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9 Conta t Homology of Hamiltonian mapping tori 0 0 2 Oliver Fabert n Mathematis hes Institut der Ludwig-Maximilians-Universität Mün hen a J Theresienstr. 39, 80333 Mün hen, Germany 3 fabertmath.lmu.de 1 January 13, 2009 ] G S . Abstra t h t In the general geometri setup for sympleM ti (cid:28)eld theory the onta t a φ manifolds an be repla ed by mapping tori of symple ti manifolds m (M,ω) φ withsymple tomorphisms . Whilethe ylindri al onta thomology M φ [ of φ is given by the Floer homologies of powers of , the other algebrai invariants of symple ti (cid:28)eld theory for Mφ provide natural generalizations 4 M of symple ti Floer homology. For symple ti ally aspheri al and Hamil- v φ 6 tonian westudythemodulispa esofrational urvesandproveatransver- 0 sality result, whi h does not need the polyfold theory by Hofer, Wyso ki 4 Manφd∼=ZeSh1n×deMr. We use our result to ompute the full onta t homology of 9 . 0 6 0 Contents / h t 1 Introdu tion and main results 2 a m 1.1 Symple ti (cid:28)eld theory in the Floer ase . . . . . . . . . . . . . . . 2 1.2 Main theorem and outline of the proof . . . . . . . . . . . . . . . . 4 : v i 2 Moduli spa es 7 X R×S1×M 2.1 Holomorphi urves in . . . . . . . . . . . . . . . . . . 7 S1 r a 2.2 How to a hieve transversalitywith -symmetry . . . . . . . . . . 11 3 Domain-dependent Hamiltonians 16 3.1 Deligne-Mumford spa e . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 De(cid:28)nition of oherent Hamiltonian perturbations . . . . . . . . . . 17 3.3 Compatibility with SFT ompa tness. . . . . . . . . . . . . . . . . 19 4 Transversality 22 4.1 Bana h spa e bundle and Cau hy-Riemann operator . . . . . . . . 22 4.2 Universal moduli spa e . . . . . . . . . . . . . . . . . . . . . . . . . 25 1 2 1 INTRODUCTION AND MAIN RESULTS 5 Cobordism 30 5.1 Moduli spa es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2 Transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6 Conta t homology 36 6.1 Chain omplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.2 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . 37 1 Introdu tion and main results 1.1 Symple ti (cid:28)eld theory in the Floer ase Thispaperis on ernedwithsymple ti (cid:28)eldtheoryintheFloer ase. Symple ti (cid:28)eld theory (SFT) is a very large proje t designed to des ribe in a uni(cid:28)ed way the theory of holomorphi urves in symple ti and onta t topology. To be more pre ise, it approa hes Gromov-Witten theory in the spirit of a topologi al quantum (cid:28)eld theory by ounting holomorphi urves in ylinders over onta t manifolds and symple ti obordisms between them. It was initiated by Eliash- berg, Givental and Hofer in their paper [EGH℄ and sin e then has found many striking appli ations in symple ti geometry and beyond. While most of the urrent appli ations lie in (cid:28)nding invariants for onta t manifolds, there exists a generalized geometri setup for symple ti (cid:28)eld theory, whi h ontains onta t manifolds as spe ial ase. (2m−1) Following[BEHWZ℄ and [CM2℄ a Hamiltonian stru ture on a losed - V ω V dimensional manifold is a losed two-form on whi h is maximally kerω = {v ∈ TV : ω(v,·) = 0} nondegenerate in the sense that is a one- dimensional distribution. Note that here we (and [CM2℄) di(cid:27)er slightly from [EKP℄. The Hamiltonian stru ture is required to be stable in the sense that λ V kerω ⊂ kerdλ λ(v) 6= 0 there exists a one-form on su h that and for all v ∈ kerω −{0} (ω,λ) . Any stable Hamiltonian stru ture de(cid:28)nes a symple ti (ξ = kerλ,ω ) ω ω ξ ξ hyperplane distribution , where is the restri tion of , and a R V R∈kerω λ(R)=1 ve tor(cid:28)eld on by requiring and whi h is alled the Reeb ve tor (cid:28)eld of the stable Hamiltonian stru ture. V (ω,λ) Examplesfor losedmanifolds withastableHamiltonianstru ture are onta t manifolds, ir le bundles and mapping tori ([BEHWZ℄,[CM2℄). For this λ V (ω :=dλ,λ) notethatwhen isa onta tformon ,thenitiseasyto he kthat is a stable Hamiltonian stru ture and the symple ti hyperplane distribution agrees (M,ω) with the onta t stru ture. For the other two ases, let be a symple ti S1 → V → M manifold. Then any prin ipal ir le bundle and any symple ti M → V → S1 V = Mφ = R×M/{(t,p) ∼ (t+ 1,φ(p))} mapping torus , i.e., φ ∈ Symp(M,ω) for arries also a stable Hamiltonian stru ture. For the ir le 1.1 Symple ti (cid:28)eld theory in the Floer ase 3 π∗ω bundle the Hamiltonian stru ture is given by the pullba k under the bundle λ S1 proje tion and the one-form is given by any - onne tion form. On the other V = M φ hand, the stable Hamiltonian stru ture on the mapping torus is given ω ∈ Ω2(M ) φ by lifting the symple ti form to via the natural (cid:29)at onne tion TV = TS1 ⊕ TM λ = dt S1 t M φ and setting for the natural - oordinate on . ξ While in the mapping torus ase is always integrable, in the ir le bundle ase ξ the hyperplane distribution may be integrable or non-integrable, even onta t. V Symple ti (cid:28)eld theory assigns algebrai invariants to losed manifolds with a stable Hamiltonian stru ture. The invariants are de(cid:28)ned by ounting J R×V -holomorphi urves in with (cid:28)nite energy, where the underlying losed Riemann surfa es are expli itly allowed to have pun tures, i.e., single points J are removed. The almost omplex stru ture on the ylindri al manifold R×V R is required to be ylindri al in the sense that it is -independent, links R×V R the two natural ve tor (cid:28)elds on , namely the Reeb ve tor (cid:28)eld and R ∂s J∂s = R the -dire tion , by , and turns the symple ti hyperplane distri- V TV ξ = TV ∩ JTV bution on into a omplex subbundle of , . It follows J R×V that a ylindri al almost omplex stru ture on is determined by its J ξ ⊂ TV ω restri tion ξ to , whi h is required to be ξ- ompatible in the sense ω (·,J ·) ξ that ξ ξ de(cid:28)nes a metri on . Note that in [CM2℄ su h almost om- J plex stru tures are alled ompatible with the stable Hamiltonian stru ture andthatthesetofthesealmost omplexstru turesisnon-emptyand ontra tible. While the pun tured urves in symple ti (cid:28)eld theory may have arbitrary genus and arbitrary numbers of positive and negative pun tures, it is shown in [EGH℄ that there exist algebrai invariants ounting only spe ial types of urves: While in rational symple ti (cid:28)eld theory one ounts pun tured urves with genus zero, onta t homology is de(cid:28)ned by further restri ting to pun tured spheres with only one positive pun ture. Further restri ting to spheres with both just one negative and one positive pun ture, i.e., ylinders, the resulting algebrai invariant is alled ylindri al onta t homology. Note however that onta t homology and ylindri al onta t homology are not always de(cid:28)ned. In order to prove the well-de(cid:28)nedness of ( ylindri al) onta t homology it however su(cid:30) es to show that there are no pun tured holomorphi urves where all pun tures are negative (or all pun tures are positive). While the existen e of holomorphi urves without positive pun tures an be ex luded for all onta t manifolds using the maximumprin iple, whi h shows that onta t homologyis well-de(cid:28)ned for all onta t manifolds, it an be seen from homologi al reasons that for mapping tori Mφ R×Mφ there annot exist holomorphi urves in arrying just one type of pun tures, whi h shows that in this ase both onta t homology and ylindri al onta t homology are de(cid:28)ned. Symple ti (cid:28)eldtheoryhen eprovidesawealthofinvariants. However,almost all omputations performed so far only use the simplest one, ylindri al onta t 4 1 INTRODUCTION AND MAIN RESULTS homology: While ylindri al onta t homology is omputed e.g. for sub riti al Stein-(cid:28)llable onta tmanifolds([Y1℄),Brieskornvarieties([K℄)andtoroidalthree- manifolds ([BC℄), omputations of the higher invariants areperformed so far only for overtwisted onta t manifolds in [Y2℄ and sket hed in [EGH℄ for prequantiza- tion spa es and in [CL℄ for unit otangent bundle of tori. 1.2 Main theorem and outline of the proof M φ While it an be seen that the ylindri al onta t homology for mapping tori φ agrees with the Floer homology of the powers of , i.e., the sub omplex for the T ∈N φT period agreeswiththeFloerhomologyof , theotheralgebrai invariants ofsymple ti (cid:28)eldtheory,inparti ular,thefull onta thomology,providenatural generalizationsof symple ti Floerhomology. While FloerhomologyforHamilto- niansymple tomorphismsoverasuitable oe(cid:30) ientringisknowntobeisomorphi tothetensorprodu tofthesingularhomologywithrational oe(cid:30) ientsoftheun- Q[H2(M)] derlying symple ti manifold with the graded group algebra generated H (M) 2 by , Q[H2(M)]={ q(A)eA :A∈H2(M),q(A)∈Q}, degeA =hc1(TM),Ai, X the ase of arbitrary symple tomorphisms is mu h more ompli ated, see [CC℄ and the referen es therein. So we restri t our attention to the Hamiltonian ase, φ where the symple tomorphism is Hamiltonian, i.e., the time-one map of the H :S1×M →R sympφleH ti (cid:29)ow of a Hamiltonian .MIn∼=thiSs1 ×asMe the Hamiltonian φ (cid:29)ow providesus with a natural di(cid:27)eomorphism , sothat we an M S1×M φ repla e by equipped with the pullba k stable Hamiltonian stru ture (ωH,λH) S1×M ωH =ω+dH∧dt λH =dt on givenby , with symple ti bundle ξH = TM RH = ∂ +XH XH and Reeb ve tor (cid:28)eld t t , where t is the symple ti H =H(t,·) t g(Rra×diMenφt,oJf) . In [EKP℄ t(hRis×iSs1a×lsoM a,lJleHd)the Floer ase. Furthermore an be identi(cid:28)ed with equipped with the pullba k ylindri aTl(aRlm×oSst1× oMmp)le=xRst2r⊕u TtuMre, whi hJiHs nonstandard inHtCh∗e(Msenφs,eJ)that the splitting is not - omplex. Let denote M φ the onta thomologyofthesymple ti mappingtorus with hosen ylindri al J R×Mφ almost omplex stru ture on . (M,ω) Main Theorem 1. Let be a losed symple ti manifold, whi h is sym- h[ω],π (M)i = 0 φ : M → M 2 ple ti ally aspheri al, , and let be a Hamiltonian symple tomorphism. Then we have HC∗(Mφ,J) ∼= S H∗−2(M,Q) ⊗Q[H2(M)], (cid:0)MN (cid:1) S where is the graded symmetri algebra fun tor. 1.2 Main theorem and outline of the proof 5 JH For the proof we observethat the ylindri al almost omplex stru ture on R×S1×M S1 isspe i(cid:28)edbythe hoi eofan -familyofalmost omplexstru tures Jt M S1 H : S1 ×M → R on and an -dependent Hamiltonian . In order to get S1 an -symmetry on moduli spa es of urves with three or more pun tures, we J H t t restri t ourselves to almost omplex stru tures and Hamiltonians , whi h t∈S1 are independent of , so that only holomorphi ylinders need to be ounted for the di(cid:27)erential in onta t homology. We a hieve transversality for all moduli spa es by onsidering domain- dependent Hamiltonian perturbations. This means that, for de(cid:28)ning the Cau hy-Riemann operator for urves, we allow the Hamiltonian to depend expli itly on points on the pun tured sphere underlying the urve whenever the pun tured sphere is stable, i.e., there are no nontrivial automorphisms, where we follow the ideas in [CM1℄. Note however that in ontrast to the Gromov-Witten ase we now have to make oherent hoi es for the di(cid:27)erent moduli spa es simultaneously, i.e., the di(cid:27)erent Hamiltonian perturbations must be ompatible with gluing of urvesin symple ti (cid:28)eld theory. For the ylindri al moduli spa es the Hamiltonian perturbation is domain-independent, and it is H known from Floer theory that in general we must allow to depend expli itly t ∈ S1 on to a hieve nondegenera y of the periodi orbits and transversality for the moduli spa es of Floer traje tories. However, the gluing ompatibility requires that also the Hamiltonian perturbation for the ylindri al moduli spa es t ∈ S1 is independent of . We solve this problem by onsidering Hamiltonians H C2 , whi h are so small in the -norm that all orbits are riti al points of H and all ylinders between these orbits orrespond to gradient (cid:29)ow lines between the underlying riti al points. Note however that we annot a hieve this with a single Hamiltonian fun tion, but have to res ale the fun tion de- T ∈ N pending on the period , whi h in turn implies that we have to ompute the onta thomologyusinganin(cid:28)nitesequen eofdi(cid:27)erentHamiltonianfun tions. RH S1 ×M Observe that the losed orbits of the Reeb ve tor (cid:28)eld on have T ∈ N integer periods, where the set of losed orbits of period is naturally T XH M identi(cid:28)ed(Aw,it∂h) the -periodi orbits of on . AIt=follows tAhTat the hAaiTn omplex for onta t homology naturally splits, T∈N , where is generated by all monomials q(x1,T1)...q(xn,Tn), with Ti-peLriodi orbits (xi,Ti) T +...+T = T 1 n and , and it is easily seen from homologi al reasons that this ∂ splitting is respe ted by the di(cid:27)erential . Furthermore, given two di(cid:27)erent H1,H2 : S1 × M → R HΦa:m(iAlto,n∂ia)n→fun(A tio,∂ns) the orresponding hain map 1 1 2 2 , de(cid:28)ned as in [EGH℄ by ounting holomorphi urves in R×S1×M JH˜ equipped with a non- ylindHr˜i :alRa×lmSo1s×t Mom→pleRx stru tHur1e H,2whi h itself an be de(cid:28)ned usAing=a homotAopTy A = AT from to , also respe ts the splittings 1 T∈N 1, 2 T∈N 2. L L 6 1 INTRODUCTION AND MAIN RESULTS TN ∈ N TN ≤ TN+1 Let be a sequen e of (maximal) periods with and limN→∞TN = ∞ HN : S1 ×M → R N ∈ N and let (A,N,∂N) Nbe∈aNsequen e of Hamil- tonianNs w∈itNh orresponding hain omplexeΦsN : (AN,∂,N) → (A.NA+s1s,u∂mNe+1th)at for every H˜wNe h:aRve×dSe1(cid:28)×neMd a→ hRain map HN HN+1 using a homotopy interpolating bAeTtweenAT and T, w∈hiN h by the above arguments restri ts to a map from N to N+1 for every . De(cid:28)ning HC≤TN(S1×M,JHN)=H (A≤TN,∂ )= H (AT,∂ ) ∗ ∗ N N ∗ N N TM≤TN (C ,Φ ) C = HC≤TN(S1×M,JHN) N N,M N ∗ we obtain a dire ted system with and Φ =Φ ◦Φ ◦...◦Φ ◦Φ N ≤M T =2N N,M N N+1 M−1 M N for . Setting we provethe S1 H : M → R main result by showing that for every -independent Hamiltonian , C2 whi h is su(cid:30) iently small in the -norm and Morse, there is an isomorphism lim HC∗≤2N(S1×M,JH/2N) ∼= S H∗−2(M,Q) ⊗Q[H2(M)]. N→∞ (cid:0)MN (cid:1) This paper is organized as follows. While we prove in 2.1 all the fundamental results about pseudoholomorphi urves in Hamiltonian mapping tori, subse tion 2.2 is devoted to explaining the S1 entral ideas of the proof the main theorem, namely how we get an -symmetry on all moduli spa esofdomain-stable urves, but still havenondegenera yforthe losedorbitsandtransversalityforallmoduli spa es. We olle talltheimportant results about the moduli spa es in theorem 2.6. After re alling the de(cid:28)nition of the Deligne-Mumford spa e of stable pun tured spheres in 3.1, we de(cid:28)ne the underlying domain-dependent Hamiltonian perturbations in 3.2 and prove in 3.3 that the onstru tion is ompatible with the SFT ompa tness theorem. After des ribing in detail the ne essary Bana h manifold setup for our Fredholm problems in 4.1, we prove in 4.2 the fundamental transversality result for the Cau hy-Riemannoperator. Sin e allourresultsonlyholdupto amaximalperiod for the asymptoti orbits, i.e., we have to res ale our Hamiltonian perturbation during the omputation of onta t homology in se tion 6, we generalize all our previousresultsto homotopiesof Hamiltonianperturbations in 5.1and 5.2. After des ribing the hain omplex underlying onta t homology in 6.1, we prove the main theorem using our previous results about moduli spa es of holomorphi R×S1×M urves in . A knowledgements This resear h was supported by the German Resear h Foundation (DFG). The author thanks U. Frauenfelder, M. Hut hings and K. Mohnke for useful onversations and their interest in his work. Spe ial thanks (cid:28)nally go to my advisor Kai Cieliebak and to Dietmar Salamon, who gave me the han e to stay at ETH Zuri h for the winter term 2006/07,for their support. Finally thanks go to the referee for his valuable omments. 7 2 Moduli spa es R×S1 ×M 2.1 Holomorphi urves in (M,ω) φ Let be a losed symple ti manifold and let be a symple tomorphism on it. As already explained in the introdu tion, the orresponding mapping torus Mφ =R×M/{(t,p)∼(t+1,φ(p))} arriesanaturalstableHamiltonianstru ture (ω,λ) ω M φ given by lifting the symple ti form to a two-form on via the (cid:29)at TM =TS1⊕TM λ=dt φ onne tion andsetting . Itfollowsthatthe orresponding ξ = kerλ TM R symple ti ve tor bundle is given by and the Reeb ve tor (cid:28)eld S1 ∂ M t φ agrees with the -dire tion on . In this paper we restri t ourselves to the h[ω],π (M)i = 0 φ 2 ase where and is Hamiltonian, i.e., the time-one map of the H :S1×M →R (cid:29)owofaHamiltonian . Inthis aseobservethattheHamiltonian φH (cid:29)ow provides us with the natural di(cid:27)eomorphism Φ:S1×M −∼=→M , (t,p)7→(t,φH(t,p)), φ M S1 × M φ so that we an repla e by equipped with the pullba k stable Hamiltonian stru ture. (ωH,λH) S1×M Proposition 2.1. The pullba k stableHamiltonian stru ture on is given by ωH = ω+dH ∧dt, λH = dt ξH RH with symple ti bundle and Reeb ve tor (cid:28)eld given by ξH = TM, RH = ∂ +XH, t t XH H =H(t,·) where t is the symple ti gradient of t . Proof: Using dΦ=(1,XH ⊗dt+dφH):TS1⊕TM →TS1⊕TM t t v =(v ,v ),v =(v ,v )∈TS1⊕TM 1 11 12 2 21 22 we ompute for , ωH(v ,v ) = ω(dΦ(v ),dΦ(v )) 1 2 1 2 = ω((XH ⊗dt)(v )+dφH(v ),(XH ⊗dt)(v )+dφH(v )) t 11 t 12 t 21 t 22 = ω(XH,XH)dt(v )dt(v )+ω(dφH(v ),dφH(v )) t t 11 21 t 12 t 22 +ω(XH,dφH(v ))dt(v )+ω(dφH(v ),XH)dt(v ) t t 22 11 t 12 t 21 = ω(v ,v )+ω(dφH(v ),XH)dt(v )−ω(dφH(v ),XH)dt(v ) 12 22 t 12 t 21 t 22 t 11 = ω(v ,v )+(dH ∧dt)(v ,v ) 1 2 1 2 8 2 MODULI SPACES λH = λ◦dΦ = dt ξH = TM and . On the other hand, it dire tly follows that , RH =∂ −XH ωH while t t spans the kernel of , ωH(·,RH) = ω(·,∂ −XH)+dH ·dt(∂ +XH)−dH(∂ +XH)·dt t t t t t t = −ω(·,XH)+dH =0 t λH(RH)=dt(∂ −XH)=1 (cid:3) with t t . J As in the introdu tion we onsider an almost omplex stru ture on the R×S1×M ylindri al manifold , whi h is required to be ylindri al in the sense R RH R that it is -independent, links the Reeb ve tor (cid:28)eld and the -dire tion ∂ J∂ = RH = ∂ +XH s, by s t t and turns the symple ti hyperplane distribution ξH = TM T(S1 × M) J into a omplex subbundle of . It follows that on R×S1×M ξH =TM is determined by its restri tion to , whi h is required to be ωξH J S1 Ht - ompatible, so that is determined by the -dependent Hamiltonian S1 ω J t and an -family of - ompatiblealmost omplex stru tures on the symple ti (M,ω) manifold . Let us re all the de(cid:28)nition of moduli spa es of holomorphi urves studied in (V,ω,λ) rational SFT in the general setup. Let be a losed manifold with stable ξ Hamiltonianstru ture with symple ti hyperplane distribution and Reeb ve tor R J R×V (cid:28)eldP+,aPnd−let be a ompatible ylindri al almoγst omplex stru ture on R . Let be two ordered sets of losed orbits of the Reeb ve tor (cid:28)eld on V γ : R → V γ(t+T) = γ(t) γ˙ = R T > 0 ,γi.e., , , M, 0w(hVe;rPe+,P−,Jd)enotes the period (oFf,(.z±T)h)en the (p{azr±a,m...e,tzr±ize}d) moduli spa e onsistCsPo1f tuples k ,where 1 n± aretwodisjointorderedsetsofpointson ,whi h F :S˙ →R×V are alled positive and negative pun tures, respSe˙ t=iveClyP.1T−he{(mz±a)p} starting from the pun tured Riemann surfa e k is required to satisfy the Cau hy-Riemann equation ∂ F =dF +J(F)·dF ·i=0 J i CP1 with the omψp±lex: Rst±ru× tSu1re→ oSn˙ . Assuming we hazv±e hosen ylindri al ψo±o(r±di∞na,tte)s=zk± F around ea h pun ture k in theks=en1s,e...t,hna±t k k, the map is additionally required to show for all the asymptoti behaviour lim (F ◦ψ±)(s,t+t )=(±∞,γ±(T±t)) s→±∞ k 0 k k t ∈ S1 γ± ∈ P± T± > 0 γ± with some 0 and the orbits k , where k denotes period of k . Aut(CP1) OMb0s(eVrv;ePt+h,aPt t−h,eJg)roup ofMoebius transformationsa tson elements in in an obvious way, ϕ.(F,(z±))=(F ◦ϕ−1,ϕ(z±)), ϕ∈Aut(CP1), k k R×S1×M 2.1 Holomorphi urves in 9 M(V;P+,P−,J) and we obtain the moduli spa e studied in symple ti (cid:28)eld theory by dividing out this a tion. It remains to identify the o uring obje ts in our spe ial ase. First, one γ RH =∂ −XH immediately veri(cid:28)es that all losed orbits of the ve tor (cid:28)eld t t on S1×M are of the form γ(t)=(t+t ,x(t)), 0 T ∈ N and therefore have natural numbers , i.e., the winding number around the S1 -fa tor, as periods. Sin e we study losed Reeb orbits up to reparametrization, t0 = 0 γ x : R/T Z → M we an set , so that an be identi(cid:28)ed with , whi h is a T -periodi orbit of the Hamiltonian ve tor (cid:28)eld, x˙(t)=XH(x(t)). t γ = (x,T) T ∈N x Hen e we will in the following write , where is the period and T H T isa -periodi orbitoftheHamiltonian . Wedenotethesetof -periodi orbits RH S1×M P(H,T) of the Reeb ve tor (cid:28)eld on by . R×S1×M For the moduli spa es of urves observe that in we an naturally F write the holomorphi map as a produ t, F =(h,u):S˙ →(R×S1)×M . F : S˙ → R×S1 ×M J h = (Phr1o,ph2o)si:tiSo˙n→2R.2×. S1 ius : S-˙ho→lomMorphi pre iselyhwhen is holomorphi and satis(cid:28)es the -dependent perturbed Cau hy-Riemann equation of Floer type, ∂ u = Λ0,1(du+XH(h ,u)⊗dh ) J,H,h 2 2 = du+XH(h ,u)⊗dh +J(h ,u)·(du+XH(h ,u)⊗dh )·i. 2 2 2 2 2 J(t,p):T(R×S1)⊕TM →T(R×S1)⊕TM Proof: Observing that is given by i 0 J(t,p)= (cid:18)∆(t,p) J (p)(cid:19) t ∆(t,p)=−XH(p)⊗ds+J (p)XH(p)⊗dt with t t t we ompute (dh,du)+J(h,u)·(dh,du)·i = (dh+i·dh·i, du+(J(h ,u)·du−XH(h ,u)⊗dh +J(h ,u)XH(h ,u)⊗dh )·i) 2 2 1 2 2 2 = (∂h,du−XH(h ,u)⊗dh ·i+J(h ,u)·(du+XH(h ,u)⊗dh )·i). 2 1 2 2 2 dh ·i=−dh ∂h=0 (cid:3) 1 2 Finally observe that if . 10 2 MODULI SPACES P± ={(x±,T±),...,(x± ,T± )} Re alling that our orbit sets are given by 1 1 n± n± , we use the rigidity ohf:hSo˙lo→moRrp×hSi 1mapsTt±o p=roTv±e+th.e..f+ollTow±ing statement about the map omponent . Let 1 n± denote the total period above and below, respe tively. h =(h ,h ) T+ = T− 1 2 Lemma 2.3. The map exists if and only if and is unique (s0,t0)∈R×S1 up a shift , h(z)=h0(z)+(s ,t ) 0 0 h0 = (h0,h0) for some (cid:28)xed map 1 2 . In parti ular, every holomorphi ylinder has a positive and a negative pun ture, there are no holomorphi planes and all holomorphi spheres are onstant. F Proof: The asymptoti behavior of the map near the pun tures implies that s→±∞ h◦ψ (s,t+t ) −→ (±∞,T t) k 0 k t0 ∈ S1 R×S1 ∼= CP1−{0,∞} h with some . Identifyihng CP1 z+,...,z+ , it follows thatT+,e.x..t,eTn+ds to a meromorphi fun tion on with 1 n+ poles of order 1 n+ z−,...,z− T−,...,T− CP1 and 1 n− zeros of order 1 n−. Sin e the zeroth Pi ard group of is trivial, i.e., every divisor of degree zero is a Tpr+in= ipTa−l divisor, we get that su h meromorphi fun tions exist pre isely when . On the other hand it follows from Liouville's theoremhth=atath·ehy0 are unaiq∈uelCy∗d∼=eteRrm×inSe1d up to a nonzehro0 :mCuPlt1ip→li aCtPiv1e fa tor, i.e.J, H with (h,u) for sohme (cid:28)xed . For every -holomorphi sphere observe that is h = (s ,t ) u J M onstant, 0 0h,[ωa]n,dπ t(hMer)eifo=re0 (cid:3)is a t0-holomorphi sphere in , whi h 2 must be onstant by . φ Note that the lemma also holds when is no longer Hamiltonian by de(cid:28)ning h=π◦F π :R×Mφ →R×S1 using the holomorphi bundle proje tion . JH (h,uIt) :foS˙llo→wsRt×haSt1×weMonSl˙y=haCvPe1t−o{s(tzu±d)y} pun tured -holomorphi urves , k with two or more pun tures, where u it remains to understand the map . Note that by proposition 2.2 the perturbed u S1 h =h0+t Cau hy-Riemannequationfor dependsonthe - omponent 2 2 0 ofthe h map . Starting with the ase of two pun tures, we make pre ise the well-known onne tion between symple ti Floer homology and symple ti (cid:28)eld theory for Hamiltonian mapping tori. JH RH P(xr+o,pTo)sition(x2−.,4T.)TheR×S-1ho×lomMorphi ylinders onne ting the -orbits and in orrespond to the Floer onne ting orbits