aa r X i v : . [ m a t h . DG ] F e b NON-TRIVIAL SMOOTH FAMILIES OF K SURFACES
DAVID BARAGLIA
Abstract.
Let X be a complex K Diff ( X ) the group of diffeo-morphisms of X and Diff ( X ) the identity component. We prove that thefundamental group of Diff ( X ) contains a free abelian group of countablyinfinite rank as a direct summand. The summand is detected using familiesSeiberg–Witten invariants. The moduli space of Einstein metrics on X is usedas a key ingredient in the proof. Introduction
There is considerable interest in understanding the topology of diffeomorphismgroups of 4-manifolds. While much remains unknown there has been some recentprogress. • Ruberman gave examples of simply-connected smooth 4-manifolds for which π ( Dif f ( X )) → π ( Homeo ( X )) is not injective [15, 16]. • Watanabe constructed many non-trivial homotopy classes in
Dif f ( S ),thereby disproving the 4-dimensional Smale conjecture [19]. • Baraglia-Konno showed that π ( Dif f ( K → π ( Homeo ( K • Smirnov showed that if X is a hypersurface in CP of degree d = 1 , π ( Dif f ( X )) is non-trivial. Note that this result excludes K
3, whichcorresponds to hypersurfaces of degree d = 4 [17].The main result of this paper is the following. Theorem 1.1.
Let X be a K surface. Then π ( Dif f ( X ) ) contains a free abeliangroup of countably infinite rank as a direct summand. The direct summand in the above theorem is detected using families Seiberg–Witten invariants. The families Seiberg–Witten invariants were originally definedin [12]. In this paper we consider a reformulation of the Seiberg–Witten invariantswhich we now outline. Let X be a compact, oriented smooth 4-manifold with b + ( X ) >
1. Let s be a spin c -structure on X and let d ( X, s ) = c ( s ) − σ ( X )4 − b ( X ) − b + ( X )be the expected dimension of the Seiberg–Witten moduli space. If d ( X, s ) ≤ − sw s : π − d ( X, s ) − ( Dif f ( X ) ) → Z . The definition, roughly, is as follows. Let f ∈ π − d ( X, s ) − ( Dif f ( X ) ). Then bythe clutching construction, f defines a family E f → S − d ( X, s ) over the sphere with Date : February 15, 2021. fibres diffeomorphic to X . The moduli space of solutions to the Seiberg–Wittenequations on the family E f with spin c -structure s is compact and has expecteddimension d ( X, s ) − d ( X, s ) = 0. For a generic perturbation the families modulispace is a compact oriented 0-manifold and sw s ( f ) is defined as a signed count ofthe points of this moduli space. If b + ( X ) ≤ − d ( X, s ) + 1, then one has to deal withwall crossing phenomena. However, we show that in the above situation there is acanonically defined chamber and we take sw s ( f ) to be the Seiberg–Witten invariantdefined with respect to this chamber. This subtlety is crucial to this paper, sincewe will be concerned with the case b + ( X ) = 3 and d ( X, s ) = − sw s . Theorem 1.2.
Let X be a compact, oriented, smooth -manifold with b + ( X ) > .Then for each spin c -structure with d ( X, s ) = − ( n + 1) ≤ − , the map sw s : π n ( Dif f ( X ) ) → Z is a group homomorphism. Theorem 1.3.
Assume that b ( X ) = 0 . Then for any given f ∈ π n ( Dif f ( X ) ) , sw s ( f ) is non-zero for only finitely many spin c -structures with d ( X, s ) = − ( n + 1) . Theorem 1.3 is essentially a consequence of the compactness properties of theSeiberg–Witten equations. However there is a subtlety due to the chamber structureand wall crossing that requires some non-trivial arguments to overcome. Fromthese two theorems it follows that (for each n ≥
1) we can put the Seiberg–Witteninvariants together into a single homomorphism sw : π n ( Dif f ( X ) ) → M s | d ( X, s )= − ( n +1) Z , x M s sw s ( x ) . Now let X be a K { α ∈ H ( X ; Z ) | α = − } be the “roots” of X . For each α ∈ ∆ we get a unique spin c -structure s α charac-terised by c ( s α ) = 2 α . Then d ( X, s α ) = − sw α : π ( Dif f ( X )) → Z .Choose an element v ∈ H ( X ; R ) such that h v, δ i 6 = 0 for all δ ∈ ∆ and define∆ ± = { δ ∈ ∆ | ± h v, δ i > } . Then∆ = ∆ + ∪ ∆ − and δ ∈ ∆ + if and only if − δ ∈ ∆ − . The reason for splitting up ∆ this way is thatthe invariants sw α and sw − α are related to one another by the charge conjugationsymmetry of the Seiberg–Witten equations. In fact, sw α = − sw − α (see Proposition2.8).In § T Ein of Einstein metrics on X , which is to be thought of as an analogue of Teichmuller space for K T Ein is a universal family E Ein → T Ein . For each δ ∈ ∆ + , we construct ahomotopy class of map g δ : S → T Ein . Let E δ → S be the family over S obtainedby pulling back the universal family under g δ . We show that E δ is associated toan element h δ ∈ π ( Dif f ( X ) ) via the clutching construction. Then, using thegeometry of T Ein , we compute the families Seiberg–Witten invariant of E δ . Thisgives the following result. ON-TRIVIAL SMOOTH FAMILIES OF K Theorem 1.4.
Let α, δ ∈ ∆ + . Then sw α ( h δ ) = ( if α = δ, otherwise . Our main theorem follows directly from this.A brief outline of the paper is as follows. In § sw s : π n ( Dif f ( X ) ) → Z and prove several properties of theseinvariants, in particular Theorems 2.6 and 2.9. In § X is a K E Ein → T Ein over the Teichmullerspace T Ein and use this to contruct classes h δ ∈ π ( Dif f ( X ) ). We then computethe Seiberg–Witten invariants of these classes and our main theorem follows.2. The families Seiberg–Witten invariant revisited
In this section we will recall the definition of the families Seiberg–Witten in-variant. We will also show that the definition of the invariant can be extended tosituations where wall-crossing phenomena is present. We show that under certainconditions a distinguished chamber exists, hence we can still obtain a well-definedinvariant.Our approach to the families Seiberg–Witten invariant follows [12] but with someadditional modifications as in [2]. Let X be a compact smooth oriented 4-manifoldand let B be a compact smooth manifold. Suppose we have a smooth fibrewiseoriented fibre bundle π : E → B whose fibres are diffeomorphic to X . Such afibre bundle will be called a smooth family over B with fibres diffeomorphic to X .We assume throughout that B is connected. Choose a basepoint p ∈ B and adiffeomorphism X p ∼ = X , where X p = π − ( p ) denotes the fibre of E over p . Then π ( B, p ) acts by monodromy on the set of spin c -structures on X . Suppose that s is a monodromy invariant spin c -structure on X . Then by monodromy invariance, s can be uniquely extended to a continuously varying family of spin c -structures˜ s = { s b } b ∈ B on the fibres of E such that ˜ s | X p ∼ = s . Note that the existence of thecontinuous family ˜ s is in general a weaker condition than requiring the existence ofa spin c -structure on the vertical tangent bundle T ( E/B ) =
Ker ( π ∗ ). However, asexplained in [1], [2], the existence of ˜ s is sufficient to contruct a families Seiberg–Witten moduli space.Let d ( X, s ) = c ( s ) − σ ( X )4 − − b + ( X ) + b ( X )be the virtual dimension of the ordinary Seiberg–Witten moduli space of X . Let g = { g b } b ∈ B be a smoothly varying family of metrics on the fibres of E . Equivalently, g is a metric on the vertical tangent bundle T ( E/B ). Then we define H + g ( X ) to bethe vector bundle on B whose fibre over b ∈ B is the space H + g b ( X b ) of g b -self-dualharmonic 2-forms. By a families perturbation η we mean a smoothly varying family η = { η b } b ∈ B of real 2-forms on the fibres of E , such that η b is g b -self-dual. Let[ η b ] ∈ H + g b ( X b ) denote the L -othogonal projection of η b to the space of self-dualharmonic forms (using the L -metric defined by g b ). The map b [ η b ] defines asection of H + g ( X ), which we denote by [ η ]. DAVID BARAGLIA
Recall that the Seiberg–Witten equations for ( X b , s b , g b ) with perturbation η b are: D A ψ = 0 ,F + A + iη b = σ ( ψ ) , where A is a spin c -connection, ψ is a positive spinor for the spin c -structure s b and σ ( ψ ) denotes the imaginary the self-dual 2-form corresponding to the trace-freepart of ψ ∗ ⊗ ψ under Clifford multiplication. Let w : B → H + g ( X ) be the section of H + g ( X ) sending b to 2 π ( c ( s b )) + gb , the orthogonal projection of 2 πc ( s b ) to H + g b ( X b )using the L -metric defined by g b . Then the η b -perturbed Seiberg–Witten equationsfor ( X b , s b , g b ) admits reducible solutions if and only if [ η b ] = w . We refer to w asthe “wall” and we say that the families perturbation η does not lie on the wall iffor all b ∈ B , we have [ η b ] = w b .We define a chamber (of the families Seiberg–Witten equations) for ( E, s ) to bea connected component of the space of pairs ( g, η ), where g is a family of metricsand η is a family of perturbations not lying on the wall. In general, there areobstructions to the existence of chambers. For instance, if b + ( X ) = 0, then theredoes not exist a chamber. On the other hand, if b + ( X ) > dim( B ) + 1, then thereexists a unique chamber.Let C be a chamber of ( E, s ). Then for a sufficiently generic element ( g, η ) ∈ C ,the moduli space M ( E, s , g, η ) of gauge equivalence classes of solutions to theSeiberg–Witten equations on the fibres of E (with respect to the spin c -structure˜ s , metric g and perturbation η ) is a smooth, compact manifold of dimension d ( X, s X ) + dim( B ) (or is empty if this number is negative). Recall that a ho-mology orientation for X is a choice of orientation of H + ( X ) ⊕ H ( X ; R ) (here H + ( X ) denotes the space of harmonic self-dual 2-forms with respect to some met-ric on X . It is straightforward to see that the homology orientations for differentchoices of metrics can be canonically identified with one another). We say that ahomology orientation is monodromy invariant if it extends to a continuously varyingorientation on the family { H + g b ( X b ) ⊕ H ( X b ; R ) } b ∈ B .Let π : M ( E, s , g, η ) → B be the projection to B . A monodromy invarianthomology orientation defines an orientation on T M ( E, s ,g,η ) ⊕ π ∗ ( T B ), hence a Gysinhomomorphism π ∗ : H j ( M ( E, s , g, η ); Z ) → H j − d ( X, s ) ( B ; Z ) . We define the families Seberg–Witten invariant SW ( E, s , C , o ) ∈ H − d ( X, s ) ( B ; Z )of E with respect to the monodromy invariant spin c -structure s , the chamber C andmonodromy invariant homology orientation o to be SW ( E, s , C , o ) = π ∗ (1) ∈ H − d ( X, s ) ( B ; Z ) . The fact that SW ( E, s , C , o ) depends only on the chamber C and not on the par-ticular choice of pair ( g, η ) ∈ C follows by much the same argument as in theunparametrised case. A generic path between pairs ( g, η ) , ( g ′ , η ′ ) determines acobordism (relative B ) of the moduli spaces M ( E, s , g, η ) and M ( E, s , g ′ , η ′ ).Let π : E → B be a smooth family over B with fibres diffeomorphic to X .Let H ( X ) denote the local system over B whose fibre over b ∈ B is H ( X ) b = H ( X b ; R ). Assume that B is simply-connected and that b + ( X ) >
1. Choose a
ON-TRIVIAL SMOOTH FAMILIES OF K basepoint p ∈ B . Then parallel translation defines a trivialisation τ : H ( X ) → B × H ( X p ; R ) . Let H ⊆ H ( X p ; R ) be a maximal positive definite subspace with respect to theintersection form and let H ⊥ denote the orthogonal complement of H (with respectto the intersection form). This defines a decomposition H ( X p ; R ) ∼ = H ⊕ H ⊥ . Let ρ H : H ( X p ; R ) → H denote the projection to the first factor.Choose a smoothly varying family of metrics g = { g b } and let H + g ( X ) be definedas before. Let ι g : H + g ( X ) → H ( X ) be the inclusion. Let pr B : B × H ( X p ; R ) → B be the projection to B . Then the composition ϕ g = ( pr B × ρ H ) ◦ τ ◦ ι g : H + g ( X ) → B × H is an isomorphism of vector bundles. This follows since τ ( ι g ( H + g ( X ))) is a positivedefinite subbundle of B × H ( X p ; R ), so meets the negative definite subbundle B × H ⊥ in the zero section.Let s be a spin c -structure on X . Since B is simply-connected, s is automaticallymonodromy invariant and so continuously extends to the fibres of E .Let w : B → H + g ( X ) be the wall with respect to the spin c -structure s . Then ϕ g ( w ) is a section of the trivial bundle B × H . Let R g = sup b ∈ B || ϕ g ( w b ) || H where || || H is the norm on H induced by the restriction to H of the intersectionform on H ( X p ; R ). Since B is compact, R g is finite. Since b + ( X ) > H is anon-zero vector space and hence there exist elements of arbitrarily large norm. Let v be any element of H with || v || H > R g . Then the constant section b b × v isdisjoint from ϕ g ( w ). Therefore, the section v g = ϕ − g ( v ) of H + g ( X ) is disjoint fromthe wall w and hence defines a chamber, depending only on g and v which we willdenote by C ( g, v ). Lemma 2.1.
The chamber C ( g, v ) does not depend on the choice of the pair ( g, v ) .Proof. First we show that for fixed g , the chamber C ( g, v ) does not depend on thechoice of v . Let R g = sup b ∈ B || ϕ g ( w b ) || H be defined as before and let v, v ′ be anytwo elements of H with || v || H , || v ′ || H > R g . The space { x ∈ H | || x || H > R g } ishomotopic to a sphere of dimension b + ( X ) − b + ( X ) >
1. Therefore we can find a continuous path { v t } t ∈ [0 , in { x ∈ H | || x || H > R g } joining v to v ′ . It follows that ( g, ( v t ) g ) ∈ C ( g, v ) for all t ∈ [0 , g, ( v ′ ) g ) ∈ C ( g, v ) and C ( g, v ) = C ( g, v ′ ).Now let g, g ′ be two different families of metrics. We will show that there existsa v ∈ H for which C ( g, v ) = C ( g ′ , v ). Together with the above shown independenceof C ( g, v ) on v , this will show that C ( g, v ) does not depend on the choice of pair( g, v ).Choose a continuous path { g t } t ∈ [0 , of families of metrics from g to g ′ . Let R = sup b ∈ B,t ∈ [0 , || ϕ g t ( w b ) || H . Compactness of B × [0 ,
1] implies that R is finite. Now choose v ∈ H such that || v || H > R . Then || v || H > sup b ∈ B || ϕ g t ( w b ) || H for each t ∈ [0 , g t , v ) defines the same chamber for all t ∈ [0 , C ( g, v ) = C ( g ′ , v ). (cid:3) DAVID BARAGLIA
Definition 2.2.
Let π : E → B be a smooth family over B with fibres diffeomorphicto X and let s be a spin c -structure on X . Assume that B is simply-connected andthat b + ( X ) >
1. For any pair ( g, v ) with v ∈ { x ∈ H | || x || H > R g } , we let C ( E, s )denote the chamber containing ( g, v g ). By Lemma 2.1, we see that C ( E, s ) does notdepend on the choice of the pair ( g, v ). Furthermore, it is clear that C ( E, s ) doesnot depend on the choice of maximal positive definite subspace H ⊆ H ( X ; R ),because the space of all such subspaces is connected. We call C ( E, s ) the canonicalchamber for ( E, s ). Definition 2.3.
Let π : E → B be a smooth family over B with fibres diffeomorphicto X . Assume that B is simply-connected and that b + ( X ) >
1. Let s be aspin c -structure on X and let o be a homology orientation for X . Then we definethe (canonical) families Seiberg–Witten invariant SW ( E, s , o ) of ( E, s , o ) to be thefamilies Seiberg–Witten invariant of ( E, s , o ) defined using the canonical chamber: SW ( E, s , o ) = SW ( E, s , C ( E, s ) , o ) ∈ H − d ( X, s ) ( B ; Z ) . Let X be a compact, oriented, smooth 4-manifold with b + ( X ) >
1. Let
Dif f ( X )the group of orientation preserving diffeomorphisms of X with the C ∞ -topology andlet Dif f ( X ) be the identity component. Let n > f ∈ π n ( Dif f ( X ) ). Using the clutching construction, f defines atopological fibre bundle E f → S n +1 over S n +1 with fibres homeomorphic to X andstructure group Dif f ( X ) . From the main theorem of [13], it follows that E f canbe made into a smooth fibre bundle with fibres diffeomorphic to X in a unique way.Let s be a spin c -structure on X and o a homology orientation. Since n > S n +1 is simply-connected and b + ( X ) >
1, the families Seiberg–Witten invariant SW ( E f , s , o ) ∈ H − d ( X, s ) ( S n +1 ; Z )is defined. If d ( X, s ) = − ( n + 1), then we can evaluate SW ( E f , s , o ) against thefundamental class of S n +1 to obtain an integer invariant. Definition 2.4.
Let X be a compact, oriented, smooth 4-manifold with b + ( X ) > s be a spin c -structure such that d ( X, s ) = − ( n + 1) for some n ≥
0. We define sw s : π n ( Dif f ( X ) ) → Z by setting sw s ( f ) = Z S n +1 SW ( E r , s , o ) , where we have chosen a homology orientation o and an orientation on S n +1 . Wehave omitted from our notation the dependence of sw s on the choice of these orien-tations. A change in either orientation has the effect of changing sw s by an overallsign. Remark . The invariant sw s ( f ) ∈ Z can be interpreted as follows. Choose ageneric pair ( g, η ) ∈ C ( E f , s ). Then the moduli space M ( E, s , g, η ) is a compact,oriented 0-manifold and sw s ( f ) is simply the number of points of M ( E f , s , g, η ),counted with sign. Theorem 2.6.
Let X be a compact, oriented, smooth -manifold with b + ( X ) > .Then for each spin c -structure with d ( X, s ) = − ( n + 1) ≤ − , the map sw s : π n ( Dif f ( X ) ) → Z is a group homomorphism. ON-TRIVIAL SMOOTH FAMILIES OF K Proof.
Let f ∈ π n ( Dif f ( X ) ) and let E f → S n +1 be the corresponding familybuilt from the clutching construction. Choose a generic pair ( g, η ) ∈ C ( E f , s ).The moduli space M ( E, s , g, η ) is a finite set of points. Let p , . . . , p m ∈ S n +1 bethe finitely many points over which M ( E f , s , g, η ) lies. Choose a point p ∈ S n +1 and an open disc D ⊂ S n +1 around p such that D is disjoint from p , . . . , p m .Using cutoff functions it is possible to construct a smooth map ψ : S n +1 → S n +1 such that ψ ( D ) = { p } and ψ : S n +1 \ D → S n +1 \ { p } is a diffeomorphism. Nowconsider the pullback ψ ∗ ( E f ) of E f under ψ . The conditions on ψ ensures that ithas degree 1 as a map of S n +1 to itself. Hence ψ is homotopic to the identity. Itfollows that ψ ∗ ( E f ) is isomorphic to E f as topological fibre bundles over S n +1 withstructure group Dif f ( X ). The main theorem of [13] then implies that ψ ∗ ( E f ) and E f are isomorphic as smooth fibre bundles. The pullback ( ψ ∗ ( g ) , ψ ∗ ( η )) is a genericpair in C ( ψ ∗ ( E f ) , s ) and the moduli space M ( ψ ∗ ( E f ) , s , ψ ∗ ( g ) , ψ ∗ ( η )) is obviouslyobtained by pulling back M ( E f , s , g, η ) by ψ . Let X p denote the fibre of E f over p and fix a diffeomorphism X p ∼ = X . Since ψ takes the constant value p on D , therestriction of ψ ∗ ( E f ) to D is the constant family ψ ∗ ( E f ) | D ∼ = D × X p ∼ = D × X .Under this trivialisation of ψ ∗ ( E f ) | D we have that ψ ∗ ( g ) , ψ ∗ ( η ) get sent to theconstant pair ( g p , η p ).Now let f ′ be another element of π n ( Dif f ( X ) ) and let E f ′ → S n +1 be the fam-ily corresponding to f ′ . Choose a generic pair ( g ′ , η ′ ) ∈ C ( E f ′ , s ). After possiblyrotating S n +1 we can assume that no point of the moduli space M ( E f ′ , s , g ′ , η ′ ) liesover p . We can further assume that D was chosen small enough that no point ofthe moduli space M ( E f ′ , s , g ′ , η ′ ) lies over D . We then obtain the pullback family ψ ∗ ( E f ′ ) with generic pair ( ψ ∗ ( g ′ ) , ψ ∗ ( η ′ )). Moreover, we have a trivialisation of ψ ∗ ( E f ′ ) | D in which ψ ∗ ( g ′ ), ψ ∗ ( η ′ ) are sent to the constant pair ( g ′ p ′ , η ′ p ′ ).Let D ′ ⊂ D be a smaller open disc around p so that the complement D \ D ′ isdiffeomorphic to S × (0 , S n +1 \ D ′ to each other usinga neck S × [0 ,
1] (where we have identified the boundary of D ′ with S ). Theresulting space ( S n +1 \ D ′ ) ∪ S ( S × [0 , ∪ S ( S n +1 \ D ′ )is just the connected sum of two copies of S n +1 , which of course is just anothercopy of S n +1 . Using the trivialisation ψ ∗ ( E f ) | D ∼ = D × X and ψ ∗ ( E f ′ ) | D ∼ = D × X , we can attach ψ ∗ ( E f ) | S n +1 \ D ′ to ψ ∗ ( E f ′ ) | S n +1 \ D ′ by taking a constant family( S × [0 , × X along the neck. Let E denote the resulting family. Since ψ ∗ ( E f )and ψ ∗ ( E f ′ ) are isomorphic to E f and E f ′ , it is clear that E is isomorphic to thefamily obtained by applying the clutching construction to f + f ′ , where + denotesthe group operation on π n ( Dif f ( X ) ).Next, since d ( X, s ) = − ( n + 1) ≤ −
2, it follows the moduli space of solutionsto the Seiberg–Witten equations for a 1-parameter family with fibres ( X, s ) hasexpected dimension d ( X, s ) + 1 = − n <
0. Therefore, for a generic path ( g t , η t )from ( g p , η p ) to ( g ′ p , η ′ p ), there are no solutions to the Seibeg–Witten equationsfor ( X, s , g t , η t ). Now we define a pair (˜ g, ˜ η ) for the family E as follows. Re-stricted to ψ ∗ ( E f ) | S n +1 \ D ′ , we take the pair to be ( ψ ∗ ( g ) , ψ ∗ ( η )). Restricted to ψ ∗ ( E f ′ ) | S n +1 \ D ′ , we take the pair to be ( ψ ∗ ( g ′ ) , ψ ∗ ( η ′ )). Restricted to the constantfamily on the neck S × [0 , g t , η t ), where t ∈ [0 ,
1] isthe coordinate for the [0 ,
1] factor of the neck. Since there are no solutions to theSeiberg–Witten equations for ( g t , η t ), it is clear that the moduli space for ( E, s , ˜ g, ˜ η ) DAVID BARAGLIA is just the disjoint union M ( ψ ∗ ( E f ) , s , ψ ∗ ( g ) , ψ ∗ ( η )) ∪ M ( ψ ∗ ( E f ′ ) , s , ψ ∗ ( g ′ ) , ψ ∗ ( η ′ ))of the corresponding moduli spaces for ψ ∗ ( E f ) and ψ ∗ ( E f ′ ). It follows that Z S n +1 SW ( E, s , ˜ g, ˜ η ) = sw s ( f ) + sw s ( f ′ ) . To complete the proof, it remains to show that R S n +1 SW ( E, s , ˜ g, ˜ η ) = sw s ( f + f ′ ). Since E is isomorphic to the family obtained from applying the clutchingconstruction to f + f ′ , we just need to show that the pair (˜ g, ˜ η ) lies in the canonicalchamber ∈ C ( E, s ).Let H + ψ ∗ ( g ) ( X ), H + ψ ∗ ( g ′ ) ( X ) and H +˜ g ( X ) be the bundles of harmonic self-dual2-forms for the families of metrics ψ ∗ ( g ) , ψ ∗ ( g ′ ), and ˜ g . Then from the con-struction of ˜ g , we have that H +˜ g ( X ) is obtained by attaching H + ψ ∗ ( g ) ( X ) | S n +1 \ D ′ and H + ψ ∗ ( g ′ ) ( X ) | S n +1 \ D ′ to the bundle H + g t ( X ) over S × [0 ,
1] whose fibre over( x, t ) ∈ S × [0 ,
1] is the space of g t -self-dual harmonic 2-forms.Let H E ( X ) denote the local system whose fibres are the degree 2 cohomologyof the fibres of E (the subscript E is just to remind us which family H E ( X ) comesfrom). Similarly define the local systems H ψ ∗ ( E f ) ( X ) , H ψ ∗ ( E f ′ ) ( X ). Then H E ( X )is obtained by attaching H ψ ∗ ( E f ) ( X ) | S n +1 \ D ′ and H ψ ∗ ( E f ′ ) ( X ) | S n +1 \ D ′ to the con-stant local system over S × [0 ,
1] with fibre H ( X p ; R ). Choose a maximal positivedefinite subspace H ⊆ H ( X p ; R ) and let ρ H : H ( X p ; R ) → H be the projection.Taking the composition of inclusion and projection to H , we obtain isomorphisms ϕ ˜ g : H +˜ g ( X ) → B × H, ϕ ψ ∗ ( g ) : H + ψ ∗ ( g ) ( X ) → B × H, ϕ ψ ∗ ( g ′ ) : H + ψ ∗ ( g ′ ) ( X ) → B × H. Similarly, we obtain an isomorphism ϕ g t : H + g t ( X ) → B × H . It is clear that therestriction of ϕ ˜ g to the first copy of S n +1 \ D ′ agrees with ϕ ψ ∗ ( g ) , the restriction of ϕ ˜ g to the second copy of S n +1 \ D ′ agrees with ϕ ψ ∗ ( g ′ ) and the restriction of ϕ ˜ g to S × [0 ,
1] agrees with ϕ g t .Let w : S n +1 → H +˜ g ( X ) denote the wall for the family E and set R = sup b ∈ S n +1 || ϕ ˜ g ( w b ) || H . Recall that we have assumed ( g, η ) ∈ C ( E f , s ). Fix an element v ∈ H such that || v || H > R . Choose an ǫ > B ( v, ǫ ) = { u ∈ H | || u − v || H < ǫ } so that each u ∈ B ( v, ǫ ) has || u || H > R ( ǫ = ( || v || H − R ) / η is chosen with ϕ g b [ η ] b ∈ B ( v, ǫ ) for all b ∈ S n +1 .Then we also have ϕ ψ ∗ ( g ) [ ψ ∗ ( η )] b ∈ B ( v, ǫ ) for all b ∈ S n +1 . Similarly, we canassume that η ′ was chosen so that ϕ g ′ b [ η ′ ] b ∈ B ( v, ǫ ) for all b ∈ S n +1 and hence ϕ ψ ∗ ( g ′ ) [ ψ ∗ ( η ′ )] b ∈ B ( v, ǫ ) for all b ∈ S n +1 as well. Lastly, we can assume that thegeneric path ( g t , η t ) joining ( g p , η p ) to ( g ′ p , η ′ p ) satisfies ϕ g t ([ η t ]) ∈ B ( v, ǫ ) for all t ∈ [0 , ϕ ˜ g [˜ η ] b ∈ B ( v, ǫ ) for all b ∈ S n +1 . Therefore we can find ahomotopy from ϕ ˜ g [˜ η ] to the constant section v and hence (˜ g, ˜ η ) lies in C ( E, s ). (cid:3) Let
Dif f ( X ) act on itself by conjugation. Since the identity element is fixed,this gives an action of Dif f ( X ) on π n ( Dif f ( X ) ). We write this action as ( f, h ) f hf − . ON-TRIVIAL SMOOTH FAMILIES OF K Proposition 2.7.
Let X be a compact, oriented, smooth -manifold with b + ( X ) > and let f : X → X be an orientation preserving diffeomorphism. Then for eachspin c -structure with d ( X, s ) = − ( n + 1) ≤ − we have sw s ( f hf − ) = sw f ∗ ( s ) ( h ) . Proof.
Let D + , D − be two copies of the unit disc on R n +1 . Attaching D + and D − on their boundary gives S n +1 . Let h ∈ π n ( Dif f ( X ) ). The family E h is obtainedby attaching D + × X to D − × X using the attaching map ∂D + × X → ∂D − × X ,( b, x ) a h ( b, x ) = ( b, ( h ( b ))( x )). Similarly E fhf − is constructed using ( b, x ) a fhf − ( b, x ) = ( b, ( f h ( b ) f − )( x )). Consider the maps ˜ f ± : D ± × X → D ± given by˜ f ± ( b, x ) = ( b, f ( x )). One finds that a fhf − ◦ ˜ f + = ˜ f − ◦ a h . This says that the maps ˜ f ± glue together to define a map ˜ f : E h → E fhf − . Themap ˜ f is an isomorphism of smooth families over S n +1 . Let ˜ s be the continuousextension of ˜ s to a family of spin c -structures on the fibres of E fhf − . Then clearly˜ f ∗ (˜ s ) is a continuous extension of f ∗ ( s ) to a family of spin c -structures on thefibres of E h . Let ( g, η ) be a generic pair for ( E fhf − , s ) lying in the canonicalchamber. Then ( ˜ f ∗ ( g ) , ˜ f ∗ ( η )) is a generic pair for ( E h , f ∗ ( s ) lying in the canonicalchamber. Clearly ˜ f induces an isomorphic between the corresponding moduli spacesfor ( E fhf − , s , g, η ) and ( E h , f ∗ ( s ) , ˜ f ∗ ( g ) , ˜ f ∗ ( η )). Hence sw s ( f hf − ) = sw f ∗ ( s ) ( h ). (cid:3) Recall that there is an involution s s on the set of spin c -structure which werefer to as charge conjugation. Recall that c ( s ) = − c ( s ). Proposition 2.8.
Let X be a compact, oriented, smooth -manifold with b + ( X ) > and let s be a spin c -structure with d ( X, s ) = − ( n + 1) ≤ − . Then sw s = ( − b +( X ) − b X ) − n sw s . Proof.
Recall that charge conjugation gives rise to a bijection from the Seiberg–Witten equations for ( X, s , g, η ) and the Seiberg–Witten equations for ( X, s , g, − η ).Fix a homology orientation for X , giving orientations on M ( X, s , g, η ) and M ( X, s , g, − η ).The charge conjugation map changes the orientation on the moduli space by a factor( − d s +1 − b ( X )+ b + ( X ) where d s = c ( s ) − σ ( X )8 . Similarly, charge conjugation gives rise to a bijection of families moduli spaces.The charge conjugation isomorphism changes the relative orientation of the modulispace (relative to the base) by the same factor ( − d s +1 − b ( X )+ b + ( X ) . Moreover, itis clear that if ( g, η ) is in the canonical chamber for ( E, s ), then ( g, − η ) is in thecanonical chamber for ( E, s ). Hence sw s = ( − u sw s where u = d s + 1 − b ( X ) + b + ( X ) . But since − n − d ( X, s ) = 2 d s + 1 − b ( X ) + b + ( X ) we see that d s = − n − b ( X ) − b + ( X )2and hence u = − n − b ( X ) − b + ( X )2 + 1 − b ( X ) + b + ( X ) = b + ( X ) − b ( X ) − n . (cid:3) Theorem 2.9.
Let X be a compact, oriented, smooth -manifold such that b + ( X ) > and b ( X ) = 0 . For a given f ∈ π n ( Dif f ( X ) ) , we have that sw s ( f ) is non-zerofor only finitely many spin c -structures with d ( X, s ) = − ( n + 1) .Proof. Let g be a metric on X and consider the Seiberg–Witten equations on X withrespect to the metric g and zero perturbation. Recall that the a priori estimates forsolutions ( A, ψ ) of the Seiberg–Witten equations (after gauge fixing) imply boundson the norms of
A, ψ in a suitable Sobolev space. A bound M ( g ) can be chosenwhich depends continuously on g and the topology of X , but does not depend onthe spin c -structure s . Hence for a smooth family E → B over a compact base B ,we obtain compactness of the families Seiberg–Witten moduli space, taken over all spin c -structures, with zero perturbation and a fixed family g = { g b } of metrics. Itfollows that the families moduli space M ( E, s , g,
0) is non-empty for only finitelymany spin c -structures, say, s , . . . , s m . For any other spin c -structure, s , the modulispace M ( E, s , g,
0) is empty. Hence η = 0 is a regular perturbation for ( E, s ) and( g,
0) defines a chamber for ( E, s ).Now let f ∈ π n ( Dif f ( X ) ) and take E → B to be the family E f → S n +1 associated to f . If b + ( X ) > n + 1, then there is only one chamber and hence wededuce that sw s ( f ) = 0 for all but finitely many spin c -structures.If b + ( X ) ≤ n + 2, then we have to consider chambers. Fix a family of metrics g . Then we have shown that for all but finitely many spin c structures s , ( g,
0) is aregular perturbation and SW ( E f , s , g,
0) = 0. However, ( g,
0) might not lie in thecanonical chamber. Hence we need to consider contributions to the Seiberg–Witteninvariant from wall crossing.For the rest of the proof, the family E f and metric g will be fixed. To simplifynotation we will write SW ( s , η ) instead of SW ( E f , s , g, η ), whenever η is a regularperturbation for ( E, s , g ). Fix a maximal positive definite subspace H of H ( X ; R ).Let ϕ : H + g ( X ) → H be the map which is the inclusion of H + g ( X ) into H ( X ; R ),followed by projection to H . For each spin c -structure s , let w ( s ) be the section of H which sends b ∈ B to ϕ (2 πc ( s ) + gb ). Given a perturbation η , let w ( η ) be thesection of H given by b ϕ ([ η ] b ). If w ( η ) and w ( s ) are disjoint, then ( g, η ) definesa chamber for ( E, s ) and hence the Seiberg–Witten invariant SW ( s , η ) is defined.Let S ( H ) denote the unit sphere in H , which has dimension b + ( X ) −
1. If w ( η ) and w ( s ) are disjoint, then ( w ( η ) − w ( s )) / || w ( η ) − w ( s ) || H defines a sectionof S ( H ). The wall crossing formula for the families Seiberg–Witten invariants [12],[3] adapted to the present setting states that SW ( s , η ) − SW ( s , η ) = Obs (cid:18) w ( η ) − w ( s ) || w ( η ) − w ( s ) || H , w ( η ) − w ( s ) || w ( η ) − w ( s ) || H (cid:19) , where Obs ( φ, ψ ) ∈ H b + ( X ) − ( B ; Z ) is the primary difference class of φ, ψ ([18, § φ, ψ : B → S ( H ) ON-TRIVIAL SMOOTH FAMILIES OF K over the b + ( X ) − B . In our case B = S n +1 and so H b + ( X ) − ( B ; Z ) = H b + ( X ) − ( S n +1 ; Z ) is zero unless b + ( X ) = n + 2.If b + ( X ) = n + 2, then the primary obstruction vanishes implying that the valueof SW ( s , η ) does not depend on the choice of chamber. But we have already seenthat for all but finitely many spin c structures s , SW ( s ,
0) = 0. Hence sw s = 0 forall but finitely many s .It remains to consider the case b + ( X ) = n + 2. In this case, the primary obstruc-tion is valued in H n +1 ( S n +1 ; Z ) ∼ = Z . Let ν ∈ H n +1 ( S n +1 ; Z ) be the generatorcorresponding to our chosen orientation on S n +1 . Then from [3, Proposition 5.7],we have Obs ( φ, ψ ) = ( − b + ( X ) − ( φ ∗ ( ν ) − ψ ∗ ( ν )) . Integrating over S n +1 , the wall crossing formula reduces to Z S n +1 SW ( s , η ) − Z S n +1 SW ( s , η ) = ( − b + ( X ) − (deg( − φ s ,η ) − deg( − φ s ,η )) , where we define φ s ,η = w ( s ) − w ( η ) || w ( s ) − w ( η ) || H for any perturbation η such that w ( η ) and w ( s ) are disjoint.Noting that deg( − φ ) = ( − b + ( X ) deg( φ ), the wall crossing formula can be re-written as Z S n +1 SW ( s , η ) − Z S n +1 SW ( s , η ) = − deg( φ s ,η ) + deg( φ s ,η ) . Now let us take η = η to be arbitrary and choose η such that w ( η ) = v is a constant such that || v || H > sup B || w ( s ) || H . Then ( g, η ) lies in the canonicalchamber. Now since || w ( η ) || H > || w ( s ) || H for all b ∈ B , we obtain a homotopy t (1 − t ) w ( s ) − w ( η ) || (1 − t ) w ( s ) − w ( η ) || H , t ∈ [0 , φ s ,η to the constant − v/ || v || H . It follows that deg( φ s ,η ) = 0 and therefore Z S n +1 SW ( s , η ) − Z S n +1 SW ( s , η ) = − deg( φ s ,η ) . But ( g, η ) lies in the canonical chamber, so R S n +1 SW ( s , η ) = sw s ( f ). Hence theabove formula reduces to(2.1) Z S n +1 SW ( s , η ) = sw s ( f ) − deg( φ s ,η ) . Now we set η = 0. Then for all but finitely many s , we have that w ( s ) is non-vanishing and that SW ( s ,
0) = 0. Hence for all but finitely many s , we find sw s ( f ) = deg( w ( s ) / || w ( s ) || H ) . To finish the proof, it remains to show that when b + ( X ) = n + 2, there areonly finitely many s such that d ( X, s ) = − ( n + 1), w ( s ) is non-vanishing and w ( s ) / || w ( s ) || H : S n +1 → S ( H ) has non-zero degree. For convenience, let us saythat a spin c -structure s is valid if d ( X, s ) = − ( n + 1) and w ( s ) is non-vanishing andlet us write deg( w ( s )) for the degree of w ( s ) / || w ( s ) || H . Then we need to show thatdeg( w ( s )) = 0 for all but finitely many valid s .We first show that there is a constant κ such that deg( w ( s )) = κ for all butfinitely many valid s . We will then argue that κ = 0. Note that if b + ( X ) = n + 2 and d ( X, s ) = − ( n + 1), then c ( s ) − σ ( X )4 − n − − n − b ( X ) = 0) and hence if s is valid, then c ( s ) = σ ( X ) + 8 . We set N = σ ( X ) + 8. If N ≥
0, then for every non-zero c ∈ H ( X ; R ) such that c = N , the projection c + g is non-zero. This is because( c + g ) ≥ ( c + g ) − | ( c − g ) | = c = N ≥ c + g ) = 0 can only occur if c = 0. So if N ≥
0, then every non-zero c ∈ H ( X ; R ) defines a non-zero map w ( c ) : S n +1 → H by taking w ( c ) = ϕ (2 πc + g ).The set { c ∈ H ( X R ) | c = 0 , c = N } is clearly connected if N >
0, since b + ( X ) >
1. Also if N = 0, then σ ( X ) = − b + ( X ) , b − ( X ) >
1. Itfollows that { c ∈ H ( X R ) | c = 0 , c = 0 } is connected. Therefore the degree of w ( c ) / || w ( c ) || H is a constant κ . Now there are only finitely many spin c structures s for which c ( s ) = 0, hence c ( s ) ∈ { c ∈ H ( X ; R ) | c = N } for all but finitely manyvalid s . So deg( w ( s )) = κ for all but finitely many valid s .Now we suppose N <
0. So σ ( X ) < − b − ( X ) > b + ( X ) > C N = { c ∈ H ( X ; R | c = N } . Then C N is homotopy equivalent to a sphere of dimension b − ( X ) −
1. For each b ∈ B , consider S b = { c ∈ C N | c + gb = 0 } . The condition c + gb = 0 means that c lies in the negative definite subspace H − g b ( X ).Therefore S b is a sphere of dimension b − ( X ) −
1. In particular S b is compact.Similarly, let S = [ b ∈ B S b = { c ∈ C N | c + gb = 0 for some b ∈ B } . Then S is a compact subset of C N (by compactness of B ). Let || || E denote anarbitrary Euclidean norm on H ( X ; R ). Then by compactness of S , we have that S is contained in some ball B R = { x ∈ H ( X ; R ) | || x || E ≤ R } of sufficiently largeradius R >
0. Since b + ( X ) , b − ( X ) >
1, it is easy to see that C N \ ( C N ∩ B R )is connected. For any c ∈ C N \ ( C N ∩ B R ), define w ( c ) : S n +1 → H as w ( c ) = ϕ (2 πc + g ). Then since c / ∈ B R , we have that w ( c ) is non-vanishing and hence thedegree of w ( c ) / || w ( c ) || H is defined. Since C N \ ( C N ∩ B R ) is connected, the degreeof w ( c ) / || w ( c ) || H is equal to a constant, κ , for every c ∈ C N \ ( C N ∩ B R ).Let s be a valid spin c -structure. Then c ( s ) ∈ C N . If c ( s ) / ∈ B R , then it followsthat deg( w ( s )) = κ . Next, we note that if c ( s ) ∈ B R , then c ( s ) ∈ B R ∩ H ( X ; Z ).But B R ∩ H ( X ; Z ) is finite because B R is compact and H ( X ; Z ) is discrete. Itfollows that for all but finitely many valid s , we have deg( w ( s )) = κ .Now we argue that κ = 0. We showed that for all c ∈ C N outside some ball B R , the degree of w ( c ) is κ . Let ψ : H ( X ; R ) → H ( X ; R ) be an isometry of theintersection form on H ( X ; R ) that sends H to itself and reverses orientation on H .Then B R ∪ ψ ( B R ) is compact so there exists a c ∈ C N such that c / ∈ B R ∪ ψ ( B R ).Hence c, ψ ( c ) ∈ C N \ ( C N ∩ B R ). Thereforedeg( w ( c )) = deg( w ( ψ ( c ))) = κ. ON-TRIVIAL SMOOTH FAMILIES OF K On the other hand, since ψ reverses orientation on H , we havedeg( w ( ψ ( c ))) = − deg( w ( c )) = − κ. This gives κ = − κ , hence κ = 0. Thus deg( w ( s )) = 0 for all but finitely many valid s and the proof is complete. (cid:3) The Einstein family
Let X be the underlying oriented smooth 4-manifold of a complex K X to construct non-trivialfamilies. The construction of this moduli space follows [8], [9] and [4].Let Ein denote the space of all Einstein metrics on X with unit volume giventhe C ∞ -topology. Every Einstein metric on X is a hyper-k¨ahler metric [10] and inparticular, Ricci flat. Then Dif f ( X ) acts on Ein by pullback. That is, for each ϕ ∈ Dif f ( X ) we define ϕ ∗ : Ein → Ein, ϕ ∗ ( g ) = ( ϕ − ) ∗ ( g ) . Let
T Dif f ( X ) denote the subgroup of Dif f ( X ) consisting of those diffeomor-phisms which act trivially on H ( X ; Z ). Let Aut ( H ( X ; Z )) be the group of auto-morphisms of the intersection form. Then we have a short exact sequence1 → T Dif f ( X ) → Dif f ( X ) → Γ → , where Γ ⊂ Aut ( H ( X ; Z )) is the subgroup of automorphisms that are induced bydiffeomorphisms of X . Note that since Γ is discrete, the identity components of T Dif f ( X ) and Dif f ( X ) are the same T Dif f ( X ) = Dif f ( X ) . As a consequence of the global Torelli theorem for K T Dif f ( X ) acts freely and properly on Ein . Let T Ein = Ein/T Dif f ( X )be the quotient. Over Ein we have the constant family X × Ein → Ein . We canequip the vertical tangent space of X × Ein with the tautological metric, namelythe metric on the fibre X × { g } is g itself. The action of Dif f ( X ) on Ein lifts to X × Ein by setting ϕ ∗ ( x, g ) = ( ϕ ( x ) , ϕ ∗ ( g )) . It is easily checked that this action preserves the fibrewise metric. Let E Ein =( X × Ein ) /T Dif f ( X ) be the quotient of X × Ein by the action of
T Dif f ( X ).As a consequence of the Ebin slice theorem [7] [6], and the deformation theory ofEinstein metrics around K¨ahler–Einstein metrics [11], the quotient E Ein is a lo-cally trivial smooth family over T Ein with fibres diffeomorphic to X . Moreover, thetautological metric descends to a metric g Ein on the vertical tangent bundle suchthat the restriction of g Ein to the fibre of E Ein over [ g ] ∈ T Ein is a representativeof the isomorphism class of Einstein metrics [ g ]. Thus E Ein is a family of Einsteinmetrics on X .Let Gr ( R , ) denote the Grassmannian of positive definite 3-planes in R , .There is a period map P : T Ein → Gr ( R , ) . Defined as follows. Fix an isometry H ( X ; R ) ∼ = R , . Then P sends an Einsteinmetric g to the 3-plane H + g ( X ). Let∆ = { δ ∈ H ( X ; Z ) | δ = − } and set W = { H ∈ Gr ( R , ) | H ⊥ ∩ ∆ = ∅} . The Grassmannian Gr ( R , ) is a contractible manifold and for each δ ∈ ∆, thesubset A δ = { H ∈ Gr ( R , ) | δ ∈ H ⊥ } is a codimension 3 embedded submanifold and W = Gr ( R , ) \ S δ ∈ ∆ A δ . It followsfrom the global Torelli theorem for K P is a homeomorphismof T Ein to the set W [5, Chapter 12, K].Choose an element v ∈ H ( X ; R ) such that h v, δ i 6 = 0 for all δ ∈ ∆ and define∆ ± = { δ ∈ ∆ | ± h v, δ i > } . Then∆ = ∆ + ∪ ∆ − and δ ∈ ∆ + if and only if − δ ∈ ∆ − .Let δ ∈ ∆ + and choose a point p ∈ A δ such that p does not lie on any A δ ′ for δ ′ ∈ ∆ + other than δ . Let H ⊂ H ( X ; R ) be the positive definite 3-planecorresponding to p . Choose a basis θ , θ , θ for H satisfying h θ i , θ j i = δ ij . Since p ∈ A δ , we have h θ j , δ i = 0 for j = 1 , ,
3. Moreover, since p does not lie on A δ ′ forany δ ′ ∈ ∆ + not equal to δ , we have h θ j , δ ′ i 6 = 0 for some j .Let B = S = { ( x , x , x ) ∈ R | x + x + x = 1 } ⊂ R be the unit 2-sphereand choose an ǫ ∈ (0 , f δ : S → Gr ( R , )defined by f δ ( x , x , x ) = span( ω , ω , ω )where for i = 1 , ,
3, we set ω i = θ i − ǫx δ/ . We choose ǫ sufficiently small so that f δ ( x , x , x ) is a positive definite subspaceof H ( X ; R ). Moreover, if ǫ is sufficiently small then f δ ( x , x , x ) is close to p forall ( x , x , x ) ∈ S and hence f δ ( x , x , x ) does not lie on A δ ′ for any δ ′ ∈ ∆ + not equal to δ . Moreover, we have( h ω , δ i , h ω , δ i , h ω , δ i ) = ǫ ( x , x , x ) . Then since ( x , x , x ) ∈ S , we see that at least one of h ω , δ i , h ω , δ i , and h ω , δ i must be non-zero. So f ( x , x , x ) does not lie on A δ . It follows that f maps to W . So there is a map g δ : B → T Ein such that f δ = P ◦ g δ . Definition 3.1.
Let E δ = g ∗ δ ( E Ein ) be the pullback of the family E Ein → T Ein bythe map g δ : S → T Ein . This is a smooth family of K S . Remark . In addition to δ , the map g δ : S → T Ein depends on a choice of p ∈ A δ \ ( A δ ∩ ( ∪ δ ′ ∈ ∆ + \{ δ } A δ ′ ) and a sufficiently small ǫ >
0. However the spaceof such pairs ( p, ǫ ) is connected, and moving ( p, ǫ ) along a path in this space onlychanges g δ by isotopy. Hence the family E δ is well defined up to isomorphism. ON-TRIVIAL SMOOTH FAMILIES OF K Remark . By construction, the family E Ein → T Ein has structure group
T Dif f ( X ),hence the same is true of the pullback family E δ . To say that E δ has struc-ture group T Dif f ( X ) amounts to saying that E δ is equipped with a trivialisation H ( X ) ∼ = H ( X ; R ) × B of the local system H ( X ).For each δ ∈ ∆ + we have constructed a homotopy class of map g δ : S → T Ein .As T Ein is simply connected [4], there is a bijection between unbased homotopyclasses of maps S → T Ein and the homotopy group π ( T Ein ). Hence g δ defines aclass [ g δ ] ∈ π ( T Ein ). Now since T Ein = Ein/T Dif f ( X ), the long exact sequenceof homotopy groups gives a map ∂ : π ( T Ein ) → π ( T Dif f ( X ) ) = π ( Dif f ( X ) ) . In particular, we may define h δ = ∂ [ g δ ] ∈ π ( Dif f ( X ) ). Applying the clutchingconstruction to h δ , we recover the family E δ .Since T Ein is simply connected, the Hurwitz theorem gives an isomorphism π ( T Ein ) = H ( T Ein ; Z ). From [8, Lemma 5.3], we have H ( T Ein ; Z ) = M δ ∈ ∆ + Z [ g δ ] . Hence π ( T Ein ) is a free abelian group with generators the maps { [ g δ ] } δ ∈ ∆ + .Recall that a K b + ( X ) = 3 , σ ( X ) = − , b ( X ) = 0 . Since X is spin and simply-connected, for each u ∈ H ( X ; Z ), there is a uniquelydetermined spin c -structure s u for which c ( s u ) = 2 u . Then d ( X, s u ) = (2 u ) + 164 − − u . Let α ∈ ∆. Then α = − d ( X, s α ) = −
2. Therefore we have theSeiberg–Witten invariant sw s α : π ( Dif f ( X ) ) → Z . To simplify notation we will write sw α for sw s α . If α ∈ ∆, then from Proposition2.8, we have sw − α = − sw α . For this reason, it suffices to only consider the homomorphisms sw α for α ∈ ∆ + .From Theorem 2.9, we have that for each f ∈ π ( Dif f ( X )), sw α ( f ) is non-zerofor only finitely many α ∈ ∆ + . Therefore, we obtain a homomorphism sw : π ( Dif f ( X ) ) → M α ∈ ∆ + Z , sw ( f ) = ⊕ α sw α ( f ) . Recall that we constructed classes h δ = ∂ [ g δ ] ∈ π ( Dif f ( X ) ) such that thefamily E δ is obtained from the clutching construction applied to h δ . Theorem 3.4.
Let α, δ ∈ ∆ + . Then sw α ( h δ ) = ( if α = δ, otherwise . Proof.
By definition, sw α ( h α ) is the Seiberg–Witten invariant of ( E δ , s α ) with re-spect to the canonical chamber, where E δ is the family obtained by the clutchingconstruction applied to h δ .We recall the construction of E δ . Choose a point p ∈ A δ such that p does not lieon any A δ ′ for δ ′ ∈ ∆ + other than δ . Let H ⊂ H ( X ; R ) be the positive definite3-plane corresponding to p . Choose a basis θ , θ , θ for H satisfying h θ i , θ j i = δ ij .Since p ∈ A δ , we have h θ j , δ i = 0 for j = 1 , ,
3. Let B = S be the unit spherein R . We take E δ → S to be the pullback of the family E Ein → T Ein by amap g δ : S → T Ein . Let P : T Ein → Gr ( R , ) be the period map. Then f δ = P ◦ g δ : S → Gr ( R , ) is defined as f δ ( x , x , x ) = span( ω , ω , ω ) , where ǫ > ω i = θ i − ǫx i δ/ , for i = 1 , , . Let ρ H : H ( X ; R ) → H be the projection to H with kernel H ⊥ . Then since θ , θ , θ ∈ H and δ ∈ H ⊥ , we have ρ H ( θ i ) = θ i , ρ H ( δ ) = 0 . In particular, this gives ρ H ( ω i ) = θ i . Let H + g ( X ) → S be the bundle of harmonic self-dual 2-forms. Then ω , ω , ω isa frame for H + g ( X ). Let ϕ : H + g ( X ) → H be inclusion into H ( X ; R ) followed by ρ H . Define w ( α ) to be the section of H given by w ( α )( b ) = ϕ (2 πc ( s α ) + gb ) = ϕ (4 πα + gb ) . Let ω ∗ , ω ∗ , ω ∗ be the dual frame of H + g b , defined by the condition h ω i , ω ∗ j i = δ ij . From ω i = θ i − ǫx i δ/
2, one finds h ω i , ω j i = δ ij − ǫ x i x j / . One can then directly check that the dual frame is given by(3.1) ω ∗ i = ω i + µx i ( x ω + x ω + x ω ) , where µ = ǫ / − ǫ / . We have α + g = h α, ω ∗ i ω + h α, ω ∗ i ω + h α, ω ∗ i ω . Applying ρ H , we get w ( α ) = 4 π ( h α, ω ∗ i θ + h α, ω ∗ i θ + h α, ω ∗ i θ ) . We use the basis θ , θ , θ to identify H with R . Then w ( α ) = 4 π ( h α, ω ∗ i , h α, ω ∗ i , h α, ω ∗ i ) . Suppose that α = δ . Then since p does not lie on A α , we have h θ j , α i 6 = 0 forsome j . From (3.1), we get w ( α ) = 4 π ( h α, ω i , h α, ω i , h α, ω i ) + O ( ǫ ) ON-TRIVIAL SMOOTH FAMILIES OF K where O ( ǫ ) denotes terms of order ǫ . Then (for sufficiently small ǫ ) it follows that w ( α ) is non-vanishing and that deg( w ( α )) = 0.Suppose instead that α = δ . In this case we find w ( α ) = w ( δ ) = 4 πǫ ( x , x , x ) + O ( ǫ ) . So (for sufficiently small ǫ ) w ( α ) is non-vanishing and deg( w ( α )) = 1 (since themap ( x , x , x ) πǫ ( x , x , x ) has degree 1). In all cases, we see that w ( α ) isnon-vanishing.The family E Ein has a fibrewise metric g Ein which is a Ricci flat Einstein metricon each fibre. Let g = g ∗ δ ( g Ein ) be the fibrewise metric on E δ obtained by pullback.Then for each b ∈ S , the metric g b is Ricci flat and in particular has zero scalarcurvature. Now consider the families Seiberg–Witten moduli space M ( E δ , s α , g, η = 0.Suppose that ( A, ψ ) ∈ M ( E δ , s α , g,
0) is a solution to the Seiberg–Witten equa-tions in the family. Then (
A, ψ ) is the solution of the Seiberg–Witten equationson some fibre X b with metric g b and zero perturbation. The Weitzenb¨ock for-mula together with the Seiberg–Witten equations and the fact that g b has zeroscalar curvature implies that ψ = 0 ([14, Corollary 2.2.6]). So every solution in M ( E δ , s α , g,
0) is reducible.On the other hand, since w ( α ) is non-vanishing, the perturbation η = 0 doesnot lie on the wall, that is, there are no reducible solutions. So the moduli space M ( E δ , s α , g,
0) is empty. Recall that in the proof of Theorem 2.9, we used the wallcrossing formula to deduce the identity Z S n +1 SW ( s , η ) = sw s ( f ) − deg( φ s ,η )(see Equation (2.1)). Taking s = s α , f = h δ and η = 0, we obtain Z S n +1 SW ( s α ,
0) = sw α ( h δ ) − deg( w ( α )) . But M ( E δ , s α , g,
0) is empty, so SW ( s α ,
0) = 0 and hence sw α ( h δ ) = deg( w ( α )) . Further, we have already shown thatdeg( w ( α )) = ( α = δ, . (cid:3) As an immediate consequence of Theorem, 3.4, we have:
Theorem 3.5.
The homomorphism sw : π ( Dif f ( X ) ) → M α ∈ ∆ + Z is surjective, hence π ( Dif f ( X ) ) contains L α ∈ ∆ + Z as a direct summand. Theorem 3.6.
The boundary map ∂ : π ( T Ein ) → π ( T Dif f ( X ) ) = π ( Dif f ( X ) ) induced by the fibration Ein → Ein/T Dif f ( X ) = T Ein admits a left inverse, givenby π ( Dif f ( X ) ) → π ( T Ein ) , x M α sw α ( x )[ g α ] . Proof.
Recall that π ( T Ein ) ∼ = H ( T Ein ; Z ) ∼ = M α ∈ ∆ + Z [ g α ]and that ∂ [ g α ] = h α . Hence, if we define t : π ( Dif f ( X ) ) → π ( T Ein ) to be givenby t ( x ) = L α ∈ ∆ + sw α ( x )[ g α ]. Then from Theorem 3.4, it follows that t ◦ ∂ = id ,so that t is a left inverse of ∂ , as claimed. (cid:3) From the homeomorphism P : T Ein → W = Gr ( R , ) \ S δ ∈ ∆ A δ , we see that T Ein is connected. Then since T Ein = Ein/T Dif f ( X ), it follows that T Dif f ( X )acts transitively on the connected components of Ein and that the components of
Ein are all homeomorphic to each other. Choose arbitrarily a basepoint p ∈ Ein .Since the components of
Ein are all homeomorphic, π ( Ein, p ) does not dependon the choice of p and we simply write π ( Ein ). From the long exact sequence inhomotopy groups associated to
Ein → T Ein we get an exact sequence · · · → π ( T Ein ) ∂ −→ π ( T Dif f ( X ) ) → π ( Ein ) → π ( T Ein ) . We have also seen that T Ein is simply-connected and that ∂ admits a left inverse,so we obtain an isomophism π ( T Dif f ( X ) ) = π ( Dif f ( X ) ) ∼ = M α ∈ ∆ + Z ! ⊕ π ( Ein ) , where the summand L α ∈ ∆ + Z is detected by the Seiberg–Witten invariants sw β . Remark . Smooth families over S with fibres diffeomorphic to X correspond,via the clutching construction, to elements of π ( Dif f ( X ) ) considered modulo theconjugation action of Dif f ( X ). For this reason we are interested in the action of Dif f ( X ) on π ( Dif f ( X ) ). The Seiberg–Witten invariants are compatible withthis action in the following sense. Let { e α } α ∈ ∆ + denote the standard basis for L α ∈ ∆ + Z . For f ∈ Dif f ( X ) and x ∈ H ( X ; R ), let us write f ∗ ( x ) = ( f − ) ∗ ( x ) sothat ( f, x ) f ∗ ( x ) is a left action. Let Dif f ( X ) act on L α ∈ ∆ + Z by setting f · e α = ( e f ∗ α if f ∗ α ∈ ∆ + , − e − f ∗ α if f ∗ α ∈ ∆ − . Then it follows easily from Propositions 2.7 and 2.8 that the map sw : π ( Dif f ( X ) ) → L α ∈ ∆ + Z is Dif f ( X )-equivariant. References
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