Positive scalar curvature and 10/8 -type inequalities on 4 -manifolds with periodic ends
aa r X i v : . [ m a t h . DG ] S e p POSITIVE SCALAR CURVATURE AND 10/8-TYPEINEQUALITIES ON -MANIFOLDS WITH PERIODIC ENDS HOKUTO KONNO AND MASAKI TANIGUCHI
Abstract.
We show 10/8-type inequalities for some end-periodic 4-manifoldswhich have positive scalar curvature metrics on the ends. As an application,we construct a new family of closed 4-manifolds which do not admit positivescalar curvature metrics.
Contents
1. Introduction 12. Preliminaries 42.1. Fredholm theory 52.2. The invariant of Mrowka–Ruberman–Saveliev 62.3. Argument due to Kametani 72.4. Moduli theory 113. Main construction 133.1. Perturbation 133.2. Kuranishi model 173.3. Spin Γ-structure on the Seiberg–Witten moduli space 183.4. Main construction 203.5. Proof of Theorem 1.1 234. Examples 24References 261.
Introduction
For a given manifold, the existence of a metric with positive scalar curvature(PSC) is a fundamental problem in Riemannian geometry. This problem was com-pletely solved for simply connected closed n -manifolds with n > • For a closed oriented spin 4-manifold X , if X admits a PSC metric, thesignature of X is zero. • For a closed oriented 4-manifold X and b + ( X ) > X admits a PSCmetric, the Seiberg-Witten invariant of X is zero. Here b + ( X ) denotes themaximal dimension of positive definite subspaces of H ( X ; R ) with respectto the intersection form.In this paper, we consider a closed oriented 4-manifold X whose homology groups(with rational coefficients) are isomorphic to that of S × S . We also assume that X contains a rational homology sphere Y as a submanifold representing a fixed generator of H ( X ; Z ). We call such X and Y a ( rational ) homology S × S anda cross section of X respectively. For such a manifold X , although one cannot useabove two obstructions, J. Lin [11], nevertheless, constructed an effective obstruc-tion to PSC metric using Seiberg–Witten theory on periodic-end 4-manifolds. Theremarkable obstruction due to Lin is described in terms of the Mrowka–Ruberman–Saveliev invariant λ SW ( X ) [16], which depends on the choice of a spin structure of X , and the Frøyshov invariant h ( Y ) [5]. (In this paper, h ( Y ) denotes the Frøyshovinvariant with respect to the (unique) spin structure on Y .) More precisely, Linproved that, if X admits a PSC metric, then the equality λ SW ( X ) = − h ( Y )(1)holds. By the use of this obstruction, Lin showed that any homology S × S whichhas Σ(2 , ,
7) as a cross section does not admit a PSC metric.In this paper, we construct an obstruction which is different from Lin’s one toPSC metric on homology S × S . To give the obstruction, we also consider Seiberg–Witten equations on periodic-end 4-manifolds as Lin’s argument. However, ourapproach is based on a quite different point of view: 10/8-type inequalities. Ourmain theorem is described as follows. Theorem 1.1.
Let X be an oriented spin rational homology S × S and Y be anoriented rational homology S embedded in X . Suppose that Y represents a fixedgenerator of H ( X ; Z ) . If Y bounds a compact spin -manifold M and if X admitsa PSC metric, then the inequality b + ( M ) ≥ − σ ( M )8 + h ( Y ) holds, where σ ( M ) is the signature of M . Moreover, if b + ( M ) is even, we have b + ( M ) ≥ − σ ( M )8 + h ( Y ) + 1 . Remark . Let M ′ be a closed spin 4-manifold and M be the complement of anembedded 4-disk in M ′ . Since S × S has a PSC metric, we can substitute X = S × S and Y = S in Theorem 1.1. Then the second inequality in Theorem 1.1recovers the original 10/8 inequality for M ′ due to M. Furuta (Theorem 1 in [6])under the assumption that b + ( M ′ ) is even.Theorem 1.1 is shown by considering the Seiberg–Witten equations on a periodic-end 4-manifold, which is obtained by gluing M and infinitely many copies of thecompact 4-manifold W defined by cutting X open along Y . In fact, the inequal-ities in Theorem 1.1 are derived as 10/8-type inequalities for this periodic-endspin 4-manifold. To show these 10/8-type inequalities, we use Y. Kametani’s ar-gument [9] which provides a 10/8-type inequality without using finite dimensionalapproximations of the Seiberg–Witten equations. On the other hand, D. Veloso [24]has considered boundedness of the Seiberg–Witten equations on a periodic-end 4-manifold under a similar PSC assumption. The authors expect that his argumentcan be also used to give similar 10/8-type inequalities.Theorem 1.1 gives a new family of homology S × S which do not admit PSCmetrics. To describe our obstruction to PSC metric, it is convenient to use thefollowing invariant. SC AND 10/8 FOR END-PERIODIC 4-MANIFOLDS 3
Definition 1.3.
For an oriented rational homology 3-sphere Y , we define a number ǫ ( Y ) ∈ Q by ǫ ( Y ) := min (cid:26) σ ( M )8 + b + ( M ) (cid:12)(cid:12)(cid:12)(cid:12) ∂M = Y and M is compact and spin (cid:27) . A similar quantity is also used in C. Manolescu [14]. Manolescu [14] constructedan invariant κ ( Y ) ∈ Q for an oriented integral homology sphere Y and showed theinequality κ ( Y ) ≤ ǫ ( Y ) + 1(2)in Theorem 1 of [14].(For a rational homology sphere Y , see Remark 4.6 of [14].)Since every spin 3-manifold bounds a spin 4-manifold and the inequality (2) givesa lower bound of ǫ ( Y ), the integer ǫ ( Y ) is well-defined. Using this invariant ǫ , wedefine an invariant ψ ( Y ) of Y by ψ ([ Y ]) := − ǫ ( Y ) + h ( Y ). When we consider Y asan oriented homology sphere, ǫ ( Y ) is a homology cobordism invariant. Then theinvariant ψ descends to a map ψ : Θ → Z . (3)Here the group Θ is the homology cobordism group of oriented homology 3-spheres.Our obstruction to PSC metric is described as follows. Corollary 1.4.
Let Y be an oriented rational homology -sphere and suppose that ψ ([ Y ]) > . Then any rational homology S × S which has Y as a cross sectiondoes not admit a PSC metric.Proof. For a given oriented rational homology 3-sphere Y , let M be a compact spin4-manifold with ∂M = Y and ǫ ( Y ) = σ ( M )8 + b + ( M ), and X be a rational homology S × S which has Y as a cross section. If X admits a PSC metric, Theorem 1.1implies that ψ ([ Y ]) = − b + ( M ) − σ ( M )8 + h ( Y ) ≤ . This proves the corollary. (cid:3)
Using Corollary 1.4, we can construct many new examples of homology S × S ’swhich do not admit PSC metrics. Such examples are given in Section 4.Lin’s method and ours obstruct PSC metric on homology S × S ’s only in termsof topological properties of cross sections of them. In our case, the obstruction isdominated by the subsemigroupΠ := (cid:8) [ Y ] ∈ Θ (cid:12)(cid:12) ψ ([ Y ]) > (cid:9) of Θ . In Section 4, we shall give many examples of elements of Π, and it is alsoeasy to see that, for any element [ Y ] ∈ Θ and [ Y ′ ] ∈ Π, there exists a naturalnumber
N >> Y ] N [ Y ′ ] ∈ Π. This may suggest that Π is a largesubsemigroup of Θ, and therefore it is natural to ask the following question:
Problem 1.5.
Study the subsemigroup Π . For example, how large is Π in Θ ?Acknowledgement. The authors would like to express their deep gratitude to YukioKametani for answering their many questions on his preprint [9]. The authorswould also like to express their appreciation to Mikio Furuta for informing them ofKametani’s preprint. The authors also wish to thank Andrei Teleman for informingthem of Veloso’s argument [24] and answering their questions on it. The authors
HOKUTO KONNO AND MASAKI TANIGUCHI also appreciate Mayuko Yamashita’s helpful comment on equivariant KO -theory.The first author was supported by JSPS KAKENHI Grant Number 16J05569 andthe Program for Leading Graduate Schools, MEXT, Japan. The second authorwas supported by JSPS KAKENHI Grant Number 17J04364 and the Program forLeading Graduate Schools, MEXT, Japan.2. Preliminaries
Let X be an oriented spin rational homology S × S and Y be an orientedrational homology S . We fix a Riemannian metric g X on X and a generatorof H ( X ; Z ), denoted by 1 ∈ H ( X ; Z ). (Note that H ( X ; Z ) is isomorphic to H ( X ; Z ), and hence to Z .) We also assume that Y is embedded into X as a crosssection of X , namely [ Y ] = 1. Let W be the homology cobordism from Y toitself obtained by cutting X open along Y . The manifold W is equipped with anorientation and a spin structure induced by that of X . We define W [ m, n ] := W m ∪ Y W m +1 ∪ Y · · · ∪ Y W n for ( m, n ) ∈ ( {−∞} ∪ Z ) × ( Z ∪ {∞} ) with m < n . Let us take a compact spin4-manifold M bounded by Y as oriented manifolds. The element 1 ∈ H ( X ; Z )corresponding to 1 ∈ H ( X ; Z ) via Poincar´e duality gives the isomorphism class ofa Z -bundle p : e X → X (4)and an identification e X ∼ = W [ −∞ , ∞ ] . (5)We can suppose that H ( M ; Z ) = 0 by surgery preserving the intersection form of M and the condition that M is spin. Assumption 2.1.
Henceforth we suppose the condition H ( M ; Z ) = 0.Then we get a non-compact manifold Z := M ∪ Y W [0 , ∞ ] equipped with a nat-ural spin structure induced by spin structures on M and W . Via the identification(5), we regard p as a map from W [ −∞ , ∞ ] to X . We set p + : W [0 , ∞ ] → X asthe restriction of p . We call an object on Z a periodic object on Z if the restrictionof the object to W [0 , ∞ ] can be identified with the pull-back of an object on X by p + . For example, we shall use a periodic connection, a periodic metric, periodicbundles and periodic differential operators. By considering pull-back by p + , theRiemannian metric g X on X induces the Riemannian metric g W [0 , ∞ ] on W [0 , ∞ ].We extend the Riemannian metric g W [0 , ∞ ] to a periodic Riemannian metric g Z on Z , and henceforth fix it. Let S + , S − be the positive and negative spinor bundlesrespectively over Z determined by the metric and the spin structure. If we fix atrivialization of the determinant line bundle of the spin structure on Z , we have thecanonical reference connection A on it corresponding to the trivial connection.To consider the weighted Sobolev norms on Z , we fix a function τ : Z → R satisfying T ∗ τ = τ + 1, where T : W [0 , ∞ ] → W [0 , ∞ ] is the deck transformdetermined by T ( W i ) = W i +1 . SC AND 10/8 FOR END-PERIODIC 4-MANIFOLDS 5
Fredholm theory.
To obtain the Fredholm property of periodic elliptic op-erators on Z , it is reasonable to work on the L k,δ -norms rather than the L k -normsfor k ≥ δ . C. Taubes [23] showed that a periodic elliptic op-erator on Z with some condition is Fredholm with respect to L k,δ -norms for generic δ ∈ R . Let D = ( D i , E i ) be a periodic elliptic complex on Z , i.e. the complex0 → Γ( Z ; E N ) D N −−→ Γ( Z ; E N − ) → . . . D −−→ Γ( Z ; E ) → • Each linear map D i is a first order periodic differential operator on Z . • The symbol sequence of (6) is exact.We consider the following norm k f k L k,δ ( Z ) := k e τδ f k L k ( Z ) by using a periodic connection and a periodic metric. We call the norm k − k L k,δ the weighted Sobolev norm with weight δ ∈ R . By extending (6) to the complex ofthe completions by the weighted Sobolev norms, we obtain the complex of boundedoperators L k + N +1 ,δ ( Z ; E N ) D N −−→ L k + N,δ ( Z ; E N − ) → . . . D −−→ L k,δ ( Z ; E )(7)for each δ ∈ R . Taubes constructed a sufficient condition for the Fredholm propertyof (7) by using the Fourier–Laplace (FL) transformation. The FL transformationreplaces the Fredholm property of the periodic operator on Z with the invertibilitiesof a family of operators on X parameterized by S . Let us described it below. Wefirst note that, since the operators in (7) are periodic differential operators, thereare differential operators ˆ D = ( ˆ D i , ˆ E i ) on X such that there is an identificationbetween p ∗ + ˆ D and D on W [0 , ∞ ]. The sufficient condition for Fredholmness is givenby invetibility of the following complexes on X . For z ∈ C , we define the complexˆ D ( z ) by 0 → Γ( X ; ˆ E N ) ˆ D N ( z ) −−−−→ Γ( X ; ˆ E N − ) → · · · ˆ D ( z ) −−−−→ Γ( X ; ˆ E ) → , (8)where the operator ˆ D i ( z ) : Γ( X ; ˆ E i ) → Γ( X ; ˆ E i − ) is give byˆ D i ( z )( f ) := e − τz ˆ D i ( e τz f ) . Theorem 2.2 (Taubes, Lemma 4.3 and Lemma 4.5 in [23]) . Suppose that thereexists z ∈ C such that the complex ˆ D ( z ) is acyclic. Then there exists a descretesubset D in R with no accumulation points such that (7) is Fredholm for each δ in R \ D . Moreover, the set D is given by D = n δ ∈ R (cid:12)(cid:12)(cid:12) ˆ D ( z ) is not invertible for some z with Re z = δ. o . Remark . The assumption of Theorem 2.2 implies that the Euler characteristicof (8) is 0 for all z . We shall consider ˆ D as the Atiyah–Hitchin–Singer complexor the spin (or spin c ) Dirac operator D + A : Γ( X ; S + ) → Γ( X ; S − ). Note that theEuler characteristic (i.e. the index) of these operators are 0 in our situation.If we consider the set D for the Atiyah–Hitchin–Singer complex on Z , one canshow that the set D does not depend on the choice of Riemannian metric on X .However if we consider the spin (or spin c ) Dirac operator on Z , the set D depends HOKUTO KONNO AND MASAKI TANIGUCHI on the choice of Riemannian metric. For this reason, we introduce some conditionof metrics. First we consider the following operator on X : D + A + f ∗ dθ : L k ( X ; S + ) → L k − ( X ; S − ) , (9)where the map f : X → S is a smooth classifying map of (4). We call g X an admissible metric on X if the kernel of (9) is 0. This condition is considered in [18].The admissibility condition does not depend on the choice of classifying map f . Remark . We can show that every PSC metric on X is an admissible metric.This is a consequence of Weitzenb¨ock formula. (see (2) in [18])Now we see that the assumption of Theorem 2.2 is satisfied for the operators inour situation. Lemma 2.5.
The assumption of Theorem 2.2 is satisfied for the following opera-tors: • The Dirac operator D + A : L k,δ ( Z ; S + ) → L k − ,δ ( Z ; S − ) for the pull-backof an admissible metric g X on X . • The Atiyah–Hitchin–Singer complex → L k +1 ,δ ( i Λ ( Z )) d −→ L k,δ ( i Λ ( Z )) d + −−→ L k − ,δ ( i Λ + ( Z )) → . • The operator d + ( d + ) ∗ : L k,δ ( i Λ + ( Z ) ⊕ i Λ ( Z )) → L k − ,δ ( i Λ ( Z )) .Proof. The Fredholm property does not depend on the choice of τ satisfying T ∗ τ = τ + 1 on W [0 , ∞ ]. Therefore we can choose a lift of f as τ . Then the operatorˆ D ( z ) | z =1 corresponding to D + A coincides with that corresponding to D + A + f ∗ dθ .Since the index of D + A + f ∗ dθ is 0, admissibility implies that ˆ D ( z ) | z =1 is acyclic.The second and third condition follow from Lemma 3.2 in [23]. (cid:3) Remark . Since D has no accumulation points, we can choose a sufficiently small δ > δ ∈ (0 , δ ) the operators in Lemma 2.5 are Fredholm.We fix the notation δ in the rest of this paper.2.2. The invariant of Mrowka–Ruberman–Saveliev.
Let X be a spin rationalhomology S × S . For such a 4-manifold X , Mrowka–Ruberman–Saveliev [16] con-structed a gauge theoretic invariant λ SW . In this section, we review the definitionof λ SW and the following result due to J. Lin [11]: the invariant − λ SW coincideswith the Frøyshov invariant of its cross section under the assumption that X admita PSC metric.For a fixed spin structure, the formal dimension of the perturbed blow-up SWmoduli space M ( X, g X , β ) of X is 0. Here β denotes some perturbation. Thereforethe formal dimension of the boundary of M ( X, g X , β ) is −
1. Mrowka–Ruberman–Saveliev showed that the space M ( X, g X , β ) has a structure of compact 0-dimensionalmanifold for a fixed generic pair of a metric and a perturbation ( g X , β ). For thegeneric pair ( g X , β ), one can define the Fredholm index of the operator D + ( Z, g X , β ) : L k ( Z ; S + ) → L k − ( Z ; S − ) . Here note that we do not use the weighted norm, however the Fredholm propertyof D + ( Z, g X , β ) is based on the choice of ( g X , β ). SC AND 10/8 FOR END-PERIODIC 4-MANIFOLDS 7
Definition 2.7 (Mrowka–Ruberman–Saveliev [16]) . If we fix an orientation of H ( X, R ), we define λ SW ( X ) := M ( X, g X , β ) − ind C D + ( Z, g X , β ) − σ ( M )8 , where M is a compact spin 4-manifold bounded by Y .Mrowka–Ruberman–Saveliev showed that λ SW ( X ) dose not depend on the choiceof metric, perturbation and M . We also use the following theorem due to Lin [11]and Lin–Ruberman–Saveliev [12]. Theorem 2.8 (Lin, Theorem 1.2 in [11], Lin–Ruberman–Saveliev, Theorem B in[12]) . Let X be an oriented spin rational homology S × S and Y be an orientedrational homology S . We fix a generator H ( X, Z ) and suppose that Y is embeddedinto X as a submanifold such that Y represents the fixed generator of H ( X, Z ) . If X has a PSC metric, then the equality λ SW ( X ) = − h ( Y ) holds. By the definition of λ SW ( X ), the following Lemma is an easy consequence of theWeitzenb¨ock formula. Lemma 2.9.
Let X be an oriented spin rational homology S × S and Y be anoriented rational homology S as in Theorem 2.8. If X admits a PSC metric, thefollowing equality holds: λ SW ( X ) = − ind C D + ( Z, g X , β ) − σ ( M )8 , where M is a compact spin 4-manifold with ∂M = Y . Argument due to Kametani.
The original proof of the 10/8-inequalitydue to Furuta [6] for closed oriented spin 4-manifolds uses properness property ofthe monopole map and the finite dimensional approximation. After the work ofFuruta, Bauer–Furuta [2] constructed a cohomotopy version of the Seiberg–Witteninvariant for closed oriented 4-manifolds by using boundedness property of themonopole map and the finite dimensional approximation. On the other hand, in[9], Kametani developed a technique to obtain the 10/8-type inequality using onlythe compactness of the Seiberg–Witten moduli space. In this section, we refer sucha technique and obtain the 10/8-type inequality in our situation. First we recallseveral definitions to formulate the theorem due to Kametani.Let G be a compact Lie group. Definition 2.10.
Let U be an oriented finite dimensional vector space over R with an inner product. The real spin G -module is the pair of a representation ρ : G → SO ( U ) and its lift ˜ ρ : G → Spin ( U ). Remark . Let X be a G -space and U be a real spin G -module. Suppose that the G -action on X is free, ( X × U ) /G → X/G become a vector bundle. By the use of thestructure of real spin G -module, one can show that P X/G := ( X × Spin ( n )) /G → X/G become a principal
Spin ( n )-bundle on X/G . The identification P X/G × π R n ∼ =( X × U ) /G induces a spin structure on ( X × U ) /G → X/G , where π is the doublecover Spin ( n ) → SO ( n ). HOKUTO KONNO AND MASAKI TANIGUCHI
We consider the Lie group
P in (2) which is the subgroup of Sp (1)( ⊂ H ) generatedby S ( ⊂ C ⊂ H ) and j ∈ H . Let ˜ R be the non-trivial representation of P in (2)defined via the non-trivial homomorphism
P in (2) → Z / Z / R . We regard H as the standard representation of P in (2)on the set of quaternions.
Lemma 2.12.
The
P in (2) -module H has a real spin P in (2) -module structure.Remark . In this paper, for a fixed positive integer m , we equip H m with astructure of a real spin P in (2)-module as the direct sum of the real spin
P in (2)-module defined in Lemma 2.12.
Proof.
Since the group
Spin ( H ) ∼ = Sp (1) × Sp (1) acts on H by v βvα where( α, β ) ∈ Sp (1) × Sp (1), the following diagram commutes P in (2) −−−−→
Spin ( H ) y y SO ( H ) SO ( H ) , (10)where the inclusion P in (2) ⊂ Sp (1) → Sp (1) × Sp (1) ∼ = Spin ( H ) is defined by g (1 , g ). This implies the conclusion. (cid:3) Let Γ be the pull-back of
P in (1) along the map
P in (2) → O (1)(see Theorem3.11 of [1]): Γ −−−−→ P in (1) y y
P in (2) −−−−→ O (1) , (11)where the map P in (2) → O (1) is the non-trivial homomorphism. The Γ-actionson H and R are induced by P in (2)-representations H and ˜ R via (23). We denotethese representations of Γ by the same notations. Lemma 2.14.
For a positive number n with n ≡ , ˜ R n has a real spin Γ -module structure.Proof. First, we suppose that n ≡ n = 4 k . In this case, wehave the following inclusion i : Spin (4) × Spin ( k ) → Spin ( n ). Define P in (2) → Spin (4) ∼ = Sp (1) × Sp (1) by g → i ( s ( g ) , ∈ Spin ( n ), where s : P in (2) → Z isthe non-trivial homomorphism. Then the diagram P in (2) −−−−→
Spin ( R n ) y y SO ( R n ) SO ( R n )(12)commutes. In the second case, we have the inclusion i : Spin (2) × Spin (2 k + 1) → Spin ( n ), where n = 4 k + 2. By the construction of Γ, there is the following SC AND 10/8 FOR END-PERIODIC 4-MANIFOLDS 9 commutative diagram: Z Z Z y y y Γ −−−−→ P in (1) −−−−→
Spin (2) y y y
P in (2) −−−−→ O (1) −−−−→ SO (2) . (13)The group homomorphism Γ → Spin ( n ) is given by the composition of the mapfrom Γ to SO (2) in (13) and i : Spin (2) × Spin (2 k + 1) → Spin ( n ). This gives theconclusion. (cid:3) Let V be a real spin G -module of dimension n . When n ≡ β ( V ) ∈ KO ∗ ( V ) which generates the total cohomology ring KO ∗ G ( V )as a KO ∗ G ( pt )-module due to Bott periodicity theorem. For a general n , we fix apositive integer m satisfying m + n ≡ β ( V ) := β ( V ⊕ R m ) ∈ KO G ( V ⊕ R m ) ∼ = KO nG ( V ). We define e ( V ) := i ∗ β ( V ) ∈ KO nG ( pt ), where the map i : pt → V is the map defined by i ( pt ) = 0 ∈ V . The class e ( V ) is called Euler classof V .We use the notation ( P, ψ ) as a spin structure on a manifold M . It meansthat ψ is a bundle isomorphism from P × π R n to T M as an SO ( n )-bundle, where π : Spin ( n ) → SO ( n ) is the double cover and P × π R n is the associated bundle for π . Definition 2.15.
Let M be a G -manifold M of dimension n , ( P, ψ ) be a spinstructure on M and m : G × P → P be a G -action on the principal Spin ( n )-bundle P on M which is a lift of G -action on M . The triple ( M, ( P, ψ ) , m ) is called a spin G -manifold ( G -manifold with an equivariant spin structure) if the followingconditions are satisfied:(1) The action m commutes with the Spin ( n )-action on P .(2) The G -action on P × π R n which is induced by the action on P coincideswith the G -action on T M via ψ . Remark . Let ( M, ( P, ψ ) , m ) be a spin G -manifold with free G -action. Thenwe have the following diagram: P q ∗ −−−−→ P/G y y M q −−−−→ M/G, (14)where q and q ∗ are quotient maps. Since the G -action is free on M , M/G has astructure of a manifold. Since the G -action m commutes with the Spin ( n ) actionon P , P in ( n ) acts on P/G . One can check that
P/G → M/G determines a spinstructure on
M/G by the second condition of the definition of spin G -manifold. Remark . Let M be a G -manifold with free G -action. We also assume that M/G has a spin structure. We denote by P M/G the principal
Spin ( n )-bundle on M/G . Then we have the diagram: q ∗ P M/G −−−−→ P M/G y y M q −−−−→ M/G. (15)Since the quotient map q : M → M/G is a G -equivariant map (the G -action on M/G is trivial), q ∗ P M/G admit a G -action m M/G which commutes with
Spin ( n )-action. By the pull-back the identification P M/G × π R n ∼ = T ( M/G ) by q , we obtainthe identification q ∗ P M/G × π R n ∼ = T M . By the definition, one can check that G -action on q ∗ P M/G and the G -action on T M coincide. Therefore, (
M, q ∗ P M/G , m
M/G )is a spin G -manifold.We setΩ spin G, free := { closed spin G -manifolds whose G -actions are free } / ∼ . The relation ∼ is given as follows: X ∼ X if there exists a compact spin G -manifold Z whose G -action is free such that ∂Z = X ∪ ( − X ) as spin manifolds.For two real spin modules U , U whose G -action on U \ { } is free, we define aninvariant w ( U , U ) in Ω spin G, free . Definition 2.18.
The element w ( U , U ) ∈ Ω spin G, free is defined by taking a smooth G -map φ : S ( U ) → U which is transverse to 0 ∈ U and setting w ( U , U ) :=[ φ − (0)].Since Ker dφ has the induced real spin G -module structure and the G -action on φ − (0) is free, w ( U , U ) determines the element in spin cobordism group with free G -action. In [9], it is shown that the class w ( U , U ) is independent of the choiceof φ .We use the following theorem due to Kametani. Theorem 2.19 (Kametani [9]) . Let G be a compact Lie group. Let U , U be tworeal spin G -modules with dim U = r and dim U = r . Suppose that G -action isfree on U \ { } . If the cobordism class w ( U , U ) ∈ Ω spin G, free is zero, there exists anelement α ∈ KO r − r G ( pt ) such that e ( U ) = αe ( U ) . (16)Furuta–Kametani [7] showed the following inequality under the divisibility of theEuler class (16). Theorem 2.20 (Furuta–Kametani [7]) . Suppose that there exists an element α ∈ KO m + l − m Γ ( pt ) such that e ( H m ) e ( ˜ R l ) = αe ( H m ) ∈ KO m + l Γ ( pt ) , (17) where the definition of Γ is given in (23) . Then the inequality ≤ m − m ) + l − holds. SC AND 10/8 FOR END-PERIODIC 4-MANIFOLDS 11
Moduli theory.
In this subsection, we review moduli theory for 4-manifoldswith periodic ends. The setting of gauge theory for such manifolds is developed byTaubes in [23]. All functional spaces appearing in this subsection are consideredon Z , and therefore we sometimes drop Z from our notation.We fix a real number δ satisfying 0 < δ < δ and an integer k ≥
3, where δ is in-troduced in Subsection 2.1. The space of connections is defined by A k,δ ( Z ) := A + L k,δ ( i Λ ( Z )). We set the configuration space by C k,δ ( Z ) := A k,δ ( Z ) ⊕ L k,δ ( S + ).The irreducible part of C k,δ ( Z ) is denoted by C ∗ k,δ ( Z ). The gauge group G k +1 ,δ forthe given spin structure is defined by G ( Z ) k +1 ,δ := (cid:8) g ∈ L k +1 , loc ( Z, S ) (cid:12)(cid:12) dg ∈ L k,δ (cid:9) . The topology of G ( Z ) k +1 ,δ is given by the metric k g − h k := k dg − dh k L k,δ + | g ( x ) − h ( x ) | , where x ∈ W is a fixed point. The space G k +1 ,δ has a structure of a BanachLie group. Let us define a normal subgroup of G ( Z ) k +1 ,δ (corresponding to theso-called based gauge group) by e G ( Z ) k +1 ,δ := { g ∈ G ( Z ) k +1 ,δ | L x ( g ) = 1 } , where L x ( g ) = lim n →∞ g ( T n ( x )). Note that we have the exact sequence1 → e G ( Z ) k +1 ,δ → G ( Z ) k +1 ,δ L x −−→ S → . The space C k,δ ( Z ) is acted by G k +1 ,δ via pull-back, and moreover one can show that G k +1 ,δ acts smoothly on C k,δ ( Z ) and e G k +1 ,δ acts freely on C k,δ ( Z ). The tangentspaces of G k +1 ,δ and e G k +1 ,δ can be described as follows. (See Lemma 7.2 in [23]) Lemma 2.21.
The following three equalities T e e G ( Z ) k +1 ,δ = n a ∈ L k +1 , loc ( i Λ ( Z )) (cid:12)(cid:12)(cid:12) da ∈ L k,δ , lim n →∞ a ( T n ( x )) = 0 o = L k +1 ,δ ( i Λ( Z )) , and T e G ( Z ) k +1 ,δ = (cid:8) a ∈ L k +1 , loc ( i Λ ( Z )) (cid:12)(cid:12) da ∈ L k,δ (cid:9) hold. We use the following notations: • B k,δ ( Z ) := C k,δ ( Z ) / G ( Z ) k +1 ,δ , • e B k,δ ( Z ) := C k,δ ( Z ) / e G ( Z ) k +1 ,δ and • B ∗ k,δ ( Z ) := C ∗ k,δ ( Z ) / G ( Z ) k +1 ,δ .As in Lemma 7.3 of [23], one can show that the spaces B ∗ k,δ ( Z ) and e B k,δ ( Z )have structures of Banach manifolds. In the proof of this fact, the following decom-position is used. Lemma 2.22.
For a fixed real number δ with < δ < δ , there is the following L δ -orthogonal decomposition L k,δ ( i Λ ( Z )) = Ker( d ∗ L δ : L k,δ ( i Λ ( Z )) → L k − ,δ ( i Λ ( Z ))) ⊕ Im( d : L k +1 ,δ ( i Λ ( Z )) → L k,δ ( i Λ ( Z ))) . Proof.
Since d + ( d + ) ∗ is Fredholm by the choice of δ (see the end of Subsec-tion 2.1), the space Im( d : L k +1 ,δ ( i Λ ( Z )) → L k,δ ( i Λ ( Z ))) is a closed subspace of L k,δ ( i Λ ( Z )). Therefore we have a decomposition L k,δ ( i Λ ( Z )) = Im( d : L k +1 ,δ ( i Λ ( Z )) → L k,δ ( i Λ ( Z ))) ⊕ { Im( d : L k +1 ,δ ( i Λ ( Z )) → L k,δ ( i Λ ( Z ))) } ⊥ L δ . On the other hand, it is straightforward to see that the spaceKer( d ∗ L δ : L k,δ ( i Λ ( Z )) → L k − ,δ ( i Λ ( Z )))is equal to { Im( d : L k +1 ,δ ( i Λ ( Z )) → L k,δ ( i Λ ( Z ))) } ⊥ L δ . (cid:3) The monopole map ν : C k,δ ( Z ) → L k − ,δ ( i Λ + ( Z ) ⊕ S − ) is defined by ν ( A, Φ) := ( F + A − σ (Φ , Φ) , D A (Φ)) , where σ (Φ , Φ) is the trace-free part of Φ ⊗ Φ ∗ and regarded as an element of L k − ,δ ( i Λ + ( Z )) via the Clifford multiplication. Recall that the map L k,δ × L k,δ → L k,δ is continuous for k > c structures, the monopolemap is a P in (2)-equivariant map. We define the monopole moduli spaces for Z by M k,δ ( Z ) := { [( A, Φ)] ∈ B k,δ ( Z ) | ν ( A, Φ) = 0 } and f M k,δ ( Z ) := n [( A, Φ)] ∈ e B k,δ ( Z ) (cid:12)(cid:12)(cid:12) ν ( A, Φ) = 0 o . At the end of this subsection, we study the local structure of dν near [( A , dν + d ∗ L δ ) ( A , : C k,δ ( Z ) → L k − ,δ ( S − ⊕ i Λ + ⊕ i Λ ) . (18) Proposition 2.23.
Suppose that X admits a PSC metric. Then there exists δ ∈ (0 , δ ) satisfying the following condition: for each δ ∈ (0 , δ ) , there exist positivenumbers l and l with l − l = 2 ind C ( D A : L k ( Z ; S + ) → L k − ( Z ; S − )) such thatthere exist isomorphims • Ker( dν + d ∗ L δ ) ( A , ∼ = H l , • Coker( dν + d ∗ L δ ) ( A , ∼ = H l ⊕ ˜ R b + ( M ) as representations of P in (2) .Proof.
It is easy to show that the operator (18) is the direct sum of d + + d ∗ L δ : L k,δ ( i Λ ) → L k − ,δ ( i Λ + ⊕ i Λ )(19)and D + A : L k,δ ( S + ) → L k − ,δ ( S − ) . (20)Taubes (Proposition 5.1 in [23]) showed that the kernel of (19) is isomorphic to R b ( M ) and the cokernel of (20) is isomorphic to R b + ( M ) for small δ . On the otherhand, since g X is a PSC metric, the operator D + ( Z, g X ,
0) : L k,δ ( S + ) → L k − ,δ ( S − ) SC AND 10/8 FOR END-PERIODIC 4-MANIFOLDS 13 is Fredholm for any δ . (Use (2) in [18]). This implies thatind C ( D A : L k ( Z ) → L k − ( Z )) = ind C ( D A : L k,δ ( Z ) → L k − ,δ ( Z ))for any δ . Therefore, using Assumption 2.1, we can see thatKer( dν + d ∗ L δ ) ( A , ∼ = H l and Coker( dν + d ∗ L δ ) ( A , ∼ = H l ⊕ R b + ( M ) as vector spaces for some l , l with l − l = 2 ind C ( D A : L k ( Z ) → L k − ( Z )).Since d + + d ∗ L δ is a P in (2)-equivariant linear map, its kernel and cokernel havestructures of
P in (2)-modules and these representations are the direct sum of H and˜ R . (cid:3) Main construction
In this section we give the proof of Theorem 1.1. We first consider combination ofthe Kuranishi model and some
P in (2)-equivariant perturbation, obtained by usingsome arguments of Y. Ruan’s virtual neighborhood technique [17]. Using it, weshall show a divisibility theorem of the Euler class following Y. Kametani [9]. Thisargument produces the 10/8-type inequality on periodic-end spin 4-manifolds.3.1.
Perturbation.
We first confirm that what is called the global slice theoremholds also for our situation. Henceforth we use this notation d ∗ for the formaladjoint of d with respect to the L δ -norm if no confusion can arise. Let us define S k,δ := Ker( d ∗ : L k,δ ( i Λ ( Z )) → L k − ,δ ( i Λ ( Z ))) × L k,δ ( S + ) . Lemma 3.1.
The map S k,δ × e G ( Z ) k +1 ,δ → C k,δ ( Z ) defined by (( a, Φ) , g ) g ∗ ( A + a, Φ) is a e G ( Z ) k +1 ,δ -equivariant diffeomorphism. In particular, we have S k,δ ∼ = e B k,δ ( Z ) . Proof.
The assertion on e G ( Z ) k +1 ,δ -equivariance is obvious. To prove that the mapgiven in the statement is a diffeomorphism, it suffices to show that the map ϕ : Ker( d ∗ : L k,δ ( i Λ ( Z )) → L k − ,δ ( i Λ ( Z ))) × e G ( Z ) k +1 ,δ → L k,δ ( i Λ ( Z ))defined by ( a, g ) a − g − dg is a diffeomorphism. Henceforth we simply denoteby Ker d ∗ the first factor of the domain of this map if there is no risk of confusion.We first show that ϕ is surjective. Take any a ∈ Ker d ∗ . Thanks to the L δ -orthogonal decomposition given in Lemma 2.22, we can find f ∈ L k +1 ,δ ( i Λ ( Z ))such that − df = a − p ( a ), where p is the L δ -orthogonal projection to Ker d ∗ from L k,δ ( i Λ ( Z )). Set g := e f . Since f decays at infinity, g ∈ e G ( Z ) k +1 ,δ holds, and weget ϕ ( p ( a ) , g ) = a .We next show that ϕ is injective. Assume that ϕ ( a, g ) = ϕ ( a ′ , g ′ ) holds for( a, g ) , ( a ′ , g ′ ) ∈ Ker d ∗ × e G ( Z ) k +1 ,δ . Then we have a − a ′ − gg ′ ) − d ( gg ′ ) = 0.Therefore, to prove that ϕ is injective, it suffices to show that, for ( a, g ) ∈ Ker d ∗ × e G ( Z ) k +1 ,δ , if ϕ ( a, g ) = 0 holds we have a = 0 and g = 1. Assume that ϕ ( a, g ) = 0. Then, since a ∈ Ker d ∗ , we have d ∗ ( g − dg ) = 0. On the other hand, d ( g − dg ) = 0also holds, and thus we can use the elliptic regularity. Therefore g − dg has theregularity of C ∞ . Since b ( Z ) = 0, there exists a function h ∈ C ∞ ( i Λ ( Z )) suchthat dh = g − dg . By the argument after Lemma 5.2 of Taubes [23], we can take h to be L δ . Since g − dg ∈ L k,δ holds, we finally get h ∈ L k +1 ,δ . Since h decays atinfinity, one can integrate by parts: 0 = ( d ∗ dh, h ) L δ = k dh k L δ , and hence dh = 0.Therefore h is constant, and moreover h is constantly zero again because of thedecay of h . Thus we get g − dg = 0, and hence g is constant and a = 0. Sincelim n →∞ g ( T n ( x )) = 1, we finally have g = 1. This completes the proof. (cid:3) Using Lemma 3.1 and restricting the map ν : C k,δ ( Z ) → L k − ,δ (Λ + ( Z ) ⊕ S − )corresponding to the Seiberg–Witten equations to the global slice, we get a mapfrom S k,δ , denoted by µ : µ : S k,δ → L k − ,δ (Λ + ( Z ) ⊕ S − ) . (21)This is a P in (2)-equivariant non-linear Fredholm map.
Remark . Note that, although the L δ -norm is P in (2)-invariant, the L k,δ -norm isnot P in (2)-invariant in general. However, by considering the average with respectto the
P in (2)-action, one can find a
P in (2)-invariant norm which is equivalentto the usual L k,δ -norm induced by the periodic metric and periodic connection.Henceforth we fix this P in (2)-invariant norm, and just call it a
P in (2)-invariant L k,δ -norm and denote it by k · k L k,δ .Via the isomorphism given in Lemma 3.1, the quotient µ − (0) /S can be identi-fied with the moduli space M k,δ ( Z ), and thus we get the following result by usingthe technique in Lin [11]. Proposition 3.3.
There exists δ > satisfying the following condition. For any δ ∈ (0 , δ ) , the space µ − (0) /S is compact.Proof. By using the identification between µ − (0) /S and M k,δ ( Z ), it is sufficientto show M k,δ ( Z ) is compact. Let { [( A n , Φ n )] } ⊂ M k,δ ( Z ) be any sequence in M k,δ ( Z ). Since ( A n , Φ n ) converges ( A ,
0) on the end for each n and, the topologicalenergy E top ( A n , Φ n ) := 14 Z Z F A tn ∧ F A tn defined in the book of Kronheimer–Mrowka [10] has a uniform bound E top ( A n , Φ n ) ≤ C . In addition, the equation (4.16) in [10] is still true in our situation: E an ( A n , Φ n ) = E top ( A n , Φ n ) + k ν ( A n , Φ n ) k L ( Z ) , where the analytic energy E an ( A, Φ) is given by E an ( A, Φ) := 14 Z X | F A | + Z X |∇ A Φ | + 14 Z X ( | Φ | + ( scal2 )) − Z X scal . Since ν ( A n , Φ n ) = 0, we have an inequality E an ( A n , Φ n ) ≤ C. We set W [ ǫ, ∞ ] := W [0 , ∞ ] \ Y × [0 , ǫ ] where Y × [0 , ǫ ] is a closed color neigh-borhood of Y × ⊂ W [0 , ∞ ]. The uniform boundedness of E an ( A n , Φ n ) implies SC AND 10/8 FOR END-PERIODIC 4-MANIFOLDS 15 that { [( A n , Φ n ) | W [ ǫ, ∞ ] ] } n has a convergent subsequence in L k,δ -topology by Theo-rem 4.5 in Lin [11]. (In Theorem 4.5 in [11], Lin imposed the boundedness of Λ q .This is because Lin considered the blow-up moduli space. On the other hand, forthe convergence in the un-blow-up moduli space, we only need the boundedness ofthe energy.) Therefore we have gauge transformations { g n } over W [ ǫ, ∞ ] such that { ( g ∗ n A n , g ∗ n Φ n ) } converges in L k,δ ( W [ ǫ, ∞ ]; i Λ ⊕ S + ). On the other hand, we alsohave energy bound E an (( A n , Φ n ) | M ∪ Y W ) ≤ C. By Theorem 5.1.1 in [10], we have gauge transformation h n on M ∪ Y Y × [0 , ǫ ′ ] suchthat { ( h ∗ n A n , h ∗ n Φ n ) } has a convergent subsequence in L k ( M ∪ Y Y × [0 , ǫ ′ ]; i Λ ⊕ S + )for ǫ < ǫ ′ . Pasting g n and h n by the use of a bump-function, we get gauge transfor-mations { g n h n } defined on the whole of Z satisfying that { ( g n h ∗ n A n , g n h n Φ n ) } has convergent subsequence in L k,δ ( Z ; i Λ ⊕ S + ). (This is a standard pasting ar-gument for gauge transformations. For example, see [3].) (cid:3) Set H k − ,δ := L k − ,δ (Λ + ( Z )) × L k − ,δ ( S − ) . For a positive real number η , we define B ( η ) := { x ∈ S k,δ | k x k L k,δ < η } Since our L k,δ -norm is P in (2)-invariant (see Remark 3.2),
P in (2) acts on S k,δ \ B ( η ). Therefore the space S k,δ \ B ( η ) has a structure of P in (2)-Hilbert manifoldwith boundary. Here let us recall we introduced positive numbers δ and δ inProposition 2.23 and in Proposition 3.3 respectively. The following lemma ensuresthat we can take a suitable and controllable perturbation of the Seiberg–Wittenequations outside a neighborhood of the reducible. Lemma 3.4.
For any δ > with < δ < min( δ , δ ) , η > and ǫ > , there existsa P in (2) -equivariant smooth map g ǫ : S k,δ \ B ( η ) → H k − ,δ satisfying the following conditions: • For every point γ ∈ ( µ | S k,δ \ B ( η ) + g ǫ ) − (0) , the differential d ( µ + g ǫ ) γ : S k,δ → H k − ,δ is surjective. • Any element of the image of g ǫ is smooth. • There exists
N > such that g ǫ ( x ) | W [ N, ∞ ] = 0 holds for any x ∈ S k,δ \ B ( η ) . • There exists a constant
C > such that k ( dg ǫ ) x : S k,δ → H k − ,δ k B ( S k,δ , H k − ,δ ) < Cǫ holds for any x ∈ S k,δ \ B ( η ) . Here k · k B ( · , · ) denotes the operator norm. • There exists a constant C ′ > such that k g ǫ ( x ) k L k − ,δ ≤ C ′ ǫ. holds for any x ∈ S k,δ \ B ( η ) . Proof.
Since S k,δ \ B ( η ) has free P in (2)-action, we have the smooth Hilbert bundle E := ( S k,δ \ B ( η )) × P in (2) H k − ,δ → ( S k,δ \ B ( η )) /P in (2) . Sicne µ : S k,δ \ B ( η ) → H k − ,δ is a P in (2)-equivariant map, µ determines a section µ ′ : ( S k,δ \ B ( η )) /P in (2) → E . The section µ ′ is a smooth Fredholm section andthe set µ ′− (0) is compact by Proposition 3.3. Now we consider a construction usedin Ruan’s virtual neighborhood [17]. Let γ ∈ µ ′− (0). The differential dµ ′ γ : T γ ( S k,δ \ B ( η )) /P in (2) → H k − ,δ is a linear Fredholm map, and therefore there exist a natural number n γ and alinear map f γ : R n γ → H k − ,δ such that dµ ′ γ + f γ : T γ ( S k,δ \ B ( η )) /P in (2) ⊕ R n γ → H k − ,δ is surjective. Concretely, we can give the map f γ as follows. Let V γ be a direct sumcomplement in H k − ,δ of the image of dµ ′ γ . By taking a basis of V γ , we get a linearembedding f γ : R n γ → V γ ⊂ H k − ,δ , where n γ = dim V γ . We here show that,by replacing f γ appropriately, we can assume that any element of Im f γ is smoothand has compact support. For each member of the fixed basis of V γ , we can takea sequence of smooth and compactly supported elements which converses to themember in L k − ,δ sense. Then we get a sequence of maps { f γ,l } l approaching f γ through the same procedure of the construction of f γ above. Since surjectivity isan open condition, for a sufficiently large l , by replacing f γ,l with f γ we can assumethat any element of Im f γ is smooth and has compact support.For each γ ∈ µ ′− (0), since surjectivity is an open condition, there exists asmall open neighborhood U γ of γ in ( S k,δ \ B ( η )) /P in (2) such that dµ ′ γ ′ + f γ is surjective for any γ ′ ∈ U γ . Since µ ′− (0) is compact, there exist finite points γ , . . . , γ p ⊂ µ ′− (0) such that µ ′− (0) ⊂ S pi =1 U γ p . Set n i := n γ i , f i := f γ i , U i := U γ i , and n := p X i =1 n i . We fix a smooth partition of unity { ρ i : U i → [0 , } i subordinate to { U i } pi =1 . Notethat, until this point, we have not used ǫ . We here define a section¯ g ǫ : ( S k,δ \ B ( η )) /P in (2) × R n → E by ¯ g ǫ ( x, v ) := ǫ p X i =1 ρ i ( x ) f i ( v i ) , (22)where ( x, v ) = ( x, ( v , . . . , v p )) ∈ ( S k,δ \ B ( η )) /P in (2) × R n = ( S k,δ \ B ( η )) /P in (2) × R n × · · · × R n p . One can easily check that, for any γ ∈ µ ′− (0), the differential d ( µ ′ + ¯ g ǫ ) ( γ, issurjective. Since any element of the image of f i ’s are smooth and has compactsupport, any element of the image of ¯ g ǫ is and does also. Note that ¯ g ǫ ( γ,
0) = 0holds for any γ ∈ µ ′− (0). Since surjectivity is an open condition, there existsan open neighborhood N of µ ′− (0) in ( S k,δ \ B ( η )) /P in (2) × R n such that, for SC AND 10/8 FOR END-PERIODIC 4-MANIFOLDS 17 any point z ∈ N , the linear map d ( µ ′ + ¯ g ǫ ) z is surjective. Because of the implicitfunction theorem, we can see that the subset U := { ( x, v ) ∈ N | ( µ + ¯ g ǫ )( x, v ) = 0 } of N , called a virtual neighborhood, has a structure of a finite dimensional manifold.By Sard’s theorem, the set of regular values of the map pr : U → R n defined as therestriction of the projection mappr : ( S k,δ \ B ( η )) /P in (2) × R n → R n is a dense subset of R n . Now we choose a regular value v ∈ R n with the sufficientlysmall norm such that { x ∈ ( S k,δ \ B ( η )) /P in (2) | ( µ + ¯ g ǫ )( x, v ) = 0 } × { v } ⊂ N . Define g ′ ǫ : ( S k,δ \ B ( η )) /P in (2) → E by g ′ ǫ ( x ) := ¯ g ǫ ( x, v ). Then we get a P in (2)-equivariant map g ǫ : S k,δ \ B ( η ) → H k − ,δ by considering the pull-back of g ′ ǫ by the quotient maps( S k,δ \ B ( η )) × H k − ,δ −−−−→ E = ( S k,δ \ B ( η )) × P in (2) H k − ,δ y y S k,δ \ B ( η ) −−−−→ ( S k,δ \ B ( η )) /P in (2)(23)and composing the projection ( S k,δ \ B ( η )) × H k − ,δ → H k − ,δ . The surjectivityof d ( µ ′ + ¯ g ǫ ) ensures that this g ǫ enjoys the first required condition in the statementof the lemma. Since any element of the image of ¯ g ǫ is smooth and has compactsupport, the map g ǫ satisfies the same condition. This implies that g ǫ meets thesecond and third conditions in the statement. The fourth and fifth conditions followfrom the expression (22). (cid:3) Kuranishi model.
To obtain the 10 / P in (2)-equivariant Kuranishi model forthis. We first recall the following well-known theorem. (For example, see TheoremA.4.3 in [15].)
Theorem 3.5 (Kuranishi model) . Let G be a compact Lie group and V and V ′ be Hilbert spaces equipped with smooth G -actions and G -invariant inner products.Suppose that there exists a G -equivariant smooth map f : V → V ′ with f (0) = 0 and df : V → V ′ is Fredholm. Then the following statements hold. (1) There exist G -invariant open subset U ⊂ V and a G -equivariant diffeomor-phism T : U → T ( U ) satisfying the following conditions: • T (0) = 0 . • There exist a G -equivariant linear isomorphism D : (Ker df ) ⊥ → Im df and a smooth G -equivariant map ˜ f : V → (Im df ) ⊥ such thatthe map f ◦ T : U → T ( U ) ⊂ V → V ′ is written as f ◦ T ( v, w ) = ( Dw, ˜ f ( v, w )) ∈ Im df ⊕ (Im df ) ⊥ = V ′ for ( v, w ) ∈ Ker df ⊕ (Ker df ) ⊥ = V . • If we define F : Ker df → (Im df ) ⊥ by F ( v ) := ˜ f ( v, , then U ∩ F − (0) can be identified with U ∩ f − (0) as G -spaces. (2) For a real number c satisfying < c ≤ / , let U c ( f ) be the open set in V defined by U c ( f ) = { x ∈ V | k pr (Ker df ) ⊥ − D − ◦ pr Im df ◦ df x k B ( V,V ) < c } , (24) where pr W is the projection to a subspace W and k − k B ( V,V ) is the operatornorm. Let Ψ : V → V be the map defined by Ψ( x ) := x + D − ◦ pr Im df ( f ( x ) − df ( x )) . Then, the image Ψ( U c ( f )) is an open subset of V and the restriction Ψ | U c ( f ) : U c ( f ) → Ψ( U c ( f )) is a diffeomorphism. (3) As the open set U in (1), we can take any open ball centered at the originand contained in Ψ( U c ( f )) . Spin Γ -structure on the Seiberg–Witten moduli space. In this subsec-tion, we show that there is a natural spin Γ-structure (equivariant spin structure) on f M k,δ ( Z ). To show this, we need several definitions related to an P in (2)-equivariantversion of family of indices for Fredholm operators. A non-equivariant version ofthe argument of this subsection is originally considered by H. Sasahira [19].Let G be a compact Lie group. Definition 3.6.
Let H and H be separable Hilbert spaces with G -linear actions.Let Fred( H , H ) be the set of Fredholm operators from H to H . We define atopology of Fred( H , H ) by the operator norm and an action of G on Fred( H , H )by f g − f g , where f ∈ Fred( H , H ) and g ∈ G .As in the non-equivariant case, for a compact G -space K , there is a map:ind K : [ K, Fred( H , H )] G → KO G ( K )via index indices of families, where [ K, Fred( H , H )] G is the set of G -homotopyclasses of G -maps from K to Fred( H , H ).In this subsection, for a fixed η >
0, we fix a perturbation g ǫ by the use ofLemma 3.4 for a fixed ǫ >
0. We also fix a
P in (2)-equivariant cut-off function ρ : S k,δ → [0 ,
1] satisfying ρ ( x ) = ( , if x ∈ B ( η ) , , if x ∈ B (2 η ) c , (25)where B (2 η ) c is the complement of B (2 η ) in S k,δ . (To construct such a function,we use a map induced by the square of the L k,δ -norm on S k,δ .) We have the P in (2)-equivariant smooth map g ǫ : S k,δ \ B ( η ) → H k − ,δ given in Lemma 3.4. We now consider the following map µ ǫ := µ + ρg ǫ : S k,δ → H k − ,δ . In our situation, we put H = S k,δ , H = H k − ,δ , G = P in (2) and K is an G -invariant topological subspace of H . The Fredholm maps d ( µ ǫ ) x : H → H for x ∈ K determine a class[ { d ( µ ǫ ) x } x ∈ K ] ∈ [ K, Fred( H , H )] P in (2) . SC AND 10/8 FOR END-PERIODIC 4-MANIFOLDS 19 If K = H , then we have an isomorphism[ H , Fred( H , H )] P in (2) ∼ = [0 , Fred( H , H )] P in (2) via
P in (2)-homotopies. This isomorphism implies that [ { d ( µ ǫ ) x } x ∈ H ] coincideswith [ { dµ } x ∈ H ] in [ H , Fred( H , H )] P in (2) since d ( µ ǫ ) = dµ .For a given P in (2)-space K and a P in (2)-module W , we denote by W theproduct P in (2)-bundle on K with fiber W . Lemma 3.7.
The class [ T ( µ − ǫ (0) \ B (2 η ))] ∈ KO P in (2) ( µ − (0) \ B (2 η )) is equalto [Ker dµ ] − [Coker dµ ] .Proof. Set K = µ − ǫ (0) \ B ( η ). The class [ T K ] ∈ KO P in (2) ( K ) is equal to the classind[ { d ( µ ǫ ) x } x ∈ K ] ∈ KO P in (2) ( K ) . On the other hand, the inclusion i : K → H induces the map i ∗ : [ H , Fred( H , H )] P in (2) → [ K, Fred( H , H )] P in (2) . Then one can check i ∗ [ { dµ } x ∈ H ] = i ∗ [ { d ( µ ǫ ) x } x ∈ H ] = [ { d ( µ ǫ ) x } x ∈ K ]. This im-plies the conclusion. (cid:3) Corollary 3.8.
Under the same assumption of Lemma 3.7, there exists a
P in (2) -module W with trivial P in (2) -action such that T ( µ − ǫ (0) \ B ( η )) ⊕ Coker dµ ⊕ W ∼ = Ker dµ ⊕ W as P in (2) -bundles.
By the use of Corollary 3.8, we can equip the Seiberg–Witten moduli space witha structure of spin Γ-manifold.
Corollary 3.9.
Under the same assumption of Lemma 3.7 and the condition b + ( M ) is even, µ − ǫ (0) \ B ( η ) has a structure of a spin Γ -manifold.Proof. By applying Corollary 3.8, we have a
P in (2)-module W with trivial P in (2)-action satisfying T ( µ − ǫ (0) \ B ( η )) ⊕ Coker dµ ⊕ W ∼ = Ker dµ ⊕ W . (26)We regard
P in (2)-modules as Γ-modules and
P in (2)-spaces as Γ-spaces via (23).Since b + ( M ) is even, the dimensions of Coker dµ and Ker dµ are even. ByLemma 2.14 and Lemma 2.12, Coker dµ and Ker dµ have spin Γ-module struc-tures. We equip W with the trivial spin G -module structure. The spin Γ-modulestructures on Coker dµ , Ker dµ and W determine spin structures on Coker dµ / Γ,Ker dµ / Γ and
W /
Γ as vector bundles on ( µ − ǫ (0) \ B ( η )) / Γ by Remark 2.11.The isomorphism (26) gives the isomorphism: T ( µ − ǫ (0) \ B ( η )) / Γ ⊕ Coker dµ / Γ ⊕ W / Γ ∼ = Ker dµ / Γ ⊕ W / Γ . Since Ker dµ / Γ ⊕ W /
Γ has a spin structure induced by the spin structures onKer dµ / Γ and
W / Γ, T ( µ − ǫ (0) \ B ( η )) / Γ also admit a spin structure. By Re-mark 2.17, we obtain a structure of a spin Γ-manifold on µ − ǫ (0) \ B ( η ). (cid:3) Remark . One can check that the spin
P in (2)-structure on µ − ǫ (0) \ B ( η ) inCorollary 3.9 does not depend on the choice of W . Main construction.
The following theorem contains the main constructionof this paper.
Theorem 3.11.
Under the assumption of Theorem 1.1 and the condition that b + ( M ) is even, there exist real spin Γ -modules U and U with Γ -invariant normsand Γ -equivariant smooth map φ : S ( U ) → U from the unit sphere of U satisfyingthe following conditions: • The group Γ acts freely on U \ { } . • The map φ : S ( U ) → U is transvers to ∈ U . • The Γ -manifold φ − (0) bounds a compact manifold acted by Γ freely, as aspin Γ -manifold. • As Γ -representation spaces, U i is isomorphic to ˜ R l i ⊕ H m i for i = 0 and ,where l = 0 , l = b + ( M ) and m − m = − ind C D A ,where the definition of Γ is given in (23) .Proof. Let X , Y , M and Z be as in Section 2. Suppose that X admits a PSC metric g X . We fix a positive number δ satisfying δ < min { δ , δ , δ } . (Recall that δ , δ ,and δ are given in Remark 2.6, Proposition 2.23, and Proposition 3.3 respectively.)We denote by D the operator dµ and put W = Ker D and W = Im D . Sincethe operator D : W ⊥ → W is an isomorphism, there exists the inverse map D − : W → W ⊥ , where ⊥ is the orthogonal complement with respect to L k,δ -norm in S k,δ . We use Theorem 3.5 for the following setting: V = S k,δ , V ′ = H k − ,δ , G = P in (2) , and φ = µ. Then we get a Kuranishi model for µ near the reducible. For this model, we usethe open subset U c ( µ ) ⊂ V for c with 0 < c < min { , k D − k − B ( W ,W ⊥ ) } definedas in (24), where k D − k B ( W ,W ⊥ ) is the operator norm of D − . We fix a positivereal number η satisfying B (4 η ) ⊂ U c ( µ ) ⊂ S k,δ (27)and also fix a P in (2)-equivariant cut-off function ρ : S k,δ → [0 ,
1] as (25). For any ǫ >
0, we have the G -equivariant smooth map g ǫ : S k,δ \ B ( η ) → H k − ,δ given in Lemma 3.4, and can consider the map µ ǫ = µ + ρg ǫ : S k,δ → H k − ,δ as in Subsection 3.3. Note that the map µ ǫ is a smooth P in (2)-equivariant Fredholmmap. Because of Lemma 3.4, the differential d ( µ ǫ ) x is surjective for any x ∈ µ − ǫ (0) ∩ B (2 η ) c , and therefore µ − ǫ (0) ∩ B (2 η ) c is a finite dimensional manifold. We also notethat µ ǫ = µ on B ( η ). We define Ψ µ ǫ : V → V byΨ µ ǫ ( x ) = x + D − ǫ ◦ pr Im( d ( µ ǫ ) ) ( µ ǫ ( x ) − d ( µ ǫ ) ( x ))= x + D − ◦ pr Im D ( µ ǫ ( x ) − D ( x )) , where D ǫ = d ( µ ǫ ) , which is nothing but D . We now use the following lemma: Lemma 3.12.
For δ with < δ < δ , the space µ − ǫ (0) /S is compact, where δ isthe constant given in Proposition 3.3. SC AND 10/8 FOR END-PERIODIC 4-MANIFOLDS 21
The proof of this lemma is given at the end of this subsection. AssumingLemma 3.12, then the space µ − ǫ (0) \ B ( η ) is also compact. Next we consider aneighborhood of the reducible. Applying Theorem 3.5 for f = µ ǫ , we obtain aKuranishi model for µ ǫ near the reducible. Here we use the open subset U c ( µ ǫ )defined as in (24) for this Kuranishi model. Fix a positive number c ′ satisfying c < c ′ < . We here choose ǫ so that k D − k B ( W ,W ⊥ ) ( C max | dρ | ǫ + C ′ ǫ ) < c ′ , where C and C ′ are the constants in Lemma 3.4. Then U c ( µ ) ⊂ U c ′ ( µ ǫ )(28)holds, because for x ∈ U c ( µ ) we have k pr (ker D ) ⊥ − D − ◦ pr Im D ◦ d ( µ ǫ ) x k B ( V,V ′ ) ≤k pr (ker D ) ⊥ − D − ◦ pr Im D ◦ ( dµ x + d ( ρg ǫ ) x ) k B ( V,V ′ ) ≤ c + k D − ◦ pr Im D ◦ d ( ρg ǫ ) x ) k B ( V,V ′ ) ≤ c + k D − k B ( W ,W ⊥ ) ( C max | dρ | ǫ + C ′ ǫ ) < c ′ . (29)Here we use the definition of U c ( µ ) given in Theorem 3.5 in the second inequalityand Lemma 3.4 in the last inequality.Next we show B (3 η ) ⊂ Ψ µ ǫ ( U c ′ ( µ ǫ )) . (30)We first note that the argument to get the inequality (29) also shows that k id − d (Φ µ ǫ ) x k < c ′ for any x ∈ B (4 η ). Going back to a proof of the inverse functiontheorem, this inequality implies that B (4 η (1 − c ′ )) ⊂ Ψ µ ǫ ( B (4 η )). (For example,see Lemma A.3.2 in [15].) Using (27), (28), we get B (4 η (1 − c ′ )) ⊂ Ψ µ ǫ ( U c ′ ( µ ǫ )).Thus we have (30).Here let us consider the P in (2)-invariant smooth map ψ : µ − ǫ (0) \ B (2 η ) → [2 η, ∞ )defined by ψ ( x ) := k x k L k,δ . Sard’s theorem implies that there exists a dense subset S in [2 η, ∞ ) such that s is a regular value of ψ for any s ∈ S . Now we choose s ∈ S with s < η , then the space ψ − ([ s, ∞ )) has a structure of a P in (2)-manifoldwith boundary. Now let us recall Theorem 3.5. Because of (30), Theorem 3.5 en-sures that there exists a
P in (2)-equivariant diffeomorphism T : B (3 η ) → T ( B (3 η ))satisfying the following conditions: • T (0) = 0. • The map µ ǫ ◦ T : B (3 η ) → T ( B (3 η )) ⊂ S k,δ → H k − ,δ is given by ( v, w ) ( Dw, ˜ µ ǫ ( v, w )) via the decompositions S k,δ = Ker d ( µ ǫ ) ⊕ (Ker d ( µ ǫ ) ) ⊥ and H k − ,δ = Im d ( µ ǫ ) ⊕ (Im d ( µ ǫ ) ) ⊥ . Here D : (Ker d ( µ ǫ ) ) ⊥ → Im d ( µ ǫ ) is a P in (2)-equivariant linear isomorphism and˜ µ ǫ : S k,δ → (Im d ( µ ǫ ) ) ⊥ is a smooth P in (2)-equivariant map. • Define µ ∗ ǫ : Ker d ( µ ǫ ) → (Im d ( µ ǫ ) ) ⊥ by µ ∗ ǫ ( v ) := ˜ µ ǫ ( v, B (3 η ) ∩ ( µ ∗ ǫ ) − (0) can be identified with B (3 η ) ∩ ( µ ǫ ) − (0) as P in (2)-spaces.Since µ ǫ ( x ) = µ ( x ) for x ∈ B ( η ), we can see thatKer d ( µ ǫ ) ∼ = Ker D A , Coker d ( µ ǫ ) ∼ = Coker D A ⊕ ˜ R b + (31)as P in (2)-modules by Proposition 2.23. We set U := Ker d ( µ ǫ ) and U := Coker d ( µ ǫ ) . We equip U with the norm defined by k v k := 1 √ s k v k L k,δ and U with that defined as the restriction of the L k − ,δ -norm. We set φ := µ ∗ ǫ | S ( U ) .We regard P in (2)-modules as Γ-modules and
P in (2)-spaces as Γ-spaces via (23).Now we check that the conclusions of Theorem 3.11 are satisfied. Since b + ( M ) iseven, dim U and dim U are also even. By Lemma 2.14 and Lemma 2.12, U and U admit real spin Γ-module structures. The map φ : S ( U ) → U (32)is transverse to 0 because of the choice of s . This implies the second condition.Since the P in (2)-action on Ker d ( µ ǫ ) = Ker D A by quaternionic multiplication,the first condition follows. By the use of Corollary 3.9, we can equip a structure ofa spin Γ-manifold on ψ − ([ s, ∞ )). On the other hand, the differential of (32) givesan isomorphism Ker dφ ⊕ U ⊕ R ∼ = U . We equip R with the trivial real spin Γ-module structure. The the vector bundles U / Γ, R / Γ and U / Γ on φ − (0) / Γ has spin structures by Remark 2.11. Therefore, φ − (0) / Γ also admit a spin structure. This induces a real spin Γ-manifold structureon φ − (0). Since the constructions are same, the structure of a spin Γ-manifold on φ − (0) coincide with that of ∂ ( ψ − ([ s, ∞ ))). This implies the third condition. Theisomorphism (31) implies the fourth condition. (cid:3) At the end of this subsection, we give the proof of Lemma 3.12.
Proof of Lemma 3.12.
The proof is similar to that of Proposition 3.3. Let { [( A n , Φ n )] } be a sequence in ν − ǫ (0) /S . For all n , the pair ( A n , Φ n ) satisfies the equation( ν + ρg ǫ )( A n , Φ n ) = 0 . Because of the property of g ǫ in Lemma 3.4, we have the inequality k g ǫ ( A n , Φ n ) k L k − ,δ < C for ( A n , Φ n ) ∈ S k,δ . Then the analytical energy (see the proof of Proposition 3.3)of ( A n , Φ n ) is bounded by some positive number (independent of n ) as in the proofof Proposition 3.3. Moreover, there exists a positive integer N >> g ǫ ( A ) | W [ N, ∞ ] = 0 for A ∈ S k,δ . Therefore ( A n , Φ n ) satisfies the usual Seiberg–Witten equation on W [ N +1 , ∞ ] for all n . There exist a subsequence { ( A n ′ , Φ n ′ ) } of { ( A n , Φ n ) } and gauge transformations g n on W [ N + 2 , ∞ ] such that { g ∗ n ( A n ′ , Φ n ′ ) } converges on W [ N + 2 , ∞ ] as in the argument in Proposition 3.3. We should showthe existence of a subsequence { ( A n ′′ , Φ n ′′ ) } of { ( A n ′ , Φ n ′ ) } and gauge transfor-mations h n on M ∪ Y W [0 , N + 3] satisfying that { h ∗ n ( A n ′′ , Φ n ′′ ) } converges on SC AND 10/8 FOR END-PERIODIC 4-MANIFOLDS 23 M ∪ W [0 , N + 3]. It can be proved by essentially the same way as in Theorem 5.1.1of [10]. The key point is the boundedness of the analytical energies of ( A n ′ , Φ n ′ ).Finally, we paste g n and h n by some bump-functions and get the conclusion. (cid:3) Proof of Theorem 1.1.
In this section, we give a proof of Theorem 1.1.
Proof of Theorem 1.1.
Let Y , X , M and Z be as in Section 2. We assume that b + ( M ) is even. By applying Theorem 3.11, there exist real spin Γ-modules U and U with Γ-invariant norms and Γ-equivariant smooth map φ : S ( U ) → U fromthe unit sphere of U satisfying the following conditions: • The group Γ acts freely on U \ { } . • The map φ : S ( U ) → U is transvers to 0 ∈ U . • The Γ-manifold φ − (0) bounds a compact manifold acted by Γ freely, as aspin Γ-manifold. • As Γ-representation spaces, U i is isomorphic to ˜ R l i ⊕ H m i for i = 0 and 1,where l = 0 , l = b + ( M ) and 2 m − m = − ind C D A .By the forth condition we can write U = H m , U = ˜ R l ⊕ H m . On the other hand, we have a smooth map φ : S ( U ) → U which is transverse to0. By the definition, we have the equality w ( U , U ) = [ φ − (0)] ∈ Ω spinΓ , free , By the third condition, the class w ( U , U ) = 0. Therefore we apply Theorem 2.19and obtain an element α ∈ KO r − r Γ ( pt ) such that e ( ˜ R l ) e ( H m ) = e ( ˜ R l ⊕ H m ) = αe ( H m ) , where dim U = 4 m = r and dim U = 4 m + l = r . Now we apply Theorem 2.20and get the inequality 0 ≤ m − m ) + l − . This implies that ind C D A + 1 ≤ b + ( M ) . By combining Lemma 2.9 and Theorem 2.8, we have h ( Y ) − σ ( M )8 + 1 ≤ b + ( M ) . (cid:3) Remark . If we take the connected sum M S × S , we can always assumethat b + ( M ) is even. Therefore, for general spin bound M of Y , we have h ( Y ) − σ ( M )8 ≤ b + ( M ) . Remark . D. Veloso [24] considered boundedness of the monopole map forperiodic-end 4-manifolds which admit PSC metric on the ends. It seems that thisargument shall also provide a similar conclusion. The authors would like to expresstheir deep gratitude to Andrei Teleman for informing them of Veloso’s argument.
Remark . In [7], Furura–Kametani showed the 10 / Examples
Using Corollary 1.4, we can construct a large family of examples of rationalhomology S × S which do not admit PSC metrics. To find explicit examples of Y in Corollary 1.4, one can consider Brieskorn 3-manifolds. In [21], N. Savelievshowed that − Σ( p, q, pqm + 1) for relatively prime numbers p, q ≥ m bounds compact spin 4-manifolds which violate the 10/8-inequalities. Example . Let ( p, q ) be a pair of relatively prime numbers satisfying A − B > A and B are natural numbers defined in [21], m be an odd positive integer, and j be a positive integer. In [21] Saveliev showed that − Σ( p, q, pqm +1) bounds a compact simply connected 4-manifold whose intersectionform is given by a ( − E ) ⊕ b (cid:18) (cid:19) for some ( a, b ) satisfying A ≤ a , b ≤ B . Onthe other hand, since − Σ( p, q, pqm + 1) bounds both negative and positive definitesimply connected 4-manifolds (see [4]), the Frøyshov invariant satisfies h ( − Σ( p, q, pqm + 1)) = 0 . (33)Thus we have ψ ([ j ( − Σ( p, q, pqm + 1))]) > ψ is the mapdefined in (3). Corollary 1.4 implies that any rational homology S × S which has j ( − Σ( p, q, pqm + 1)) as a cross section does not admit a PSC metric.For example, for j > m , let Y be a 3-manifold given as one of j ( − Σ(4 , , m + 1)) , j ( − Σ(4 , , m + 1)) , and j ( − Σ(4 , , m + 1)) . (34)Then any rational homology S × S which has Y as a cross section does not admita PSC metric. Remark . Lin [11] showed the equality (1) under the assumption that X admits aPSC metric and Y is a cross section of X . On the other hand, the mod 2 reductionof λ SW ( X ) coincides with the Rochlin invariant µ ( Y ) of Y [16]. Therefore theequality h ( Y ) ≡ µ ( Y ) mod 2(35)holds if X admits a PSC metric and Y is a cross section of X , hence this equality(35) also gives an obstruction to PSC metric on X . For example, any rationalhomology S × S which has j ( ± Σ( p, q, pqm + 1)) for p, q and m satisfying − m ( p − q − ≡ j ≡ − m ( p − q −
1) is theCasson invariant λ (Σ( p, q, pqm + 1)) (see Example 3.30 in [22]). Since the Frøyshovinvariant of Σ( p, q, pqm +1) is equal to 0 for all j , p , q and m , j ( ± Σ( p, q, pqm +1))with (36) does not satisfy (35). On the other hand, note that one cannot show thenon-existence of a PSC metric on any rational homology S × S which has Y givenin (34) as a cross section using the obstruction obtained from the equality (35) since h ( Y ) ≡ µ ( Y ) ≡ SC AND 10/8 FOR END-PERIODIC 4-MANIFOLDS 25
Here we give examples of 4-manifolds for which we can show the non-existenceof PSC metrics using our obstruction, but for which one cannot show it using otherknown ways. Although the equality (35) obviously gives only reduced informationsince we take mod 2, the equality (1) gives the full information obtained from Lin’sobstruction. We here exhibit pairs of (
X, Y ) which satisfy (1) but X do not admitPSC metrics. Example . Let ( q, r ) be a pair of relatively prime odd numbers. Let T ( q, r )denote ( q, r )-torus knot. Note that the double branched cover of T ( q, r ) is theSeifert manifold Σ(2 , q, r ). It is known that the signature of T ( q, r ) is equal to8 λ (Σ(2 , q, r )) (see Example 5.10 of [22]). Let K be ( − T ( q, qk + 1)) lT (3 , q, k and l are positive integers satisfying q ≡ , k ≡ , l > kq = 16 l + k. The the double branched cover Σ( K ) of K is − Σ(2 , q, qk + 1) l Σ(2 , , − K ) = − Σ( K ) and Σ( K J ) = Σ( K ) J ).) Let τ be theinvolution of the branched cover. We set X ( K ) as the mapping torus of τ . InTheorem C of [13], Lin–Ruberman–Saveliev showed − λ SW ( X ( K )) = sign( K )8 , where sign( K ) is the signature of K . Therefore we have λ SW ( X ( K )) = − sign( K )8= sign( T ( q, qk + 1))8 − l sign( T (3 , λ (Σ(2 , q, qk + 1)) − lλ (Σ(2 , , − k ( q −
1) + 2 l = 0by the choice of k , l and q . We use Y ( K ) = − Σ(2 , q, qk + 1) l Σ(2 , ,
11) as across section of X ( K ). Since h (Σ( p, q, pqk + 1)) = 0 for a pair of relatively primenumbers ( p, q ) and a positive integer k , we get h ( Y ( K )) = 0. Therefore the pair( X ( K ) , Y ( K )) satisfies (1). Moreover, Saveliev [20] and Manolescu [14] constructedthe following spin boundings: • − Σ(2 , q, qk + 1) = ∂ ( − (cid:0) q +14 (cid:1) E ⊕ (cid:18) (cid:19) ) and • Σ(2 , ,
11) = ∂ ( − E ⊕ (cid:18) (cid:19) ).Corresponding these results, we can show ψ ( − Σ(2 , q, qk + 1)) ≥ (cid:18) q + 14 (cid:19) − ψ (Σ(2 , , ≥ . (37)This implies that, for q ≥ ψ ( Y ( K )) is positive. We can see X ( K ) does notadmit a PSC metric by using Corollary 1.4.It is easy to show that for any element [ Y ] ∈ Θ and [ M ] ∈ Π, there exists
N >> Y ] N [ M ] ∈ Π. This property of ψ gives the following example. Example . For any homology sphere Y , there exists N >> • ψ ( Y − Σ(2 , q, q + 1))) > • ψ ( Y q ( − Σ(4 , , > q > N . This is shown by using ψ ( − Σ(4 , , > S × S ’s X and X , we fix an embedding f i from S to X i representing a fixed generator of H ( X i ; Q ) for i = 1 ,
2. We also fixa tubular neighborhood of f i ( S ) and its identification g i : S × D → X i of f i . Ifwe choose a diffeomorphism ξ from S × S to itself, we obtain a diffeomorphism ξ ∗ := g | S × ∂D ◦ ξ ◦ ( g | g ( S × ∂D ) ) − : g ( S × ∂D ) → g ( S × ∂D ) . We define connected sum of X and X along S via ξ ∗ by X ξ ∗ X := ( X \ int(Im g )) ∪ ξ ∗ ( X \ int(Im g )) , where int(Im g i ) is the interior of Im g i in X i for i = 1 ,
2. We can show that X ξ ∗ X is a rational homology S × S .Example 4.4 implies the following fact. Corollary 4.5.
For any rational homology S × S X which has some homology S as a cross section, there exists N >> such that X ξ ∗ ( S × ( − Σ(2 , q, q + 1))) and X ξ ∗ ( S × ( q ( − Σ(4 , , do not admit PSC metrics for q > N and some ξ ∗ .Proof. Let Y be a homology S with [ Y ] = 1 ∈ H ( X, Q ). Then Example 4.4implies that there exists N >> • ψ ( Y − Σ(2 , q, q + 1))) > • ψ ( Y q ( − Σ(4 , , > q > N . Note that we can choose Y − Σ(2 , q, q +1)) (resp. Y q ( − Σ(4 , , X ξ ∗ ( S × ( − Σ(2 , q, q +1))) (resp. X ξ ∗ ( S × ( q ( − Σ(4 , , ξ ∗ . Now we use Corollary 1.4, we obtain the conclusion. (cid:3) References [1] M. F. Atiyah, R. Bott, and A. Shapiro,
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