ ⊕ U ′ ) = j X l =0 c S ,l ( V ′ ) w j − l +1 ( H + ) . (cid:3) Combining Lemmas 4.1 and 4.3, we obtain: Sq j ( η ) = j X l =0 c S ,l ( V ′ ) w j − l ( H + ) η = j X l =0 l X k =0 x l − k (cid:18) a ′ − kl − k (cid:19) s k ( D ) w j − l ( H + ) η = j X l =0 x l j − l X k =0 (cid:18) a ′ − kl (cid:19) s k ( D ) w j − l − k ( H + ) η. Similarly, Sq j +1 ( η ) = j X l =0 x l j − l X k =0 (cid:18) a ′ − kl (cid:19) s k ( D ) w j − l − k +1 ( H + ) η. Then using Equation (4.2) to substitute for η , we find: Sq j ( η ) = a − X m =0 j X l =0 j − l X k =0 (cid:18) a ′ − kl (cid:19) s k ( D ) w j − k − l ( H + ) SW m ( f, φ ) x a − − m + l = j X l =0 a − − l X m = − l j − l X k =0 (cid:18) a ′ − kl (cid:19) s k ( D ) w j − k − l ( H + ) SW m + l ( f, φ ) x a − − m = j X l =0 a − X m =0 j − l X k =0 (cid:18) a ′ − kl (cid:19) s k ( D ) w j − k − l ( H + ) SW m + l ( f, φ ) x a − − m , where the last line is obtained by using the fact that V is a trivial bundle of rank a to deduce that x i = 0 if i ≥ a and also the fact that SW i ( f, φ ) = 0 if i ≥ a , so thatthe sum over m in the range − l ≤ m ≤ a − − l gives the same result as summingover m in the range 0 ≤ m ≤ a −
1. We can repeat the same calculation for the odd Steenrod squares Sq j +1 . In summary, we have obtained:(4.3) Sq j ( η ) = a − X m =0 j X l =0 j − l X k =0 (cid:18) a ′ − kl (cid:19) s k ( D ) w j − k − l ( H + ) SW m + l ( f, φ ) x a − − m ,Sq j +1 ( η ) = a − X m =0 j X l =0 j − l X k =0 (cid:18) a ′ − kl (cid:19) s k ( D ) w j − k − l +1 ( H + ) SW m + l ( f, φ ) x a − − m . On the other hand applying Sq j or Sq j +1 directly to Equation (4.2) and usingthe Cartan formula, we find:(4.4) Sq j ( η ) = a − X m =0 j X l =0 Sq j − l ( SW m + l ( f, φ )) (cid:18) a − − m − ll (cid:19) x a − − m ,Sq j +1 ( η ) = a − X m =0 j X l =0 Sq j − l +1 ( SW m + l ( f, φ )) (cid:18) a − − m − ll (cid:19) x a − − m . Equating powers of x in Equations (4.3) and (4.4), we immediately find: Lemma 4.4.
The Steenrod squares of the (mod reductions of the) Seiberg-Wittenclasses SW m ( f, φ ) satisfy the following recursive equations: j X l =0 Sq j − l ( SW m + l ( f, φ )) (cid:18) a − − m − ll (cid:19) = j X l =0 j − l X k =0 (cid:18) a ′ − kl (cid:19) s k ( D ) w j − k − l ( H + ) SW m + l ( f, φ ) , j X l =0 Sq j − l +1 ( SW m + l ( f, φ )) (cid:18) a − − m − ll (cid:19) = j X l =0 j − l X k =0 (cid:18) a ′ − kl (cid:19) s k ( D ) w j − k − l +1 ( H + ) SW m + l ( f, φ ) . Remark . Before we proceed to solve this recursion, let us make some observa-tions: • For j = 0, the above equations reduce to the trivial identites: Sq ( SW m ( f, φ )) = SW m ( f, φ ) , Sq ( SW m ( f, φ )) = w ( H + ) SW m ( f, φ ) , the latter being a simple consequence of the fact that Sq is the Bocksteinhomomorphism and that the classes SW m ( f, φ ) lift to integral cohomologyclasses but with values in the local coefficient system Z w ( H + ) . • Recall that SW i ( f, φ ) = 0 if i ≥ a and s k ( D ) = 0 if k > a ′ . Thus all non-zero terms in the above formula involve only binomial coefficients whoseupper index is non-negative. Lemma 4.6.
Suppose there are classes θ jm ∈ H m +2 j − (2 d − b + − ( Z ) for all j, m ≥ satisfying the recursive relation (4.5) j X l =0 θ j − lm + l (cid:18) a − − m − ll (cid:19) = j X l =0 j − l X k =0 (cid:18) a ′ − kl (cid:19) s k ( D ) w j − k − l ( H + ) SW m + l ( f, φ ) for all j, m ≥ . Then Sq j ( SW m ( f, φ )) = θ jm . N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 45
Similarly, if there are classes π jm ∈ H m +2 j +1 − (2 d − b + − ( Z ) for all j, m ≥ satisfying (4.6) j X l =0 π j − lm + l (cid:18) a − − m − ll (cid:19) = j X l =0 j − l X k =0 (cid:18) a ′ − kl (cid:19) s k ( D ) w j − k − l +1 ( H + ) SW m + l ( f, φ ) , for all j, m ≥ . Then Sq j +1 ( SW m ( f, φ )) = π jm .Proof. Follows easily from Lemma 4.4 by induction on j . (cid:3) We now look for solutions of the recursive equations (4.5), (4.6). Experimentingwith small values of j , one is naturally lead to look for solutions of the form:(4.7) θ jm = j X l =0 j − l X k =0 f mk,l s k ( D ) w j − l − k ( H + ) SW m + l ( f, φ ) ,π jm = j X l =0 j − l X k =0 f mk,l s k ( D ) w j − l − k +1 ( H + ) SW m + l ( f, φ )for some coefficients f j,mk,l ∈ Z , where k, l, m ≥ Lemma 4.7.
Let θ jm , π jm be defined as in Equation (4.7) for some coefficients f mk,l .Suppose that f mk,l satisfies the following recursion: (4.8) l X l ′ =0 f m + l ′ k,l − l ′ (cid:18) a ′ + d − − m − l ′ l ′ (cid:19) = (cid:18) a ′ − kl (cid:19) (mod 2) for all k, l, m ≥ and all a ′ ≥ max ( k, − d + 1 + m + l ) . Then θ jm , π jm satisfyEquations (4.5) and (4.6) for all j, m ≥ .Proof. Follows by direct substitution of (4.7) into (4.5) and (4.6). (cid:3)
Lemma 4.8.
For all integers u and all j ≥ , we have: j X l =0 (cid:18) u + lj − l (cid:19)(cid:18) l − l (cid:19) = (cid:18) u + 1 j (cid:19) (mod 2) . Proof.
The identity can be easily verified for j = 0 , ,
2, so we will assume that j ≥
3. If j ≥
3, then we need to show that j X l =2 (cid:18) lj − l (cid:19)(cid:18) l − l (cid:19) = (cid:18) u + 1 j (cid:19) + (cid:18) uj (cid:19) + (cid:18) u + 1 j − (cid:19) (mod 2) for all j ≥ . Using Pascal’s formula the right hand side simplifies mod 2 to (cid:0) uj − (cid:1) . So we needto show that:(4.9) j X l =2 (cid:18) lj − l (cid:19)(cid:18) l − l (cid:19) = (cid:18) uj − (cid:19) (mod 2) for all j ≥ . Assume l ≥
2. Using Pascal’s formula, we find (cid:18) ll (cid:19) = (cid:18) l − l (cid:19) + (cid:18) l − l − (cid:19) = 2 (cid:18) l − l (cid:19) . Hence (cid:0) l − l (cid:1) is even unless (cid:0) ll (cid:1) = 2 (mod 4). However it is known that the numberof factors of 2 in (cid:0) ll (cid:1) is equal to the number of 1s in the binary expansion of l .Thus (cid:0) l − l (cid:1) is odd if and only if l is a power of 2, say l = 2 k . So (4.9) is equivalentto showing that:(4.10) X k ≥ k ≤ j (cid:18) u + 2 k j − k (cid:19) = (cid:18) uj − (cid:19) (mod 2) for all j ≥ . Recall that if positive integers a, b have binary expansions a = a + 2 a + · · · + 2 u a u , b = b + 2 b + · · · + 2 u b u , then:(4.11) (cid:18) ab (cid:19) = u Y i =0 (cid:18) a i b i (cid:19) (mod 2) . Now suppose that j = j + 2 j + · · · + 2 u j u , where j u = 1. Then the sum on theleft hand side of (4.10) runs from k = 1 , . . . , u . The Vandermonde identity gives: (cid:18) u + 2 k j − k (cid:19) = X r,s ≥ r + s = j − k (cid:18) ur (cid:19)(cid:18) k s (cid:19) . But using (4.11), we see that (cid:0) k s (cid:1) is even, unless s = 0 or 2 k . Hence (cid:18) u + 2 k j − k (cid:19) = ((cid:0) uj − k (cid:1) + (cid:0) uj − k +1 (cid:1) (mod 2) for 1 ≤ k ≤ u − , (cid:0) uj − k (cid:1) (mod 2) for k = u. Substituting this into the left hand side of (4.10) gives: u X k =1 (cid:18) uj − k (cid:19) + u − X k =1 (cid:18) uj − k +1 (cid:19) . The terms in these sums cancel in pairs (mod 2), except for the k = 1 term in thefirst sum. Hence the above equals (cid:0) uj − (cid:1) mod 2. This proves (4.10) and the proofof the lemma is complete. (cid:3) Lemma 4.9.
For all k, l, m ≥ , let f mk,l = (cid:18) d − − m + l + kl (cid:19) . Then f mk,l satisfies Equation (4.8) for all k, l, m, a ′ ≥ .Proof. Note that f mk,l is a binomial coefficient whose upper index is an integer whichmay be negative. Set v = a ′ + ( d − − m ), u = k + ( d − − m ). Thus it suffices toshow that:(4.12) j X l =0 (cid:18) u + j − lj − l (cid:19)(cid:18) v − ll (cid:19) = (cid:18) v − uj (cid:19) (mod 2) N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 47 for all u, v, j with j ≥ u, v any integers. Note that(4.13) (cid:18) ab (cid:19) = a ( a − · · · ( a − b + 1) b != ± ( b − a − b − a − · · · ab != ± (cid:18) b − a − b (cid:19) = (cid:18) b − a − b (cid:19) (mod 2) . Hence (4.12) can be re-written as: j X l =0 (cid:18) − u + l − j − l (cid:19)(cid:18) v − ll (cid:19) = (cid:18) v − uj (cid:19) (mod 2) , or replacing u with − ( u + 1), this becomes(4.14) j X l =0 (cid:18) u + lj − l (cid:19)(cid:18) v − ll (cid:19) = (cid:18) v + u + 1 j (cid:19) (mod 2)for all j ≥ u, v . We will prove that (4.14) holds by an inductiveargument. For any integers u, v and any j ≥
0, let us set δ ( u, v, j ) = j X l =0 (cid:18) u + lj − l (cid:19)(cid:18) v − ll (cid:19) + (cid:18) v + u + 1 j (cid:19) ∈ Z . So we aim to show that δ ( u, v, j ) for all u, v and all j ≥
0. For any u, v and any j ≥
0, let P ( j, r ) be the proposition that Equation (4.14) holds, or equivalently,that δ ( u, v, j ) = 0. Recall that Pascal’s formula (cid:0) ab (cid:1) = (cid:0) a − b (cid:1) + (cid:0) a − b − (cid:1) holds for anynon-negative integer b and any number a . Hence we find: δ ( u, v, j ) = j X l =0 (cid:18) u + lj − l (cid:19)(cid:18) v − ll (cid:19) + (cid:18) v + u + 1 j (cid:19) = j X l =0 (cid:18) u − lj − l (cid:19)(cid:18) v − ll (cid:19) + j − X l =0 (cid:18) u − lj − − l (cid:19)(cid:18) v − ll (cid:19) + (cid:18) v + uj (cid:19) + (cid:18) v + uj − (cid:19) = δ ( u − , v, j ) + δ ( u − , v, j − . Hence if any two of P ( u, v, j ) , P ( u − , v, j ) and P ( u − , v, j −
1) hold, then sodoes the third. Applying Pascal’s formula instead to the second binomial factor in δ ( u, v, j ) gives: δ ( u, v, j ) = j X l =0 (cid:18) u + lj − l (cid:19)(cid:18) v − ll (cid:19) + (cid:18) v + u + 1 j (cid:19) = j X l =0 (cid:18) u + lj − l (cid:19)(cid:18) v − − ll (cid:19) + j X l =1 (cid:18) u + lj − l (cid:19)(cid:18) v − − ll − (cid:19) + (cid:18) v + uj (cid:19) + (cid:18) v + uj − (cid:19) = δ ( u, v − , j ) + j X l =1 (cid:18) ( u + 1) + ( l − j − − ( l − (cid:19)(cid:18) ( v − − ( l − l − (cid:19) + (cid:18) ( v −
2) + ( u + 1) + 1 j − (cid:19) = δ ( u, v − , j ) + δ ( u + 1 , v − , j − . Hence if any two of P ( u, v, j ) , P ( u, v − , j ) and P ( u + 1 , v − , j −
1) hold, then sodoes the third. Next, using (4.13), we find: δ ( u, v, j ) = j X l =0 (cid:18) u + lj − l (cid:19)(cid:18) v − ll (cid:19) + (cid:18) v + u + 1 j (cid:19) = j X l =0 (cid:18) u + j − ll (cid:19)(cid:18) v − j + lj − l (cid:19) + (cid:18) v + u + 1 j (cid:19) = δ ( v − j, u + j, j ) . Hence P ( u, v, j ) holds if and only if P ( v − j, u + j, j ) holds.Setting j = 0 in (4.14), we see that P ( u, v,
0) holds trivially for all u, v . Next,we consider the case v = −
1. Using (4.13), we find: j X l =0 (cid:18) u + lj − l (cid:19)(cid:18) − − ll (cid:19) = j X l =0 (cid:18) u + lj − l (cid:19)(cid:18) ll (cid:19) . It is well known that (cid:0) ll (cid:1) is even for all l >
0, hence the above sum equals (cid:0) uj (cid:1) mod2. This shows that P ( u, − , j ) holds for all j ≥ u . Similarly, weconsider the case v = 0. Using (4.13), we find j X l =0 (cid:18) u + lj − l (cid:19)(cid:18) − ll (cid:19) = j X l =0 (cid:18) u + lj − l (cid:19)(cid:18) l − l (cid:19) . So P ( u, , j ) is equivalent to j X l =0 (cid:18) u + lj − l (cid:19)(cid:18) l − l (cid:19) = (cid:18) u + 1 j (cid:19) (mod 2) . But this is precisely what was shown in Lemma 4.8. Hence we have verified P ( u, , j )for all u and all j ≥ P ( u, v, j ) for all u, v and all j ≥
0, wewill now prove it for all u , all v ≥ − j ≥
0. For v ≥ − j ≥ Q ( v, j ) be the Proposition that P ( u, v, j ) holds for all values of u . We haveshown that Q ( v,
0) holds for all v ≥ −
1, that Q ( − , j ) holds for all j ≥ Q (0 , j ) holds for all j ≥
0. Now suppose that j ≥ v ≥
1. Recall that wehave shown that if any two of P ( u, v, j ) , P ( u, v − , j ) and P ( u + 1 , v − , j − P ( u, v, j ) is implied by P ( u, v − , j ) and P ( u + 1 , v − , j − Q ( v, j ) holds for all v ≥ − j ≥
0, orequivalently that P ( u, v, j ) holds for all u , for all v ≥ − j ≥ P ( u, v, j ) holds if and only if P ( v − j, u + j, j ) holds. Thus we haveshown that P ( u, v, j ) holds for all u, v, j such that j ≥ u + j ≥ −
1. Let λ = − u − j . So we have shown P ( u, v, j ) holds whenever j ≥ λ ≤
1. Nowwe proceed by induction on λ (with λ ≤ P ( u ′ , v ′ , j ′ ) holds whenever j ′ ≥ − u ′ − j ′ ≤ λ for some λ ≥
1. Now let u, v, j be such that j ≥ − u − j = λ + 1. Recall thatwe have shown that if any two of P ( u ′ , v ′ , j ′ ) , P ( u ′ − , v ′ , j ′ ) and P ( u ′ − , v ′ , j ′ − P ( u + 1 , v, j + 1) and P ( u, v, j + 1) imply P ( u, v, j ). But note that P ( u + 1 , v, j + 1) holds by induction because − ( u + 1) − ( j + 1) = − u − j − λ − P ( u, v, j + 1) holds by induction because N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 49 − u − ( j + 1) = − u − j + 1 = λ . This shows that P ( u, v, j ) also holds and so theinductive step is complete. This completes the proof of the result. (cid:3) Theorem 4.10.
The Steenrod squares of the families Seiberg-Witten invariants aregiven by: Sq j ( SW m ( f, φ )) = j X l =0 j − l X k =0 (cid:18) d − − m + l + kl (cid:19) s k ( D ) w j − l − k ( H + ) SW m + l ( f, φ ) ,Sq j +1 ( SW m ( f, φ )) = j X l =0 j − l X k =0 (cid:18) d − − m + l + kl (cid:19) s k ( D ) w j − l − k +1 ( H + ) SW m + l ( f, φ ) . for all m, j ≥ .Proof. By Lemma 4.6, it suffices to show that θ jm , π jm satisfy Equations (4.5), (4.6),where θ jm , π jm are given by Equation (4.7) with f mk,l = (cid:0) d − − m + l + kl (cid:1) . But Lemmas4.7 and 4.9 show that equations (4.5), (4.6) are indeed satisfied. (cid:3) For convenience, we give the explicit formulas for the first few even Steenrodsquares: Sq ( SW m ( f, φ )) =( d + m ) SW m +1 ( f, φ ) + ( s ( D ) + w ( H + )) SW m ( f, φ ) .Sq ( SW m ( f, φ )) = (cid:18) d − m + 12 (cid:19) SW m +2 ( f, φ ) + (cid:0) ( d + m + 1) s ( D ) + ( d + m ) w ( H + ) (cid:1) SW m +1 ( f, φ )+ (cid:0) s ( D ) + s ( D ) w ( H + ) + w ( H + ) (cid:1) SW m ( f, φ ) .Sq ( SW m ( f, φ )) = (cid:18) d − m + 23 (cid:19) SW m +3 ( f, φ ) + (cid:18)(cid:18) d − m + 22 (cid:19) s ( D ) + (cid:18) d − m + 12 (cid:19) w ( H + ) (cid:19) SW m +2 ( f, φ )+ (cid:0) ( d + m ) s ( D ) + ( d + m + 1) s ( D ) w ( H + ) + ( d + m ) w ( H + ) (cid:1) SW m +1 ( f, φ )+ (cid:0) s ( D ) + s ( D ) w ( H + ) + s ( D ) w ( H + ) + w ( H + ) (cid:1) SW m ( f, φ ) . Similar formulas hold for the odd Steenrod squares (to go from Sq j ( SW m ( f, φ ))to Sq j +1 ( SW m ( f, φ )) just replace each occurrence of w i ( H + ) with w i +1 ( H + )). Corollary 4.11.
Suppose that j > m − (2 d − b + − . Then: j X l =0 j − l X k =0 (cid:18) d − − m + l + kl (cid:19) s k ( D ) w j − l − k ( H + ) SW m + l ( f, φ ) = 0 (mod 2) , j X l =0 j − l X k =0 (cid:18) d − − m + l + kl (cid:19) s k ( D ) w j − l − k +1 ( H + ) SW m + l ( f, φ ) = 0 (mod 2) . Proof.
Start with Theorem 4.10 and recall that Sq j ( x ) = 0 whenever j > deg ( x ). (cid:3) For instance if 2 m < d − b + + 1, then:( d + m ) SW m +1 ( f, φ ) + ( s ( D ) + w ( H + )) SW m ( f, φ ) = 0 (mod 2) . and if 2 m < d − b + + 1, then: (cid:18) d − m + 12 (cid:19) SW m +2 ( f, φ ) + (cid:0) ( d + m + 1) s ( D ) + ( d + m ) w ( H + ) (cid:1) SW m +1 ( f, φ )+ (cid:0) s ( D ) + s ( D ) w ( H + ) + w ( H + ) (cid:1) SW m ( f, φ ) = 0 (mod 2) . Corollary 4.12.
Suppose that b + = 2 p + 1 is odd and suppose that for some j > we have that H l ( B, Z ) = 0 for < l < j . Then (cid:18) p + jj (cid:19) SW m + j ( f, φ ) + (cid:0) w j ( H + ) + s j ( D ) (cid:1) SW m ( f, φ ) = 0 (mod 2) . where m = d − p − .Remark . Suppose that f is a finite dimensional approximation of the Seiberg-Witten monopole map of a spin c family of 4-manifolds with fibres ( X, s X ) with b + ( X ) = 2 p + 1 >
1. Then for m = d − p −
1, we see that SW m ( f, φ ) ∈ H ( B, Z ) ∼ = Z is the mod 2 reduction of the ordinary Seiberg-Witten invariant SW ( X, s X ) of X . Proof.
Note that 2 d − b + − d − p −
1) and hence 2 j > m − (2 d − b + − m = d − p −
1. So j X l =0 j − l X k =0 (cid:18) d − − m + l + kl (cid:19) s k ( D ) w j − l − k ( H + ) SW m + l ( f, φ ) = 0 (mod 2) . But H l ( B, Z ) = 0 for 0 < l < j , so all terms in the above sum are zero except for( l, k ) = (0 , , (0 , j ) or ( j, (cid:3) Theorem 4.14.
Suppose that b + = 2 p + 1 is odd and that the Stiefel-Whitneyclasses of H + are trivial. Write p as p = 2 a p ′ , where a ≥ and p ′ is odd. Let m = d − p − and note that SW m ( f, φ ) ∈ H ( B ; Z ) ∼ = Z . Then for ≤ b < a wehave SW m +2 b ( f, φ ) = s b ( D ) SW m ( f, φ ) (mod 2) for all b such that ≤ b < a . Furthermore, we have s a ( D ) SW m ( f, φ ) = 0 (mod 2) . Proof.
Since the Stiefel-Whitney classes of H + are trivial, Theorem 4.10 gives(4.15) j X l =0 (cid:18) p + jl (cid:19) s j − l ( D ) SW m + l ( f, φ ) = 0 (mod 2) . Next we note that if p = 2 a p ′ , where p ′ is odd and if 0 ≤ b < a , then (cid:0) p +2 b l (cid:1) is evenfor l in the range 0 ≤ l ≤ b , except when l = 0 or 2 b in which case it is odd. ThusEquation (4.15) for j = 2 b simplifies to: s b ( D ) SW m ( f, φ ) + SW m +2 b ( f, φ ) = 0 (mod 2) . Similarly (cid:0) p +2 a l (cid:1) is even for l in the range 0 ≤ l ≤ a , except when l = 0 in whichcase it is odd. So Equation (4.15) for j = 2 a simplifies to: s a ( D ) SW m ( f, φ ) = 0 (mod 2) . (cid:3) N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 51
Theorem 4.15.
Suppose that H + is orientable and that d − b + − . Supposealso that SW ( f, φ ) ∈ H ( B ; Z ) ∼ = Z is odd. • If SW j ( f, φ ) = 0 (mod 2) for all j > , then w ( H + ) = c ( D ) (mod 2) . • Conversely if w ( H + ) = c ( D ) (mod 2) , then SW j ( f, φ ) = 0 (mod 2) forsome j > .Proof. Assume that SW j ( f, φ ) vanishes mod 2 for all j > SW ( f, φ ) isodd. We will show that w ( H + ) = c ( D ). The second result follows by taking thecontrapositive. From the Sq j and Sq j +1 cases of Theorem 4.10 with m = 0 weobtain: j X k =0 s k ( D ) w j − k ( H + ) = 0 (mod 2) , j X k =0 s k ( D ) w j − k +1 ( H + ) = 0 (mod 2) . this says that the 2 j and 2 j + 1-th Stiefel-Whitney classes of the virtual bundle H + − D vanish. Thus w ( H + − D ) = 1. But w ( H + − D ) = w ( H + ) c ( − D ) = w ( H + ) s ( D ) (mod 2). Multiplying both sides of w ( H + ) s ( D ) = 1 (mod 2) by c ( D ),we get w ( H + ) = c ( D ) (mod 2). (cid:3) Application to K surfaces. In this Subsection we give an application ofthe Steenrod squares computation to K B be a topological manifold. An n -microbundle ξ on B consists of [18]: ξ = { B i −→ E p −→ B } where: • E is a topological space • i : B → E is called the zero-section • p : E → B is called the projection • p ◦ i = id • E is locally trivial around the zero section. That is, for each b ∈ B thereexists a neighbourhood U of b in B , a neighbourhood V of i ( b ) in E and ahomeomorphism V → U × R n such that the restriction of p to V coincideswith the projection U × R n → U .Two microbundles ξ, ξ ′ are considered isomorphic if there exists a homeomor-phism between neighbourhoods of the zero sections of ξ, ξ ′ respecting the zerosections and projection maps.Every topological manifold X has a tangent microbundle τ X , given by τ X = { X ∆ −→ X × X pr −→ X } where ∆ is the diagonal and pr is the projection to the first factor. If X is smooththen τ X is isomorphic to the tangent bundle: T X = { X i −→ T X p −→ X } . Theorem 4.16 (Kister-Mazur [12]) . Let ξ = { B i −→ E p −→ B } be an n -microbundleover B . Then there exists an open neighbourhood U ⊆ E of the zero section suchthat p | U : U → B is a locally trivial fibre bundle with fibre R n . The fibre bundle isunique up to isomorphism. Thus, by restriction every n -microbundle ξ can be represented by an honest fibrebundle p : E → B with fibres homeomorphic to R n and equipped with a section i : B → E . This fibre bundle is unique up to fibre bundle isomorphism. Let T op ( n )denote the group of homeomorphisms of R n preserving the origin. Then the fibrebundle p : E → B has structure group T op ( n ) and thus determines a principal T op ( n ) bundle F ( ξ ) → B . This principal bundle is well-defined up to principalbundle isomorphism and we call it the frame bundle of ξ .There is an obvious notion of an oriented microbundle. We can define the ori-ented frame bundle F + ( ξ ) of an oriented n -microbundle ξ , which is a principal ST op ( n )-bundle where ST op ( n ) is the subgroup of T op ( n ) preserving the orienta-tion of R n .It is known that the natural inclusion SO ( n ) → ST op ( n ) induces an isomorphismof fundamental groups, and both groups are connected. Hence there is a uniqueconnected double covering φ : SpinT op ( n ) → ST op ( n ). A spin structure on anoriented n -microbundle ξ is a double cover ˜ F + ( ξ ) → F + ( ξ ) whose restriction toeach fibre is isomorphic to φ as covering spaces. If ξ is represented by an orientedvector bundle V , this is equivalent to the usual notion of a spin structure on V .Let X be a topological manifold. We define a topological spin structure on X tobe a spin structure on τ X .Let E → B be a continuous locally trivial fibre bundle with fibres homeomorphicto X . We define the vertical tangent microbundle of E to be the microbundle τ ( E/B ) = { E ∆ −→ E × B E pr −→ E } . If E, B are smooth manifolds and E → B a smooth fibre bundle, then τ ( E/B ) isisomorphic to the vertical tangent bundle T ( E/B ).Suppose that E → B is fibrewise oriented, or equivalently, the transition func-tions are valued in Homeo + ( X ), the group of orientation preserving homeomor-phisms of X . Then associated to τ ( E/B ) is its principal
ST op ( n )-oriented framebundle F + ( τ ( E/B )) → E . We define a families topological spin structure for E → B to be a spin structure on τ ( E/B ), that is, a double covering ˜ F + ( τ ( E/B )) →F + ( τ ( E/B )) which restricts to φ on each fibre. Lemma 4.17.
Let ( Y, g ) be a smooth compact oriented Riemannian n -manifold and f : Y → Y an orientation preserving isometry which fixes a point p ∈ Y . Thenthere exists a diffeomorphism f : Y → Y and an open neighbourhood U ⊂ Y × Y of the diagonal such that: (1) pr : U → Y is a locally trivial fibre bundle with fibres homeomorphic to R n . (2) There exists a smooth isotopy f t from f to f such that f t × f t preserves U for all t ∈ [0 , . (3) There is an open neighbourhood of p in Y on which f acts trivially.Proof. Since Y is compact, there exists a δ > y ∈ Y , theexponential map with respect to g gives a diffeomorphism exp y : { v ∈ T y Y | || v || < δ } → U y ⊂ Y from the open disc in T y Y of radius δ to some open neighbourhood of y in Y . Nowdefine U ⊂ Y × Y to be U = { ( y, exp y ( v )) ∈ Y × Y | y ∈ Y, v ∈ T y Y, || v || < δ } . N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 53
Let p : D δ ( T Y ) → Y denote the open unit disc bundle in T Y of radius δ about thezero section. Then the exponential map gives a homeomorphism exp : D δ ( T Y ) → U such that p = pr ◦ exp . This proves (1), since we can identify U with the fibre bundle p : D δ ( T Y ) → Y in a way that preserves zero sections. Moreover f preserves U because f is an isometry.We have that f is an isometry which fixes p ∈ Y . So it follows that f preserves U p and via the exponential map, the restriction f : U p → U p corresponds to thelinear isometry G : T p Y → T p Y , G = df ( y ). Also, G is orientation preservingsince f is. Since the group SO ( n ) is connected, we can find a smooth path G t with G = Id , G = G . Let τ : [0 , → [0 ,
1] be a smooth function such that τ ( t ) = 0for 0 ≤ t ≤ / τ ( t ) = 1 for 2 / ≤ t ≤
1. Now we define f t : Y → Y as follows.If y / ∈ U p , set f t ( y ) = f ( y ). If y ∈ U p then y = exp p ( v ) for a unique v ∈ T p Y with || v || < δ . In this case we set f t ( y ) = exp p (cid:0) G − tτ (1 −|| v || /δ ) v (cid:1) . Then f t is a smooth isotopy from f to f , where f fixes pointwise the imageunder the exponential map of { v ∈ T p Y | || v || < δ/ } . It is easy to see that f t × f t preserves U for all t and hence we have shown (2) and (3). (cid:3) Let
X, Y be topological n -manifolds. Here we make an observation concerningthe tangent microbundle of the connected sum X Y . Let p X : U X ⊂ X × X , p Y : U Y ⊂ Y × Y be open neighbourhoods of the diagonals such that U X → X , U Y → Y are locally trivial fibre bundles with fibres R n . Choose points x ∈ X and y ∈ Y . Let B x ⊂ X , B y ⊂ Y be open neighbourhoods of x, y such that B x is givenby the unit open disc in some coordinate system centered at x and similarly for B y .The connected sum X Y is constructed by identifying the boundary of X \ B x with the boundary of Y \ B y in an orientation reversing way.Assume the coordinates around x, y are such that we can define open balls B ′ x , B ′ y around x, y with radius slightly larger than 1. Shrinking the coordinate systemsif necessary, we can assume that there exist trivialisations of U X , U Y over B ′ x , B ′ y .Then U X \ p − X ( B x ) is a locally trivial fibre bundle over U X \ B x whose restrictionover the boundary S n − -sphere of U X \ B x is the trivial bundle S n − × R n . A similarstatement holds for Y . Hence we can attach U X \ p − X ( B x ) to U Y \ p − Y ( B y ) alongtheir common boundary to form a locally trivial fibre bundle U X Y → X Y . Theattachment is unique up to isomorphism since ST op ( n ) is connected. It is easilyseen that U X Y represents the tangent microbundle of X Y .Let Y be an oriented topological n -manifold. Let ( Y , g ) . . . ( Y k , g k ) be com-pact, smooth, Riemannian oriented n -manifolds. For each j ∈ { , . . . , k } supposethat f j, : Y j → Y j is an orientation preserving isometry which fixes a point p j ∈ Y j .Then by Lemma 4.17 there exist diffeomorphisms f j, : Y j → Y j , open neighbour-hoods U j ⊂ Y j × Y j (0 ≤ j ≤ k ) such that: • For each j , pr : U j → Y j is a locally trivial fibre bundle with fibres home-omorphic to R n . • There exists smooth isotopies f j,t from f j, to f j, such that f j,t × f j,t preserves U j for all t ∈ [0 , • There exist open neighbourhoods N p j of p j in Y j on which f j, acts trivially. Let Y = Y Y · · · Y k be the connected sum, where we attach each Y j for j > Y such that the attaching ball in Y j is contained in U j . As describedabove we can glue the microbundles U , U , . . . , U k together to form a representative U → Y of the tangent microbundle of Y . In particular, the frame bundle for U is a representative for the tangent ST op ( n ) frame bundle F + ( τ Y ). Then each f j, extends to a homeomorphism h j : Y → Y such that h j × h j preserves U . Clearlythe h j commute. Moreover since each h j preserves U , it follows that h j admits alift to a principal bundle automorphism h ′ j : F + ( τ Y ) → F + ( τ Y ) and the h ′ j againcommute.Consider now the mapping torus E = Y × h ,...,h k R k → T k associated to thecommuting homeomorphisms h , . . . , h k . From the above construction it is clearthat the vertical tangent microbundle F + ( τ ( E/T k )) can be identified with themapping torus F + ( τ Y ) × h ′ ,...,h ′ k R k → T k of the lifts h ′ , . . . , h ′ k . Proposition 4.18.
Let E → T k be the mapping torus constructed as above. Sup-pose that Y , . . . , Y k are equipped with spin structures s , . . . , s k . Let s denote theconnected sum spin structure on Y . If f j, : Y j → Y j preserves s j for each j , then s extends to a spin structure on τ ( E/T k ) .Proof. The spin structure s determines a double cover ˜ F + ( τ Y ) → ˜ F + ( τ Y ) whichrestricts to SpinT op ( n ) → ST op ( n ) on each fibre. The assumption that f j, : Y j → Y j preserves s j implies that f j, also preserves s j and it follows that h j preserves s .Thus h ′ j can be lifted to ˜ h ′ j : ˜ F + ( τ Y ) → ˜ F + ( τ Y ).We claim that h ′ i , h ′ j commute for any i, j . To see this, we note that commuta-tivity of h i , h j implies that [ h ′ i , h ′ j ] is a deck transformation of ˜ F + ( τ Y ) → ˜ F + ( τ Y ).Now consider the fibre of ˜ F + ( τ Y ) → ˜ F + ( τ Y ) over some point y ∈ Y . Since h i , h j both act trivially in a neighbourhood of y , we see that h ′ i , h ′ j act as decktransformations on the fibre ˜ F + ( τ Y ) y → ˜ F + ( τ Y ) y over y . But the group ofsuch deck transformations is abelian, hence the commutator [ h ′ i , h ′ j ] acts triviallyon ˜ F + ( τ Y ) y . But this implies [ h ′ i , h ′ j ] = 1, since [ h ′ i , h ′ j ] is a deck transformation.We therefore obtain a double cover of F + ( τ ( E/T k )) by taking the mapping torusof ˜ h ′ , . . . , ˜ h ′ k . Clearly this gives a spin structure on τ ( E/T k ) which extends s . (cid:3) Let
Homeo + ( X ) denote the group of orientation preserving homeomorphismsof X with the C -topology and let Dif f + ( X ) denote the group of orientationpreserving diffeomorphisms of X with the C ∞ -topology. Note that, if X is a closedoriented 4-manifold with non-zero signature, we have Homeo + ( X ) = Homeo ( X )and Dif f + ( X ) = Dif f ( X ). Remark . Let f , f ∈ Dif f ( X ). If there is a continuous map F • : X × [0 , → X satisfying that F = f , F = f , and F t ∈ Dif f ( X ) for each t ∈ [0 , F • by a smooth family, one can show that there exists a smoothmap F ′• : X × [0 , → X satisfying that F ′ = f , F ′ = f , and F ′ t ∈ Dif f ( X ) foreach t ∈ [0 , Definition 4.20.
Let B be a smooth manifold. • For an oriented topological manifold X , a continuous family over B withfibres homeomorphic to X is a fibre bundle E → B whose fibres are home-omorphic to X and whose transitions functions are valued in Homeo + ( X )(so E → B is fibrewise oriented). N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 55 • For an oriented smooth manifold X , a smooth family over B with fibresdiffeomorphic to X is a fibre bundle E → B whose fibres are diffeomorphicto X and whose transition functions are valued in Dif f + ( X ). • For an oriented topological manifold X admitting a smooth structure, acontinuous family E → B over B with fibres homeomorphic to X is called smoothable if there exists a smooth family E ′ → B with fibres diffeomorphicto X , equipped with a smooth structure, such that E is isomorphic to theunderlying continuous family associated to E ′ . Remark . If E → B is a smooth family with fibres diffeomorphic to X in theabove sense, by replacing E with an isomorphic bundle, we may assume that thetotal space of E is a smooth manifold and the projection E → B is a surjectivesubmersion. This is based on M¨uller–Wockel [20] as follows. A priori, the transitionfunctions of E are continuous maps from double-indexed open sets { U ij } ij of B to Dif f + ( M ), regarded as a Fr´echet manifold. However, the main theorem of [20]implies that there exists a bundle E ′ → B such that E ′ is isomorphic to E andthat the transition functions U ij → Dif f ( X ) of E ′ are smooth for some choiceof local trivializations of E ′ . Recall that the evaluation map X × Dif f ( X ) → X is a smooth map. This follows from the fact that Dif f ( X ) is an open set of thesmooth mapping space M ap ( X, X ) with the C ∞ -topology, and the evaluation map M ap ( X, X ) × X → X is smooth (see, for example, 42.13. Theorem in p.444 of[13]). Therefore we can deduce that E ′ admits a smooth manifold structure andthe projection E ′ → B is a surjective submersion.For any continuous family E → B with fibres X , the bundle H + → B is well-defined up to isomorphism (we can define it to be a maximal positive definitesub-bundle of the bundle with fibres H ( X ; R )).For any smooth family with fibres diffeomorphic to X , equipped with a familiesspin structure, we may define the virtual index bundle D ∈ K ( B ) of the Diracoperator.Let X be the underlying topological 4-manifold of a K • π ( Homeo + ( X )) is isomorphic to the isometry group of H ( X ; Z ) by resultsof Freedman [8] and Quinn [22]. • For the standard smooth structure of X , the image of π ( Dif f + ( X )) in π ( Homeo + ( X )) ∼ = Aut ( H ( X ; Z )) is precisely the index 2 subgroup ofelements which preserve the orientation of H + ( X ) [17, 5]. Proposition 4.22.
Let E → B be a continuous family with fibres X and supposethat E admits a families topological spin structure. If E is smoothable for somesmooth structure on X , then w ( H + ) = 0 .Proof. Suppose E is smoothable for some smooth structure on X . Then E ad-mits a families spin structure. Let s be the unique spin c structure on X of zerocharacteristic. Then SW ( X, s ) is odd by [19]. By Corollary 1.3, we have c ( D ) + w ( H + ) = 0 ∈ H ( B ; Z ) . To prove the proposition, it suffices to show that c ( D ) = 0. But D is the familiesindex of a spin structure on a family of 4-manifolds, so D lies in the image of KO − ( B ) → K ( B ). The group KO − ( B ) can be identified with the Grothendieckgroup of quaternionic vector bundles on B . In particular, all such bundles havetrivial first Chern class. (cid:3) Up to homeomorphism, we may identify X with the connected sum X = 2( − E ) S × S ) . We label the three copies of S × S as ( S × S ) i , i = 1 , , f on S × S which acts as − H ( S × S ; Z ) and such that f acts as the identity on some disc U . To be specific,we start with the diffeomorphism which acts as a reflection about the equator oneach factor of S and after performing an isotopy as in the proof of Lemma 4.17,we can assume the diffeomorphism acts as the identity in a neighbourhood of afixed point. Now in the connected sum decomposition X = 2( − E ) S × S ),we assume that each copy of ( S × S ) is attached to 2( − E ) via a handle thatends in the fixed disc. Then we set f = id − E ) f f id, f = id − E ) id f f. Then f , f : X → X are commuting, orientation preserving homeomorphisms. Let E f ,f → T be the mapping cylinder. Lemma 4.23.
The family E → T admits a families topological spin structure.Proof. This is just a special case of Proposition 4.18. (cid:3)
Theorem 4.24.
The family E → T is not smoothable, but its restriction to any -dimensional embedded submanifold S ⊂ T is smoothable.Proof. Let E ∈ H ( S × S ; Z ) be the class Poincar´e dual to the diagonal S → S × S . Then f ( E ) = − E . For i = 1 , ,
3, let E i denote the corresponding classin H (( S × S ) i ; Z ). Then E , E , E form a basis for H + ( X ) and f ( E ) = − E , f ( E ) = − E , f ( E ) = E ,f ( E ) = E , f ( E ) = − E , f ( E ) = − E . Clearly f , f preserve orientation on H + . Let x, y ∈ H ( T ; Z ) be a basis corre-sponding to the two S -factors of T . Using the splitting of H + into the real linebundles spanned by E , E , E we find: w ( H + ) = (1 + x )(1 + x + y )(1 + y )= (1 + y + xy )(1 + y )= 1 + xy. Hence w ( H + ) = xy = 0. This shows that E → B is not smoothable by Proposi-tion 4.22. On the other hand, consider the restriction of E → B to some embeddedcircle j : S → T . Then j ∗ ( E ) → S is the mapping cylinder of the homeomor-phism g = f a f b , where the underlying homology class of j ( S ) is ( a, b ). But since f , f preserve orientation of H + ( X ), so does g . As remarked earlier, this meansthat g is continuously isotopic to a diffeomorphism and hence j ∗ ( E ) is smoothablewith respect to the standard smooth structure on X = K (cid:3) Proposition 4.25.
The total space of the family E is smoothable as a manifold.Proof. The proof is totally the same with Section 6 of [10]. (cid:3)
N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 57 Wall crossing formula for families Seiberg-Witten invariants
In this section we use the monopole map to give a new derivation of the wallcrossing formula for families Seiberg-Witten invariants. The families wall cross-ing formula was originally proven in [16] using parametrised Kuranishi models andobstruction bundles. In contrast our approach is purely cohomological and consid-erably simpler.5.1.
Wall crossing formula.
As usual our starting point is a finite dimensionalmonopole map f : ( S V,U , B
V,U ) → ( S V ′ ,U ′ , B V ′ ,U ′ ). For the wall crossing formulait is most convenient to stabilise f such that the bundles V ′ , U ′ are trivial, say V ′ = C a ′ , U ′ = R b ′ . According to Assumption 1, we may assume that U ′ = U ⊕ H + and that f | U : U → U ′ is the inclusion U → U ′ . Recall that the set CH ( f ) ofchambers for f is the set of homotopy class of sections of U ′ \ U , or equivalentlyhomotopy classes of sections of S ( H + ), the unit sphere bundle in H + with respectto some choice of metric on H + . Let [ φ ] ∈ CH ( f ) be a chamber represented by asection φ : B → S ( H + ). Since φ avoids U , we obtain a pushforward map: φ ∗ : H jS ( B ; Z ) → H j +2 a ′ + b ′ S ( S V ′ ,U ′ , S U ; Z ) . Recall that for m ≥ m -th Seiberg-Witten invariant of ( f, φ ) is a cohomologyclass SW m ( f, φ ) ∈ H m − (2 d − b + − ( B, Z w ) where w = w ( H + ). By Theorem 3.6, SW m ( f, φ ) is given by: SW m ( f, φ ) = ( π Y ) ∗ ( x m ` f ∗ φ ∗ (1)) . Let [ ψ ] ∈ CH ( f ) be a second chamber for f represented by a section ψ : B → S ( H + ). Then similarly SW m ( f, ψ ) = ( π Y ) ∗ ( x m ` f ∗ ψ ∗ (1)) . It follows that the difference SW m ( f, φ ) − SW m ( f, ψ ) can be computed provided weknow the difference ψ ∗ (1) − ψ ∗ (1) ∈ H a ′ + b ′ S ( S V ′ ,U ′ , S U ; Z ). As before, we use thefollowing notation: e τ φV ′ ,U ′ = φ ∗ (1), e τ ψV ′ ,U ′ = ψ ∗ (1). Since φ and ψ are homotopicwithin the total space of U , we have that the images of e τ φV ′ ,U ′ and e τ ψV ′ ,U ′ under thenatural map H a ′ + b ′ S ( S V ′ ,U ′ , S U ; Z ) → H a ′ + b ′ S ( S V ′ ,U ′ , B V ′ ,U ′ ; Z ) coincide. Thenfrom Proposition 3.2 we have that e τ φV ′ ,U ′ − e τ ψV ′ ,U ′ = e θ ( φ, ψ ) ` δτ ,U for some e θ ( φ, ψ ) ∈ H a ′ + b + − S ( B, Z w ).Next we observe that φ : B → S V ′ ,U ′ factors as: B φ −→ S ( H + ) ι −→ S V ′ ,U ′ . A similar factorisation holds for ψ and therefore we have:(5.1) e τ φV ′ ,U ′ − e τ ψV ′ ,U ′ = φ ∗ (1) − ψ ∗ (1) = ι ∗ ( φ ∗ (1) − ψ ∗ (1)) . Lemma 5.1.
We have ι ∗ (1) = x a ′ δτ ,U .Proof. We factor S ( H + ) ι −→ S V ′ ,U ′ as follows: S ( H + ) ι −→ S ,U ′ ι −→ S V ′ ,U ′ . Then it suffices to show that ( ι ) ∗ (1) = δτ ,U and ( ι ) ∗ ( δτ ,U ) = x a ′ δτ ,U . Firstconsider ι . Since S acts trivially on S ( H + ) and S ,U ′ , it suffices to compute( ι ) ∗ (1) in non-equivariant cohomlogy. We view ( ι ) ∗ as a homomorphism( ι ) ∗ : H ∗ ( S ( H + ); Z ) → H ∗ + b +1 ( S ,U ′ , S U ; Z w ) . But any element of H ∗ ( S ,U ′ , S U ; Z w ) has the form α ` δτ ,U + β ` e τ φ ,U ′ forsome α ∈ H ∗ ( B ; Z ) and some β ∈ H ∗ ( B ; Z w ). In particular, we have ( ι ) ∗ (1) = α ` δτ ,U + β ` e τ φ ,U ′ for some α ∈ H ( B ; Z ) = Z and some β ∈ H − b + ( B ; Z w ).We claim that the image of ( ι ) ∗ (1) under the natural map H ∗ ( S ,U ′ , S U ; Z w ) → H ∗ ( S ,U ′ , B ,U ′ ; Z w ) is zero. To see this note that ( ι ) ∗ (1) is Poincar´e dual to S ( H + ) ⊂ S ,U ′ . But S ( H + ) is the boundary of the unit disc bundle D ( H + ) and D ( H + ) is disjoint from B ,U ′ . This proves the claim. Thus ( ι ) ∗ (1) = αδτ ,U forsome α ∈ Z . To compute α , it suffices to restrict the the fibres of S ( H + ) and S ,U ′ over a point in B . Upon restriction, ι is the just the inclusion map of spheres: S ( R b + ) → S ( R ⊕ R b + ⊕ R b ) . Then it is straightforward to verify in this situation that α = 1.Next, we verify that ( ι ) ∗ ( δτ ,U ) = x a ′ δτ ,U . But ( ι ) ∗ ( δτ ,U ) = ι ∗ (1) is Poincar´edual to S ( H + ), so arguing as we did for ι , this implies that the image of ( ι ) ∗ ( δτ ,U )under the map H ∗ S ( S V ′ ,U ′ , S U ; Z w ) → H ∗ S ( S V ′ ,U ′ , B V ′ ,U ′ ; Z w ) is zero. Thus( ι ) ∗ ( δτ ,U ) = γ ` δτ ,U for some γ ∈ H a ′ S ( B ; Z ). But ( ι ) ∗ ( ι ) ∗ ( δτ ,U ) = e V ′ ` δτ ,U , where e V ′ denotes the equivariant Euler class of V ′ . It follows that γ = e V ′ .Then since we are assuming V ′ is a trivial bundle, we have e V ′ = x a ′ and the resultfollows. (cid:3) Let π : S ( H + ) → B denote the projection to B . From π ◦ φ = id , we see that π ∗ ( φ ∗ (1)) = 1 and similarly π ∗ ( ψ ∗ (1)) = 1. It follows that π ∗ ( φ ∗ (1) − ψ ∗ (1)) = 0.The Gysin sequence for S ( H + ) → B then implies that(5.2) φ ∗ (1) − ψ ∗ (1) = π ∗ ( θ ( φ, ψ ))for some θ ( φ, ψ ) ∈ H b + − ( B ; Z w ). Moreover since S ( H + ) admits sections, theGysin sequence splits and the pullback map π ∗ : H ∗ ( B ; Z w ) → H ∗ ( S ( H + ); Z w )is injective, so that θ ( φ, ψ ) is uniquely characterised by Equation (5.2). Theorem 5.2 (Wall crossing formula for families Seiberg-Witten invariants) . Given φ, ψ ∈ CH ( f ) , let θ ( φ, ψ ) ∈ H b + − ( B ; Z w ) be defined as in Equation (5.2). Then: SW m ( f, φ ) − SW m ( f, ψ ) = ( if m < d − ,θ ( φ, ψ ) s m − ( d − ( D ) if m ≥ d − . Proof.
From Equations (5.1) and (5.2) we have: e τ φV ′ ,U ′ − e τ ψV ′ ,U ′ = ι ∗ ( φ ∗ (1) − ψ ∗ (1))= ι ∗ ( π ∗ ( θ ( φ, ψ )))= θ ( φ, ψ ) · ι ∗ (1)= θ ( φ, ψ ) · x a ′ δτ ,U , N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 59 where the last equality follows from Lemma 5.1. Combined with Theorem 3.6, wenow have: SW m ( f, φ ) − SW m ( f, ψ ) = ( π Y ) ∗ ( x m ` f ∗ ( e τ φV ′ ,U ′ − e τ ψV ′ ,U ′ ))= ( π Y ) ∗ ( x m + a ′ ` f ∗ ( θ ( φ, ψ )) · δτ ,U ))= θ ( φ, ψ ) · ( π Y ) ∗ ( x m + a ′ ` δτ ,U ) . The result now follows by applying Proposition 3.5. (cid:3)
Relative obstruction class.
In the previous subsection we gave a wall cross-ing formula for families Seiberg-Witten invariants in terms of a class θ ( φ, ψ ). Inthis subsection we examine the class θ ( φ, ψ ) more carefully and identify with theprimary obstruction class for the existence of a homotopy between φ and ψ .Let [ φ ] , [ ψ ] ∈ CH ( f ) be two chambers, which we represent by sections φ, ψ : B → S ( H + ) of the unit sphere bundle S ( H + ) associated to H + . By definition, φ and ψ represent the same chamber if and only if there is a homotopy betweenthem. While the general problem of computing homotopy classes of sections ofsphere bundles is complicated, we can use obstruction theory to gain some insights.Since the fibres of S ( H + ) → B are spheres of dimension b + −
1, we see there is noobstruction to finding a homotopy between φ and ψ on the ( b + − B .Extending such a homotopy to the ( b + − b + − primary difference in [25, § Obs ( φ, ψ ) ∈ H b + − ( B ; Z w ) . One may define
Obs ( φ, ψ ) as follows. Consider the pullback of H + to B × I , where I = [0 ,
1] is the unit interval. The pair ( φ, ψ ) define a non-vanishing section of H + over B × ,
1. In such a situation we have a relative Euler class [11] e rel ( H + , φ, ψ ) ∈ H b + ( B × I, B × { , } ; Z w ) . Let ι : B → B × I be given by ι ( b ) = ( b, / ι ∗ : H ∗− ( B ; Z w ) → H ∗ ( B × I, B × { , } ; Z w )is easily seen to be an isomorphism. The class can Obs ( φ, ψ ) may be defined bythe relation (c.f. [16, Lemma 3.8]): ι ∗ ( Obs ( φ, ψ )) = e rel ( H + , φ, ψ ) . Note that ι ∗ coincides with the coboundary map δ : H ∗− ( B ; Z w ) → H ∗ ( B × I, B × { , } ; Z w ) in the long exact sequence of the pair ( B × I, B × { , } ).A more geometric definition of e rel ( H + , φ, ψ ) is as follows: choose a section ˜ s ofthe pullback of H + to B × I such that ˜ s | B ×{ } = φ , ˜ s | B ×{ } = ψ and such that˜ s is transverse to the zero section. Then the zero locus ˜ s − (0) is Poincar´e dual to e rel ( H + , φ, ψ ). Lemma 5.3.
Let − ψ denote the composition of ψ with the antipodal map. Sup-pose that φ and − ψ meet transversally. Then Obs ( φ, ψ ) is Poincar´e dual to theintersection locus S = { b ∈ B | φ ( b ) = − ψ ( b ) } .Proof. Let us define a section ˜ s of the pullback of H + to B × I by:˜ s ( b, t ) = (1 − t ) φ + tψ. Then ˜ s | B ×{ } = φ , ˜ s | B ×{ } = ψ . Moreover, since φ and ψ are both sections of theunit sphere bundle S ( H + ), we see that ˜ s ( b, t ) = 0 only when t = 1 /
2. Moreover˜ s ( b, /
2) = ( φ + ψ ), so ˜ s ( b, /
2) = 0 if and only if φ ( b ) = − ψ ( b ). So the zerolocus of ˜ s is precisely S × { / } = ι ( S ). If φ and − ψ intersect transversally thenit is easy to see that ˜ s meets the zero section transversally. Thus ι ( S ) is Poincar´edual to e rel ( H + , φ, ψ ). Then since ι ∗ ( Obs ( φ, ψ )) = e rel ( H + , φ, ψ ) we have that S is Poincar´e dual to Obs ( φ, ψ ). (cid:3) The above lemma translates into the following cohomological formula for
Obs ( φ, ψ ):(5.3) Obs ( φ, ψ ) = π ∗ ( φ ∗ (1) ` ( − ψ ) ∗ (1)) . Indeed φ ∗ (1) ` ( − ψ ) ∗ (1) is Poincar´e dual to the intersection locus of φ ( B ) and − ψ ( B ) in S ( H + ) and thus π ∗ ( φ ∗ (1) ` ( − ψ ) ∗ (1)) is Poincar´e dual to S = { b ∈ B | φ ( b ) = − ψ ( b ) } . Proposition 5.4.
We have an equality
Obs ( φ, ψ ) = θ ( φ, ψ ) .Proof. Recall that θ ( φ, ψ ) is characterised by the identity π ∗ ( θ ( φ, ψ )) = φ ∗ (1) − ψ ∗ (1) while on the other hand Obs ( φ, ψ ) is given by Equation (5.3). To relatethese two expressions we will need to carry out some cohomological computationson S ( H + ). Consider the Gysin sequence for S ( H + ) → B : · · · → H ∗ ( B ; Z w ) π ∗ −→ H ∗ ( S ( H + ); Z w ) π ∗ −→ H ∗− ( b + − ( B ; Z ) → · · · This sequence may be split using a section of S ( H + ). For instance if we take thesection φ then φ ∗ : H ∗− ( b + − ( B ; Z ) → H ∗ ( S ( H + ); Z w ) is a right inverse to π ∗ and φ ∗ : H ∗ ( S ( H + ); Z w ) → H ∗ ( B ; Z w ) is a left inverse of π ∗ . Thus φ induces anisomorphism H ∗ ( S ( H + ); Z w ) ∼ = H ∗ ( B ; Z w ) ⊕ H ∗− ( b + − ( B ; Z ) , which can moreover be thought of as an isomorphism of H ∗ ( B ; Z )-modules. Itfollows in particular that to each section φ there is a uniquely determined class τ φ ∈ H b + − ( S ( H + ); Z ) satisfying π ∗ ( τ φ ) = 1 and φ ∗ ( τ φ ) = 0. By the above splittingwe also have that φ ∗ (1) = π ∗ ( α φ ) τ φ + π ∗ ( β φ )for uniquely determined classes α φ ∈ H ( B ; Z ) ∼ = Z and β φ ∈ H b + − ( B ; Z w ).Then since π ◦ φ = id , we have1 = π ∗ φ ∗ (1) = α φ π ∗ ( τ φ ) + π ∗ ( π ∗ ( β φ )) = α φ . So α φ = 1. Next, let e φ ∈ H b + − ( B ; Z w ) denote the Euler class of h φ i ⊥ , theorthogonal complement of h φ i in H + . Then since h φ i ⊥ can be identified with thenormal bundle of φ ( B ), we have e φ = φ ∗ ( φ ∗ (1)) = φ ∗ ( τ φ ) + φ ∗ π ∗ ( β φ ) = β φ . So β φ = e φ and we have shown that(5.4) φ ∗ (1) = τ φ + π ∗ ( e φ ) . Now if ψ is another section we have that π ∗ ( τ φ − τ ψ ) = 0 and hence(5.5) τ φ − τ ψ = π ∗ ( λ ( φ, ψ )) N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 61 for a uniquely determined class λ ( φ, ψ ) ∈ H b + − ( B ; Z w ). Combining equations(5.4) and (5.5) with the definition of θ ( φ, ψ ) we see that:(5.6) θ ( φ, ψ ) = λ ( φ, ψ ) + e φ − e ψ . Next, consider τ φ . By the Gysin sequence we have τ φ = π ∗ ( a φ ) ` τ φ + π ∗ ( b φ )for uniquely determined classes a φ ∈ H b + − ( B ; Z w ) and b φ ∈ H b + − ( B ; Z ).From φ ∗ ( τ φ ) = 0 we see that b φ = 0. To compute a φ we first note that τ φ isPoincar´e dual to the self-intersection of φ ( B ). But the normal bundle to φ ( B ) isisomorphic to h φ i ⊥ , which implies that π ∗ ( φ ∗ (1) ` φ ∗ (1)) = e φ . On the other hand, using (5.4), we find e φ = π ∗ ( φ ∗ (1) ` φ ∗ (1))= π ∗ (( τ φ + π ∗ ( e φ )) ` ( τ φ + π ∗ ( e φ )))= π ∗ (cid:16) τ φ + π ∗ ( e φ ) ` τ φ + ( − b + − π ∗ ( e φ ) ` τ φ + π ∗ ( e φ ) (cid:17) = a φ + e φ + ( − b + − e φ . Therefore a φ = ( − b + e φ and we have shown that: τ φ = ( − b + π ∗ ( e φ ) ` τ φ . Let ( −
1) : S ( H + ) → S ( H + ) denote the antipodal map. Note first of all that ( − b + . Thus( − ∗ ◦ π ∗ ( x ) = ( − ∗ ( π ∗ ( x ) `
1) = π ∗ ( x ) ` ( − ∗ (1) = ( − b + π ∗ ( x )for any x ∈ H ∗ ( B ; Z w ). Using the splitting of the Gysin sequence we have that( − ∗ τ φ = c φ τ φ + π ∗ ( γ φ )for some c φ ∈ Z and some γ φ ∈ H b + − ( B ; Z w ). Since π ◦ ( −
1) = π , we have that π ∗ ◦ ( − ∗ τ φ = π ∗ τ φ = 1 and hence c φ = 1. To compute γ φ , we note that φ ( B ) and − φ ( B ) are disjoint so φ ∗ (1) ` − φ ∗ (1) = 0. Therefore,0 = π ∗ ( φ ∗ (1) ` − φ ∗ (1))= π ∗ (( τ φ + π ∗ ( e φ )) ` ( − ∗ ( τ φ + π ∗ ( e φ )))= π ∗ (cid:16) ( τ φ + π ∗ ( e φ )) ` ( τ φ + π ∗ ( γ φ + ( − b + e φ )) (cid:17) = π ∗ (cid:16) τ φ + π ∗ ( e φ ) ` τ φ + ( − b + − π ∗ ( γ φ + ( − b + e φ ) ` τ φ (cid:17) = ( − b + e φ + e φ + ( − b + − ( γ φ + ( − b + e φ )= ( − b + e φ + ( − b + − γ φ . Hence γ φ = e φ and we have shown that(5.7) ( − ∗ τ φ = τ φ + π ∗ ( e φ ) . Putting all of these calculations together, we have:
Obs ( φ, ψ ) = π ∗ ( φ ∗ (1) ` − ψ ∗ (1))= π ∗ (( τ φ + π ∗ ( e φ )) ` ( − ∗ ( τ ψ + π ∗ ( e ψ )))= π ∗ (cid:16) ( τ φ + π ∗ ( e φ )) ` ( τ ψ + π ∗ ( e ψ + ( − b + e ψ )) (cid:17) = π ∗ (cid:16) τ φ τ ψ + π ∗ ( e φ ) ` τ ψ + ( − b + − π ∗ ( e ψ ) ` τ φ − π ∗ ( e ψ ) ` τ φ (cid:17) = π ∗ (( τ ψ + π ∗ ( λ ( φ, ψ )) ` τ ψ ) + e φ + ( − b + − e ψ − e ψ = π ∗ ( τ ψ ) + λ ( φ, ψ ) + e φ + ( − b + − e ψ − e ψ = ( − b + e ψ + λ ( φ, ψ ) + e φ + ( − b + − e ψ − e ψ = λ ( φ, ψ ) + e φ − e ψ . Comparing with Equation (5.6), we see that
Obs ( φ, ψ ) = θ ( φ, ψ ). (cid:3) Using this result, the wall crossing formula for the families Seiberg-Witten invari-ants (Theorem 5.2) can be re-written in terms of the obstruction class
Obs ( φ, ψ ): Corollary 5.5.
Given φ, ψ ∈ CH ( f ) , we have: SW m ( f, φ ) − SW m ( f, ψ ) = ( if m < d − ,Obs ( φ, ψ ) s m − ( d − ( D ) if m ≥ d − . The following proposition gives some further properties of the obstruction class
Obs ( φ, ψ ) depending on the parity of b + : Proposition 5.6.
We have the following: • Suppose b + is even. Then e φ = e ψ for any two sections φ, ψ and hence Obs ( φ, ψ ) = λ ( φ, ψ ) . • Suppose b + is odd. Then λ ( φ, ψ ) = − e φ + e ψ and · Obs ( φ, ψ ) = e φ − e ψ .Thus if H b + − ( B ; Z w ) has no -torsion we may write Obs ( φ, ψ ) = ( e φ − e ψ ) .Proof. Recall that τ φ = τ ψ + π ∗ ( λ ( φ, ψ )). Applying ( − ∗ and using (5.7) we find τ φ + π ∗ ( e φ ) = τ ψ + π ∗ ( e ψ ) + ( − b + π ∗ ( λ ( φ, ψ )= τ φ + π ∗ ( e ψ + ( − b + λ ( φ, ψ ) − λ ( φ, ψ ))and hence − λ ( φ, ψ ) + ( − b + λ ( φ, ψ ) = e φ − e ψ . If b + is even, this gives e φ = e ψ and thus Obs ( φ, ψ ) = λ ( φ, ψ ) + e φ − e ψ = λ ( φ, ψ ).If b + is odd, then we get 2 λ ( φ, ψ ) = − e φ + e ψ and hence2 · Obs ( φ, ψ ) = 2 λ ( φ, ψ ) + 2( e φ − e ψ ) = e φ − e ψ as claimed. (cid:3) Proposition 5.7.
Suppose H + is trivialisable. Choose a trivialisation and identifysections φ, ψ with maps φ, ψ : B → S b + − . Choose an orientation on H + and henceon S b + − so that Obs ( φ, ψ ) may be considered as a class in H b + − ( B ; Z ) . Then Obs ( φ, ψ ) = ( − b + − ( φ ∗ ( ν ) − ψ ∗ ( ν )) where ν ∈ H b + − ( S b + − ; Z ) is the generator compatible with the chosen orientation. N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 63
Proof.
Using S ( H + ) = B × S b + − one easily checks that τ φ = ν − φ ∗ ( ν ). Therefore λ ( φ, ψ ) = − φ ∗ ( ν ) + ψ ∗ ( ν ). For any section φ , the orthogonal complement h φ i ⊥ of φ in H + is isomorphic to the pullback of the tangent bundle of S b + − under φ : B → S b + − . Thus e φ = φ ∗ ( e ( T S b + − )), which is zero if b + is even and 2 φ ∗ ( ν )is b + is odd. Thus is b + is odd we find Obs ( φ, ψ ) = λ ( φ, ψ ) + e φ − e ψ = − φ ∗ ( ν ) + ψ ∗ ( ν ) + 2 φ ∗ ( ν ) − ψ ∗ ( ν ) = φ ∗ ( ν ) − ψ ∗ ( ν ) . On the other hand if b + is even, then Obs ( φ, ψ ) = λ ( φ, ψ ) = − φ ∗ ( ν ) + ψ ∗ ( ν ) andthe result has been proven. (cid:3) Unparametrised wall crossing formula.
In this subsection we will see howto recover the general wall crossing formula for the unparametrised Seiberg-Witteninvariants of a 4-manifold with b even, as in [14].Let X be a compact, oriented smooth 4-manifold with b + ( X ) = 1 and b ( X )even. Let s denote a spin c -structure on X . Fix an orientation of H + and let ω ∈ H + denote the unique class for which a reducible solution exists. Having fixedsuch an orientation, the set CH ( f ) consist of two chambers which we denote by+ , − and define as follows: we say that a class η ∈ H + is in the + chamber if η − ω agrees with the given orientation on H + and similarly η is in the − chamber if η − ω gives the opposite orientation. Choose a metric g and a generic perturbation η representing the + chamber. Let M + denote the moduli space of solutionsto the Seiberg-Witten equations with respect to ( X, s , g, η ) and let L denote thenatural line bundle over M + . Then M + is a smooth compact manifold of dimension2 d − b + ( X ) − b ( X ) = 2( d − b ( X ) / d = c ( s ) − σ ( X )8 . Assume that thisdimension is non-negative, i.e. that ( d −
1) + b ( X ) / ≥
0. Let T denote the torusof gauge equivalence classes of U (1)-connections on the determinant line bundle of s with harmonic curvature. This is a torsor for the Jacobian Jac ( X ) = H ( X ; R / Z ),so in particular T is a torus of dimension b ( X ). The inclusion of M + into theconfiguration space defines a natural map π : M + → T and this map correspondsto the fact that when we take the finite dimensional approximation of the monopolemap f , the base space is taken as B = T .The ordinary Seiberg-Witten invariant SW + ( X, s ) of ( X, s ) with respect to thechamber + is given by: SW + ( X, s ) = Z M + c ( L ) ( d − b ( X ) / = Z T π ∗ ( c ( L ) ( d − b ( X ) / )= Z T SW ( d − b ( X ) / ( f, +) , where f denotes a finite dimensional approximation of the monopole map. A similarequality holds for the − chamber.A short calculation shows that Obs (+ , − ) = 1. Now the wall-crossing formula(Theorem 5.2) implies that: SW + ( X, s ) − SW − ( X, s ) = Z T s b ( X ) / ( D ) , where s b ( X ) / ( D ) is the b ( X ) / D ∈ K ( T )of the natural family of Dirac operators on the fibres of X × T → T . Let { y i } denote a basis of H ( X ; Z ) and let { x i } be the corresponding dual basisof H ( T ; Z ) ∼ = Hom ( H ( X, Z ); Z ). Let Ω = P i x i ` y i ∈ H ( X × T ; Z ) denotethe first Chern class of the Poincar´e line bundle. Then the families index theoremgives: Ch ( D ) = Z X e ( c ( s ))+Ω ∧ (cid:18) − σ ( X )8 vol X (cid:19) , where R X is understood to mean integration over the fibres of X × T → T . Next werecall that if X is a compact oriented 4-manifold with b + ( X ) = 1 then y ` y ` y ` y = 0 for any y , y , y , y ∈ H ( X ; Z ) (see [14, Lemma 2.4]). It follows thatΩ = 0 and thus Ch ( D ) = Z X (cid:18) c ( s )2 + c ( s ) (cid:19) (cid:18) + 16 Ω (cid:19) (cid:18) − σ ( X )8 vol X (cid:19) = Z X (cid:18) c ( s )2 + c ( s ) (cid:19) (cid:18) (cid:19) (cid:18) − σ ( X )8 vol X (cid:19) = c ( s ) − σ ( X )8 + 14 Z X c · Ω = d + 14 Z X c · Ω . Hence c ( D ) = d , c ( D ) = R X c ( s ) · Ω and Ch i ( D ) = 0 for i > Ch i ( D )denotes the degree 2 i part of Ch ( D )). Using the splitting principle one can verifythat if Ch ( V ) = X n ≥ n ! p n , where p n ∈ H n ( T ; Z )for a virtual bundle V , then s ( V ) = exp X n ≥ ( − n n p n . In particular in the case V = D , we immediately get s j ( D ) = ( − j j ! α j , where α = R X c ( s )2 · Ω . Putting j = b ( X ) / SW + ( X, s ) − SW − ( X, s ) = 1( b ( X ) / Z T ( − α ) b ( X ) / , which is precisely the general wall crossing formula [14, Theorem 1.2].6. Divisibility conditions on families Seiberg-Witten invariants from K -theory We have seen how mod 2 conditions on the families Seiberg-Witten invariantsarise via computation of their Steenrod squares. In this section we consider adifferent approach using K -theory and the Chern character, which yields variousdivisibility conditions on the families Seiberg-Witten invariants modulo torsion.This approach was used by Bauer and Furuta in [3] to obtain divisibility conditionsof the Seiberg-Witten invariants in the unparametrised setting. N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 65 K -theoretic Seiberg-Witten invariants and divisibility conditions. We will work with the S -equivariant K-theory and cohomology of various spaces.Recall that H ∗ S ( pt ; Z ) is isomorphic to Z [ x ], the ring of polynomials in x with inte-ger coefficients. Let b H ∗ S ( pt ; Z ) denote the completion of H ∗ S ( pt ; Z ) with respect tothe filtration by degree. Then b H ∗ S ( pt ; Z ) = b Z [ x ], the ring of formal power series in x with integer coefficients. Let R [ S ] = K ∗ S ( pt ) denote the S -equivariant K -theoryof a point, which is also the representation ring of S . We let S act on C by scalarmultiplication. This defines a representation of S and we let ξ = [ C ] ∈ R [ S ].Then R [ S ] = Z [ ξ, ξ − ] is the ring of Laurent polynomials in ξ .For any space X and abelian group A , we let b H ∗ ( X ; A ) denote the completionof H ∗ ( X ; A ) with respect to the filtration by degree. We use similar notation forthe completion of the equivariant cohomology groups of a G -space. The Cherncharacter defines a ring homomorphism Ch : K ∗ ( X ) → b H ev/odd ( X ; Q ). We usethis to define the equivariant Chern character as follows. For any G -space X , let X G = X × G EG denote the homotopy quotient. We define the equivariant Cherncharacter as the composition: K ∗ G ( X ) π ∗ −→ K ∗ G ( X × EG ) = K ∗ ( X G ) Ch −→ b H ev/odd ( X G ; Q ) = b H ev/oddG ( X ; Q )where π : X × EG → X is the projection to X . Example 6.1.
For G = S and X = pt , we obtain a Chern character homomor-phism Ch : R [ S ] → b H ∗ S ( pt ; Q ). Using BS = CP ∞ , one easily checks that theChern character is given by: Ch : Z [ ξ, ξ − ] → b Q [ x ] , Ch ( ξ k ) = e kx . Let S act trivially on B and recall that we defined H ∗ = H ∗ S ( B ; Z ) = H ∗ ( B ; Z )[ x ].Similarly we define K ∗ = K ∗ S ( B ) = R [ S ] ⊗ K ∗ ( B ) = K ∗ ( B )[ ξ, ξ − ] . The equivariant Chern character for B takes the form: Ch : K ∗ → b H ev/odd Q Ch ( α ⊗ ξ k ) = Ch ( α ) e kx , α ∈ K ∗ ( B ) , where b H ev/odd Q = b H ev/odd ⊗ Q .Recall the cohomological description of the families Seiberg-Witten invariants:we have a finite dimensional monopole map f : ( S V,U , B
V,U ) → ( S V ′ ,U ′ , B V ′ ,U ′ )satisfying Assumption 1. Let [ φ ] ∈ CH ( f ) be a chamber represented by a section φ : B → U ′ whose image is disjoint from U . Assume also that H + is orientable andfix an orientation. Since φ avoids U , we obtain a pushforward map: φ ∗ : H jS ( B ; Z ) → H j +2 a ′ + b ′ S ( S V ′ ,U ′ , S U ; Z ) . Then for m ≥
0, the m -th Seiberg-Witten invariant of ( f, φ ) is given by: SW m ( f, φ ) = ( π Y ) ∗ ( x m ` f ∗ φ ∗ (1)) ∈ H m − (2 d − b + − ( B, Z ) . Imitating this definition in K -theory, we will obtain K -theoretic Seiberg-Witteninvariants. To do this, we will assume that H + admits a spin c -structure. Thenwe can also assume that the bundles U and U ′ admit spin c -structures such that U ′ = H + ⊕ U holds at the level of spin c vector bundles. Then since the normal bundle of φ ( B ) ⊂ S V ′ ,U ′ is equivariantly K -oriented, we obtain a pushforward mapin equivariant K -theory: φ ∗ : K jS ( B ) → K j + b ′ S ( S V ′ ,U ′ , S U ) . Similarly, the vertical tangent bundle of π Y : Y → B has a spin c -structure and weobtain a pushforward map( π Y ) ∗ : K ∗ ( Y, ∂Y ) → K ∗ + b − ( B ) . Definition 6.2.
Suppose that H + is equipped with a spin c -structure. Then foreach m ≥
0, we define the m -th K -theoretic Seiberg-Witten invariant of ( f, φ ) by: SW Km ( f, φ ) = ( π Y ) ∗ ( ξ m ` f ∗ φ ∗ (1)) ∈ K b + − ( B ) . Remark . We make some remarks on this definition: • It is possible to define SW Km ( f, φ ) even when H + is not spin c providedone works with twisted K -theory . For simplicity we will not discuss thisgeneralisation. • One can check that SW Km ( f, φ ) depends only on the stable homotopy classof f . The proof is analogous to that of Proposition 3.8. • One can give a more geometric interpretation of SM Km ( f, φ ) analogousto the cohomological Seiberg-Witten invariants. Let M φ be the familiesSeiberg-Witten moduli space (after perturbing f to obtain a smooth mod-uli space) and let π M φ : M φ → B be the projection. Then arguing in muchthe same way as we did for the cohomological invariants, we see that: SW Km ( f, φ ) = ( π M φ ) ∗ ( L m )where L → M φ is the natural line bundle over M φ . Note that a spin c -structure on H + induces one on M φ . • In the case that B = { pt } is a point and b + is odd, it follows that SW Km ( f, φ ) ∈ K ( pt ) = Z is the index of the spin c Dirac operator on M φ coupled to theline bundle L m .For a complex vector bundle V over B , let T d S ( V ) ∈ b H ev Q denote the S -equivariant Todd class of V , where as usual S acts on V by scalar multiplication.This can be expanded as a formal power series in x : T d S ( V ) = X j ≥ T d j ( V ) x j for some characteristic classes T d j ( V ) ∈ H ev ( B ; Q ).Recall that Spin c ( n ) = Spin ( n ) × ± U (1) and therefore we obtain a homomor-phism ϕ : Spin c ( n ) → U (1) which sends ( g, z ) ∈ Spin ( n ) × U (1) to z . Supposethat E is a rank n vector bundle with a spin c structure. The homomorphism ϕ applied to the principal Spin c ( n )-bundle of E determines a line bundle. We callthis the spin c line bundle of E . Theorem 6.4.
Let κ ∈ H ( B ; Z ) be the first Chern class of the spin c line bundleassociated to H + . Then the K -theoretic and cohomological Seiberg-Witten invari-ants are related by: Ch ( SW Km ( f, φ )) = e − κ/ ˆ A ( H + ) − X j ≥ T d j ( D ) X k ≥ m k k ! SW j + k ( f, φ ) ∈ H ∗ ( B ; Q ) . N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 67
Proof.
Recall from Subsection 3.1 the definition of the spaces Y and e Y . Let π Y : Y → B and π e Y : e Y → B be the projections to B . Let V = Ker (( π Y ) ∗ ) denote thekernel of ( π Y ) and similarly let e V = Ker (( π e Y ) ∗ ). By definition e Y is the complementof a tubular neighbourhood of S U in S V,U . Let ˆ A S ( E ) denote the S -equivariant ˆ A -genus of an equivariant vector bundle E . Then ˆ A S ( e V ) = ˆ A S ( T vert S V,U ) | e X , where T vert S V,U denotes the vertical tangent bundle of S V,U → B . But T vert S V,U ⊕ R ∼ = V ⊕ U ⊕ R and so it follows that ˆ A S ( e V ) = ˆ A S ( V ⊕ U ⊕ R ) = ˆ A S ( V ⊕ U ). Nextrecall that Y = e Y /S is the quotient of e X by a free S -action. It follows thatˆ A ( V ) = ˆ A S ( e V ) = ˆ A S ( V ⊕ U )where we have identified S -equivariant cohomology on e Y with ordinary cohomol-ogy on Y .Since assuming that H + admits a spin c structure, we can after stabilisationassume that U ′ = H + ⊕ U as spin c vector bundles. For a spin c vector bundle W we let κ W denote the first Chern class of the associated line bundle of the spin c -structure. In particular κ = κ H + and κ U ′ = κ + κ U . Recall that SW Km ( f, φ ) is defined by SW Km ( f, φ ) = ( π Y ) ∗ ( ξ m ` f ∗ φ ∗ (1)). Now weproceed by direct calculation using Grothendieck-Riemann Roch: Ch ( SW Km ( f, φ )) = Ch (( π Y ) ∗ ( ξ m ` f ∗ φ ∗ (1)))= ( π Y ) ∗ (cid:16) Ch ( ξ m ` f ∗ φ ∗ (1)) e κ V / ˆ A ( V ) (cid:17) = ( π Y ) ∗ (cid:16) Ch ( ξ m ` f ∗ φ ∗ (1)) T d S ( V ) e κ U / ˆ A ( U ) (cid:17) = e κ U / ˆ A ( U ) · ( π Y ) ∗ (cid:16) e mx ` f ∗ ( Ch ( φ ∗ (1))) T d S ( V ) (cid:17) = e κ U / ˆ A ( U ) · ( π Y ) ∗ (cid:16) e mx ` f ∗ ( φ ∗ (1)) T d S ( V ′ ) − e − κ U ′ / ˆ A ( U ′ ) − T d S ( V ) (cid:17) = e ( κ U − κ U ′ ) / ˆ A ( U ) ˆ A ( U ′ ) − · ( π Y ) ∗ (cid:16) e mx ` f ∗ ( φ ∗ (1)) T d S ( V ) T d S ( V ′ ) − (cid:17) = e − κ/ ˆ A ( H + ) − · ( π Y ) ∗ (cid:16) e mx ` f ∗ ( φ ∗ (1)) T d S ( D ) (cid:17) = e − κ/ ˆ A ( H + ) − · X j ≥ T d j ( D ) X k ≥ m k k ! · ( π Y ) ∗ (cid:0) x j + k ` f ∗ ( φ ∗ (1)) (cid:1) = e − κ/ ˆ A ( H + ) − · X j ≥ T d j ( D ) X k ≥ m k k ! · SW j + k ( f, φ ) . (cid:3) Example 6.5.
Let X be a compact, oriented, smooth 4-manifold with b ( X ) = 0and b + ( X ) = 2 p + 1 odd. Let s be a spin c -structure on X and put d = c ( s ) − σ ( X )8 .Then we take B = { pt } . So ˆ A ( H + ) = 1, κ = 0 and T d S ( D ) = (cid:18) x − e − x (cid:19) d . So T d j ( D ) = c j,d , where c j,d is the coefficient of x j in (cid:16) x − e − x (cid:17) d . Assume that d ≥ p + 1. The only non-zero Seiberg-Witten invariant is SW d − p − ( f, φ ) is the ordinarySeiberg-Witten invariant of X (with respect to the chamber φ if b + ( X ) = 1). Onthe other hand SW Km ( f, φ ) ∈ Z is the index of the spin c Dirac operator coupledto L m on the Seiberg-Witten moduli space, which we denote as Ind ( D ⊗ L m ).Theorem 6.4 gives: SW Km ( f, φ ) = d − p − X j =0 c d − p − − j,d m j j ! SW ( X, s , φ ) . However, we notice that P d − p − j =0 c d − p − − j,d m j j ! is the x d − p − -coefficient of (cid:18) x − e − x (cid:19) d e mx . But this is the same as: Z CP d − (cid:18) x − e − x (cid:19) d x p e mx = Z CP d − T d ( CP d − ) x p Ch ( O ( m ))where now x = c ( O (1)). Let us denote this quantity by n ( d, m, p ). Clearly when p = 0, n ( d, m, p ) is the Dolbeault index of O ( m ), which is (cid:0) m + d − m (cid:1) .Define rational numbers a p,l to be the coefficients of the Taylor expansion (c.f.[3, Theorem 3.7]): log (1 − y ) p = ∞ X l =0 a p,l y p + l . Then x p = ( − p X l ≥ a p,l (1 − e − x ) p + l and thus n ( d, m, p ) = ( − p X l ≥ a p,l Z CP d − T d ( CP d − ) Ch ((1 − ξ − ) p + l ξ m )= ( − p X l ≥ a p,l Res | x =0 (cid:18) − e − x ) d (1 − e − x ) p + l e mx (cid:19) = ( − p d − p − X l =0 a p,l Res | x =0 (cid:18) − e − x ) d − p − l e mx (cid:19) = ( − p d − p − X l =0 a p,l (cid:18) m + d − p − l − m (cid:19) . So we have shown that:
Ind ( D ⊗ L m ) = ( − p d − p − X l =0 a p,l (cid:18) m + d − p − l − m (cid:19)! SW ( X, s , φ ) . Let us set n = d − p − q ( m ) = Ind ( D ⊗ L m ), so that q ( m ) = ( − p n X l =0 a p,n − l (cid:18) m + ll (cid:19)! SW ( X, s , φ ) . N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 69
Evidently q ( m ) is a polynomial in m with rational coefficients. Moreover q ( m ) ∈ Z whenever m is a non-negative integer. Introduce the difference operator ∆ onpolynomial functions of m by: (∆ f )( m ) = f ( m +1) − f ( m ). Then ∆ (cid:0) m + kj (cid:1) = (cid:0) m + kj − (cid:1) by Pascal’s identity. Iterating, we get that ∆ k (cid:0) m + kk (cid:1) = 1 and ∆ r (cid:0) m + kk (cid:1) = 0 for all r > k . Now since ∆ l q ( m ) is integer valued for all l, m ≥
0, we see by induction on l that a p,n − l SW ( X, s , φ ) ∈ Z for l = 0 , , . . . , n . Thus SW ( X, s , φ ) is divisible bythe denominators of a p,l for l = 0 , , . . . , n . This is precisely the statement of [3,Theorem 3.7]. Proposition 6.6.
Suppose that H + is equipped with a spin c -structure and let κ bethe first Chern class of the spin c line bundle of H + . Suppose that the Chern classesof D are torsion. Then e κ/ ˆ A ( H + ) Ch ( SW Km ( f, φ )) = X r ≥ ( − d − r − r X l =0 a d − r − ,r − l (cid:18) m + ll (cid:19) SW r ( f, φ ) ∈ H ∗ ( B ; Q ) where the rational numbers a p,l are defined as in Example 6.5 (note that a p,l isdefined even when p < ).Proof. Since the Chen classes of D are torsion it follows that T d S ( D ) = (cid:16) x − e − x (cid:17) d and thus T d j ( D ) = c j,d , where c j,d are defined as in Example 6.5. Substituting intoTheorem 6.4 we get: e κ/ ˆ A ( H + ) Ch ( SW Km ( f, φ )) = X j,k ≥ c j,d m k k ! SW j + k ( f, φ )= X r ≥ X j + k = r c j,d m k k ! SW r ( f, φ )= X r ≥ n ( d, m, d − r − SW r ( f, φ )where the rational numbers n ( d, m, p ) are defined by n ( d, m, p ) = X j + k = d − p − c d − p − − j,d m j j ! . But from Example 6.5, we find that n ( d, m, d − r −
1) = ( − d − r − r X l =0 a d − r − ,r − l (cid:18) m + ll (cid:19) and thus e κ/ ˆ A ( H + ) Ch ( SW Km ( f, φ )) = X r ≥ ( − d − r − r X l =0 a d − r − ,r − l (cid:18) m + ll (cid:19) SW r ( f, φ ) . (cid:3) Corollary 6.7.
Suppose that B = S r is an even dimensional sphere and that f is a finite dimensional approximation of the monopole map of a spin c family of -manifolds π : E → B whose fibres are diffeomorphic to a -manifold X with b ( X ) = 0 and b + ( X ) = 2 p + 1 odd. Suppose that n = r + d − p − ≥ and that c r ( D ) = 0 . Then SW n ( f, φ ) ∈ H r ( S r ; Z ) ∼ = Z is divisible by the denominators of a p − r,l , for l = 0 , , . . . n .Proof. Since B is simply-connected, it follows that the bundle H = R π ∗ R over B whose fibres are degree 2 cohomology of the fibres of E → B is trivial. It followsthen that H + is also trivial. Thus H + admits a spin structure and therefore aspin c -structure with κ = 0. Also ˆ A ( H + ) = 1 since H + is trivial. From Proposition6.6 we get: Ch ( SW Km ( f, φ )) = d X j =0 ( − d − j − j X l =0 a d − j − ,j − l (cid:18) m + ll (cid:19) SW j ( f, φ ) ∈ H ∗ ( S r ; Q ) . But for a sphere, the Chern character takes values in integral cohomology. Inparticular, extracting the degree 2 r part, we find: n X l =0 a p − r,n − l (cid:18) m + ll (cid:19) SW n ( f, φ ) ∈ H r ( S r ; Z )for all m ≥
0, where n = r + d − p −
1. Arguing as in Example 6.5, this impliesthat SW n ( f, φ ) is divisible by the denominators of a p − r,l for l = 0 , , . . . , n . (cid:3) Remark . In the setting of Corollary 6.7, if r ≥
5, then it follows by the familiesindex theorem that c r ( D ) = 0 holds automatically. Remark . To be in the range where there is no wall crossing, we need b + >dim ( B ) + 1, which in the above corollary is equivalent to p > r . On the other handif r > p then the primary difference class Obs ( φ, ψ ) is easily seen to vanish, so therecan only be wall crossing if p = r .6.2. K -theoretic wall crossing formula. In this subsection we state a wallcrossing formula for the K -theoretic Seiberg-Witten invariants. As usual, let f :( S V,U , B
V,U ) → ( S V ′ ,U ′ , B V ′ ,U ′ ) be a finite dimensional monopole map. Let U ′ = U ⊕ H + and suppose that H + is equipped with a spin c -structure. As in the previ-ous subsection, let κ ∈ H ( B ; Z ) denote the first Chern class of the associated linebundle of this spin c -structure.Let [ φ ] , [ ψ ] ∈ CH ( f ) be chambers of f , represented by maps φ, ψ : B → S ( H + ).By the K -theoretic Gysin sequence for π : S ( H + ) → B , there exists a unique class Obs K ( φ, ψ ) ∈ K b + − ( B ) such that φ K ∗ (1) − ψ K ∗ (1) = π ∗ ( Obs K ( φ, ψ ))where we use superscript K to indicate pushforwards in K -theory. Clearly Obs K ( φ, ψ )is the K -theoretic analogue of the class Obs ( φ, ψ ) defined in § Proposition 6.10.
For any φ, ψ : B → S ( H + ) , we have Ch ( Obs K ( φ, ψ )) = e − κ/ ˆ A ( H + ) − Obs ( φ, ψ ) ∈ H ∗ ( B ; Q ) . Proof.
The normal bundle of φ ( B ) in S ( H + ) is isomorphic to h φ i ⊥ , the orthog-onal complement of φ in H + . But this is stably equivalent to H + . ThereforeGrothendieck-Riemann-Roch gives: Ch ( φ K ∗ (1)) = φ ∗ ( Ch (1) e − κ/ ˆ A ( H + ) − ) = e − κ/ ˆ A ( H + ) φ ∗ (1) . N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 71
Similarly, Ch ( ψ K ∗ (1)) = e − κ/ ˆ A ( H + ) ψ ∗ (1) . Taking differences gives: π ∗ ( Ch ( Obs K ( φ, ψ ))) = Ch ( π ∗ ( Obs K ( φ, ψ )))= Ch ( φ K ∗ (1) − ψ K ∗ (1))= e − κ/ ˆ A ( H + ) − ( φ ∗ (1) − ψ ∗ (1))= e − κ/ ˆ A ( H + ) − π ∗ ( Obs ( φ, ψ )) . But π ∗ : K ∗ ( B ) → K ∗ ( S ( H + )) is injective since S ( H + ) admits sections. Hence Ch ( Obs K ( φ, ψ )) = e − κ/ ˆ A ( H + ) − Obs ( φ, ψ ). (cid:3) Lemma 6.11.
Let V → B be a complex vector bundle over B of rank a ≥ and let π P ( V ) : P ( V ) → B be the associated projective bundle. Set ξ = O V (1) . Let a ′ ≥ be a non-negative integer and set d = a − a ′ . For any W ∈ K ( B ) and any integer m , define S + m ( W ) , S − m ( W ) ∈ K ( B ) by: S + m ( W ) = ( Sym m ( W ∗ ) if m ≥ , if m < , S − m ( W ) = ( if m > − d, ( − d − Sym − m − d ( W ) if m ≤ − d. Also let S m ( W ) = S + m ( W ) + S − m ( W ) . Then: ( π P ( V ) ) K ∗ ( ξ m (1 − ξ − ) a ′ ) = S m ( V − C a ′ ) . Proof.
For any integer m , let us set q m = ( π P ( V ) ) K ∗ ( ξ m ) and h m = ( π P ( V ) ) K ∗ ( ξ m (1 − ξ − ) a ′ ). Then by expanding (1 − ξ − ) a ′ , we find:(6.1) h m = a ′ X j =0 ( − j q m − j (cid:18) a ′ j (cid:19) . Next we observe that q m = ( π P ( V ) ) K ∗ ( O V ( m )) is the families index of O V ( m ). But P ( V ) → B has a fibrewise complex structure and the families index correspondsto taking fibrewise index of the Dolbeault complex coupled to O V ( m ). Hence for m ≥ q m is represented by the vector bundle over B whose fibres consist ofhomogeneous degree m polynomials, so q m = Sym m ( V ∗ ) for m ≥
0. Combinedwith Serre duality we find that for any m ∈ Z q m = q + m + q − m , where q + m = ( Sym m ( V ∗ ) if m ≥
00 if m < , q − m = ( m > − a, ( − a − Sym − m − a ( V ) if m ≤ − a. The from Equation (6.1) we have h m = h + m + h − m , where(6.2) h ± m = a ′ X j =0 ( − j q ± m − j (cid:18) a ′ j (cid:19) . To complete the lemma, it suffices to show that h ± m = S ± m ( V − C a ′ ). Consider the+ case first. Introduce the following formal power series in t with coefficients in K ( B ): Q + = X m ≥ q + m t m = X m ≥ Sym m ( V ∗ ) t m . By the splitting principle we may write V = L + · · · + L a for some line bundles.Then we have Q + = a Y j =1 − tL ∗ j ) . Next we note that since q + m = 0 for m <
0, then h + m = 0 for m < − a ′ . Hence setting H + = X m ∈ Z h + m t m , we have that H + is a formal Laurent series in t with coefficients in K ( B ). Thenfrom Equation (6.2), one finds that H + = Q + (1 − t ) a ′ = a Y j =1 − tL ∗ j ) (1 − t ) a ′ . Equating coefficients one sees that h + m = S + m ( V − C a ′ ).Now we consider the − case. Introduce the formal power series Q − = X m< q − m t − m = ( − a − t a X m ≥ Sym m ( V ) t m = ( − a − t a a Y j =1 − tL j ) . Similar to the + case, we find that h − m = 0 for m > − a , so we may define thefollowing formal Laurent series: H − = X m ∈ Z h − m t − m . Then from Equation (6.2), one finds that H − = Q − (1 − t − ) a ′ = ( − a − t a a Y j =1 − tL j ) ( − a ′ t − a ′ (1 − t ) a ′ = ( − d − t d a Y j =1 − tL j ) (1 − t ) a ′ . Equating coefficients, one sees that h − m = S − m ( V − C a ′ ). (cid:3) Theorem 6.12 ( K -theoretic wall crossing formula) . For any φ, ψ ∈ CH ( f ) andany integer m , we have: SW Km ( f, φ ) − SW Km ( f, ψ ) = Obs k ( φ, ψ ) S m ( D ) . where S m is defined as in Lemma 6.11.Proof. Since the proof is very similar to the cohomological wall crossing formulawe will give only a brief sketch. By stabilisation, we can assume U ′ , V ′ are trivial.Then U and U ′ are both spin c . We can also assume the rank of V is at least 2.Repeating the arguments of § SW Km ( f, φ ) − SW Km ( f, ψ ) = ( π Y ) K ∗ ( ξ m · f ∗ ( φ K ∗ (1) − ψ K ∗ (1))) N THE BAUER-FURUTA AND SEIBERG-WITTEN INVARIANTS 73 where φ, ψ : ( B, ∅ ) → ( S V ′ ,U ′ , S U ) are representatives for the chambers. We canassume that φ, ψ take values in S ( H + ). Then as before we find φ K ∗ (1) − ψ K ∗ (1) = ι K ∗ (( φ ) K ∗ (1) − ( ψ ) K ∗ (1)) = Obs K ( φ, ψ ) ι K ∗ (1)where φ , ψ are the corresponding maps φ , ψ : B → S ( H + ) and ι : ( S ( H + ) , ∅ ) → ( S V ′ ,U ′ , S U ) is the inclusion. The K -theoretic analogue of Lemma 5.1 is ι K ∗ (1) =(1 − ξ − ) a ′ δτ K ,U . Therefore we have: SW Km ( f, φ ) − SW Km ( f, ψ ) = Obs K ( φ, ψ )( π Y ) K ∗ ( ξ m (1 − ξ − ) a ′ δτ K ,U )= Obs K ( φ, ψ )( π P ( V ) ) K ∗ ( ξ m (1 − ξ − ) a ′ )= Obs K ( φ, ψ ) S m ( V − C a ′ )= Obs K ( φ, ψ ) S m ( D ) , where we used Lemma 6.11. (cid:3) References
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