THE KUNNETH FORMULA IN FLOER HOMOLOGY FOR MANIFOLDS WITH RESTRICTED CONTACT TYPE
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THE KUNNETH FORMULA IN FLOER HOMOLOGY FOR MANIFOLDS WITH RESTRICTED CONTACT TYPE BOUNDARY ALEXANDRU OANCEA Department of Mathematics, ETHZ, Ramistrasse 101, 8092 Zurich (CH). Email: oancea@ math.ethz.ch Abstract. We prove the Kunneth formula in Floer (co)homology for mani- folds with restricted contact type boundary. We use Viterbo's definition of Floer homology, involving the symplectic completion by adding a positive cone over the boundary. The Kunneth formula implies the vanishing of Floer (co)homology for subcritical Stein manifolds. Other applications include the Weinstein conjecture in certain product manifolds, obstructions to exact La- grangian embeddings, existence of holomorphic curves with Lagrangian bound- ary condition, as well as symplectic capacities. 1. Introduction The present paper is concerned with the Floer homology groups FH?(M) of a compact symplectic manifold (M, ?) with contact type boundary, as well as with their cohomological dual analogues FH?(M). The latter were defined by Viterbo in [V] and are invariants that take into account the topology of the underlying manifold and, through an algebraic limit process, all closed characteristics on ∂M . Their definition is closely related to the Symplectic homology groups of Floer, Hofer, Cieliebak and Wysocki [FH, CFH, FHW, CFHW, C1]. Throughout this paper we will assume that ? is exact, and in particular ??, pi2(M)? = 0.
Voir
- kunneth exact
- construction can
- lagrangian embedding
- conley-zehnder index modulo
- m?
- compact region
- m? ?
- valued conley-zehnder
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THE K UNNETH FORMULA IN FLOER HOMOLOGY FOR MANIFOLDS WITH RESTRICTED CONTACT TYPE BOUNDARY
ALEXANDRU OANCEA Department of Mathematics, ETHZ, Ramistrasse101,8092Zurich(CH). Email:oancea @ math.ethz.ch
Abstract.prove the Kunneth formula in Floer (co)homology for mani-We folds with restricted contact type boundary. We use Viterbo’s denition of Floer homology, involving the symplectic completion by adding a positive cone over the boundary. The Kunneth formula implies the vanishing of Floer (co)homology for subcritical Stein manifolds. Other applications include the Weinstein conjecture in certain product manifolds, obstructions to exact La-grangian embeddings, existence of holomorphic curves with Lagrangian bound-ary condition, as well as symplectic capacities.
1.Introduction
The present paper is concerned with the Floer homology groupsF H(M) of a compact symplectic manifold (M, ω) with contact type boundary, as well as with their cohomological dual analoguesF H(MbroweredenedbyViteT.)alehrett in [V] and are invariants that take into account the topology of the underlying manifoldand, through an algebraic limit process, all closed characteristics on∂M. Their denition is closely related to the Symplectic homology groups of Floer, Hofer, Cieliebak and Wysocki [FH, CFH, FHW, CFHW, C1]. Throughout this paper we will assume thatωis exact, and in particular hω, 2(M)i This last condition will be referred to as= 0.symplectic asphericity. The groupsF H(M) are invariant with respect to deformations of the symplectic formωthat preserve the contact type character of the boundary and the condi-tionhω, 2(M)i groups= 0. TheF H(M) actually depend only on thesymplectic c c completionMofM manifold. TheMis obtained by gluing a positive cone along the boundary∂Mand carries a symplectic formbωwhich is canonically determined c byωehtdnalvectorconformaleodnM shall often write. WeF H(M) instead c ofF H(M grading on). TheF H(M) is given by minus the Conley-Zehnder index modulo 2, withthe minimal Chern number ofM. There exist canonical maps Hn+(M, ∂M) c→F H(Mc), F H(Mc) c→Hn+(M, ∂M) which shift the grading byn=2dimM.
Date: 24 August 2005. 2000MathematicsSubjectClassication: 53D40, 37J45, 32Q28. The author is currently supported by the Forschungsinstitut fur Mathematik at ETH Zurich. 1
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ALEXANDRU OANCEA
Theorem A (Kunneth formula).Let(M2m, ω)and(N2n, )be compact sym-plectic manifolds withrestricted the minimal Chern Denotecontact type boundary. numbers ofM,NandMNbyM,NandMN= gcd (M, N)respectively. (a)For any ringAncewequesplihichstntiecoeexrehestohsastsistcaxetrfco noncanonically (1)Lrb+bs=kF Hrb(M ω)F Hbs(N )F Hk(MN ω)
Lbr+sb=k 1Tor1A`F Hrb(M ω) F Hs(N )´ b The morphismcinduces a morphism of exact sequences whose source is the Kunneth exact sequence of the product(M, ∂M)(N, ∂N)and whose target is (1). (b)oFaryneldKismmosihproerehnasiiecstntooefc (2)Lb b b)F Hk(MN ω) r+bs=kF Hr(M ω)F Hs(N The morphismcinduces a commutative diagram with respect to the Kunneth isomorphism in cohomology for(M, ∂M)(N, ∂N).
In the above notation we havek∈Z/2MNZ, 0r2M 1, 0s2N 1 and thebsymbol associates to an integer its class in the correspondingZ/2Z ring. The reader can consult [D, VI.12.16] for a construction of the Kunneth exact sequence in singular homology. The algebraic properties of the mapc thestrongly in
uence symplectic topology of the underlying manifold. Our applications are based on the following theorem, which summarizes part of the results in [V]. Theorem (Viterbo[V]).Let(M2m, ω)be a manifold with contact type boundary such thathω, 2(M)i= 0 the map. Assumec:F H(M) →H2m(M, ∂M)is notsurjective. Then the following hold. (a)The same is true for any hypersurface of restricted contact typeM bounding a compact region; (b)Any hypersurface of contact typeMbounding a compact region carries a closed characteristic (Weinstein conjecture); (c)There is no exact Lagrangian embeddingLM(hereMis assumed to be exact by denition); (d)For any Lagrangian embeddingLMthere is a loop onLwhich is con-tractible inM, has strictly positive area and whose Maslov number is at most equal tom+ 1; (e)For any Lagrangian embeddingLMand any compatible almost complex structureJthere is a nonconstantJ-holomorphic curveS(of unknown genus) with non-empty boundary∂SL.
Viterbo [V] introduces the following denition, whose interest is obvious in the light of the above theorem. Denition 1.(Viterbo)A symplectic manifold(M2m, ω)which veries hω, 2(M)i= 0is said to satisfy the Strong Algebraic Weinstein Conjecture (SAWC) if the composed morphism below is not surjective ) pr→H2m(M, ∂M). F H(M) c→H(M, ∂M
