Table Of Content9
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 holo-
morphi
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
