Continuous Hamiltonian dynamics and area-preserving homeomorphism group of $D^2$

0.39 MB·English

by  Yong-Geun Oh

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S CONTINUOUS HAMILTONIAN DYNAMICS AND
9 AREA-PRESERVING HOMEOMORPHISM GROUP OF D2
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] YONG-GEUNOH
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S
Abstract. The main purpose of this paper is to propose a scheme of a
h. proof of the nonsimpleness of the group HomeoΩ(D2,∂D2) of area preserv-
t ing homeomorphisms of the 2-disc D2. We first establish the existence of
a
Alexander isotopyinthecategory ofHamiltonianhomeomorphisms. Thisre-
m
duces the question of extendability of the well-known Calabi homomorphism
[ Cal:DiffΩ(D1,∂D2)→RtoahomomorphismCal:Hameo(D2,∂D2)→R

tothatofthevanishingofthebasicphasefunctionfF,aFloertheoreticgraph
2
selectorconstructedin[Oh1],thatisassociatedtothegraphofthetopological
v Hamiltonian loop and its normalized Hamiltonian F on S2 that is obtained
7 viathe natural embedding D2 ֒→S2. HereHameo(D2,∂D2)isthe group of
0
HamiltonianhomeomorphismsintroducedbyMu¨llerandtheauthor[OM]. We
3
thenprovideanevidenceofthisvanishingconjecturebyprovingtheconjecture
4 forthespecialclassofweakly graphical topologicalHamiltonianloopsonD2.
0
.
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MSC2010: 53D05, 53D35, 53D40; 37E30.
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: Contents
v
i
X 1. Introduction and statements of main results 2
1.1. Calabi homomorphism on D2 2
r
a 1.2. Basic phase function and Calabi invariant 4
1.3. GraphicalHamiltonian diffeomorphism onD2 andits Calabiinvariant 6
Part 1. Calabi invariant and basic phase function 7
path
2. Calabi homomorphism Cal on the path space 7
2.1. Hamiltonian topology and hamiltonian homotopy 8
2.2. Calabi invariants of topological Hamiltonian paths in D2 10
3. Alexander isotopy of loops in Pham(Sympeo(D2,∂D2),id) 11
4. Reduction to the engulfed case and its Lagrangianization 14
5. Basic phase function f and its axioms 16
H
Date:January2015;revisedinSeptember 2015.
Keywordsandphrases. Area-preservinghomeomorphismgroup,Calabiinvariant,Lagrangian
submanifolds,generatingfunction,basicphasefunction,topologicalHamiltonianloop.
ThepresentworkissupportedbytheIBSproject#IBS-R003-D1.
1
2 YONG-GEUNOH
6. Calabi homomorphism and basic phase function 18
7. Extension of Calabi homomorphism 21
Part 2. Weakly graphical topological Hamiltonian loops on D2 22
8. Geometry of graphical symplectic diffeomorphisms in 2-dimension 22
9. Weakly graphical Hamiltonian diffeomorphism and Alexander isotopy 25
10. Vanishing of basic phase function for the graphical case on D2 27
Appendix A. Homotopy invariance of basic phase function 31
Appendix B. Timewise basic phase function as a solution to Hamilton-
Jacobi equation 32
References 34
1. Introduction and statements of main results
1.1. Calabi homomorphism on D2. Denote by DiffΩ(D2,∂D2) the group of
area-preservingdiffeomorphismssupportedintheinteriorofD2 withrespecttothe
standard area form Ω=dq∧dp on D2 ⊂R2. For any φ∈DiffΩ(D2,∂D2)
φ∗Ω=Ω
by definition. Write Ω=dα for some choice of α. Then this equation leads to the
statement φ∗α−α is closed. Furthermore since φ is supported in the interior, the
one-form
φ∗α−α
vanishesnear∂D2andsodefinesadeRhamcohomologyclasslyinginH1(D2,∂D2).
Sincethelattergroupistrivial,wecanfindafunctionh supportedintheinterior
φ,α
such that
dh =φ∗α−α. (1.1)
φ,α
Then the following is the well-known definition of Calabi invariant [Ca].
Definition 1.1 (Calabi invariant). We define
1
Cal(φ)= h .
φ,α
2
ZD2
One can show that this value does not depend on the choice of the one-form
α but depends only on the diffeomorphism. We will fix one such form α and so
suppress the dependence α from our notation, and just denote h =h .
φ φ,α
Another equivalentdefinition does notinvolvethe choice ofone-formα but uses
the ‘past history’ of the diffeomorphism in the setting of Hamiltonian dynamics
[Ba]. More precisely, this definition implicitly relies on the following three facts:
(1) Ω on two dimensional surface is a symplectic form and hence
DiffΩ(D2,∂D2)=Symp (D2,∂D2)
ω
where ω =Ω.
(2) D2 is simply connected, which in turn implies that any symplectic isotopy
is a Hamiltonian isotopy.
(3) The group DiffΩ(D2,∂D2) is contractible. (For this matter, finiteness of
π (DiffΩ(D2,∂D2),id)∼={0} is enough.)
1
AREA-PRESERVING HOMEOMOPHISM GROUP OF D2 3
It is well-known(see [GG], [Oh6] for example) and easy to construct a sequence
φ ∈DiffΩ(D2,∂D2) such that φ →id in C0 topology but
i i
Cal(φ )=1
i
for all i’s. This implies that Cal cannotbe continuously extended to the full group
HomeoΩ(D2,∂D2) of area-preservinghomeomorphisms.
However here is the main conjecture of the paper concerning nonsimpleness of
the group HomeoΩ(D2,∂D2). The author learned from A. Fathi in our discussion
on the groupHameo(D2,∂D2) [F] that the following will be important in relation
to the study of nonsimpleness conjecture. (We refer to [Oh5] for the argument
needed to complete this nonsimpleness proof out of this conjecture.)
Conjecture 1.2. Let Hameo(D2,∂D2) ⊂ HomeoΩ(D2,∂D2) be the subgroup of
Hamiltonian homeomorphisms on the two-disc. Then the Calabi homomorphism
Cal : DiffΩ(D2,∂D2) → R extends continuously to Hameo(D2,∂D2) in Hamil-
tonian topology in the sense of [OM].
For the study towards this conjecture, as we did in [Oh5], we first define a
homomorphism on the path spaces
Calpath(λ):Pham(Symp(D2,∂D2),id)→R
by
1
Calpath(λ)= H(t,x)Ωdt. (1.2)
Z0 ZD2
WewillalsodenotethisaveragebyCal(H)dependingonthecircumstances. Based
on these facts (1) and (2), we can represent φ = φ1 for the time-one map φ1
H H
of a time-dependent Hamiltonian H = H(t,x) supported in the interior. Then
based on (3) and some standard calculations in Hamiltonian geometry using the
integrationby parts,one provesthatthis integraldoesnot depend onthe choice of
HamiltonianH 7→φ. ThereforeitdescendstoHam(D2,∂D2)=DiffΩ(D2,∂D2).
ThenanotherapplicationofStokes’formula,onecanprovethatthislatterdefinition
indeed coincides with that of Definition 1.1. (See [Ba] for its proof.)
It is via this second definition how the author attempts to extend the classi-
cal Calabi homomorphism Cal : Ham(D2,∂D2) → R to its topological analog
Cal:Hameo(D2,∂D2)→R. In [Oh5], the definition (1.2) is extended to a homo-
morphism
Calpath :Pham(Sympeo(D2,∂D2),id)→R
onthe setPham(Sympeo(D2,∂D2),id)oftopologicalHamiltonianpaths. (Seesec-
tion2forthe precisedefinition.) Herefollowingthe notationfrom[OM], wedenote
by Sympeo(D2,∂D2) the C0-closure of Symp(D2,∂D2). Gromov-Eliashberg’s C0
symplectic rigidity theorem [El] states
Diff(D2,∂D2)∩Sympeo(D2,∂D2)=Symp(D2,∂D2).
path
In [Oh5, Oh6], it is shown that a proof of descent of Cal to the group
Hameo(D2,∂D2):=ev (Pham(Sympeo(M,ω),id))
1
ofHamiltonianhomeomorphisms(ormore succinctly hameomorphisms) is reduced
to the following extension result of Calabi homomorphism.
4 YONG-GEUNOH
One important ingredient in our scheme towards the proof of Conjecture 1.2,
which itself has its own interest, is the existence of the Alexander isotopy in the
topological Hamiltonian category. Recall that the well-known Alexander isotopy
on the disc D2 exists in the homeomorphism category but not in the differentiable
category. We will establish that such an Alexander isotopy defines contractions
of topological Hamiltonian loops to the identity constant loop in the topological
Hamiltonian category.
Theorem1.3(Alexanderisotopy;Theorem3.3). AnytopologicalHamiltonianloop
in Hameo(D2,∂D2) is contractible to the identity loop via topological Hamiltonian
homotopy of loops.
1.2. Basic phase function and Calabi invariant. The scheme of the proof of
Conjecture 1.2 we propose is based on the following conjectural result of the basic
phase function introduced in [Oh1]. This conjecture is also a crucial ingredient
neededintheproofofhomotopyinvarianceofthespectralinvarianceoftopological
Hamiltonianpathslaidoutin[Oh7]. Explanationofthisconjectureisnowinorder.
Recallthe classicalactionfunctional onT∗N for anarbitrarycompactmanifold
N is defined as
1
Acl(γ)= γ∗θ− H(t,γ(t))dt
H
Z Z0
on the space P(T∗N) of paths γ : [0,1] → T∗N, and its first variation formula is
given by
1
dAcl(γ)(ξ)= ω(γ˙−X (t,γ(t)),ξ(t))dt−hθ(γ(0)),ξ(0)i+hθ(γ(1)),ξ(1)i. (1.3)
H H
Z0
The basic phase function graph selector is canonical in that the assignment
H 7→f ; C∞([0,1]×T∗N;R)→C0(N)
H
varies continuously in (weak) Hamiltonian topology of C∞([0,1]×T∗N;R) [OM,
Oh9]. The construction f in [Oh1] is given by considering the Lagrangianpair
H
(o ,T∗N), q ∈N
N q
and its associated Floer complex CF(H;o ,T∗N) generated by the Hamiltonian
N q
trajectory z :[0,1]→T∗N satisfying
z˙ =X (t,z(t)), z(0)∈o , z(1)∈T∗N. (1.4)
H N q
Denote by Chord(H;o ,T∗N) the set of solutions of (1.4). The differential ∂
N q (H,J)
on CF(H;o ,T∗N) is provided by the moduli space of solutions of the perturbed
N q
Cauchy-Riemann equation
∂u +J ∂u −X (u) =0
∂τ ∂t H (1.5)
(u(τ,0)∈(cid:0) oN, u(τ,1)∈(cid:1) Tq∗N.
The resulting spectral invariantρlag(H;[q]) is to be defined by the mini-max value
ρlag(H;[q])= inf λ (α)
H
α∈[q]
where[q]isageneratorofthehomologygroupHF(oN,Tq∗N)∼=Z. Thebasicphase
function f :N →R is then defined by f (q)=ρlag(H;[q]) firstfor genericq ∈N
H H
and then extending to the rest of M by continuity. (See [Oh1] for the detailed
construction and section 5 of the present paper for a summary.)
AREA-PRESERVING HOMEOMOPHISM GROUP OF D2 5
NextwerelatethebasicphasefunctiontotheCalabiinvariantonthetwo-discas
follows. Let F be a topological Hamiltonian generating a topological Hamiltonian
path φ on the 2-disc D2 with suppF ⊂ IntD2. We consider an approximating
F
sequenceF withsuppF ⊂IntD2. We embedD2 intoS2 astheupper hemisphere
i i
and then extend F canonically to whole S2 by zero.
i
We now specialize the above discussion on the basic phase function to the cases
of the Lagrangianizationof symplectic diffeomorphisms, i.e., consider their graphs
Graphφ={(φ(x),x)|x∈S2}⊂S2×S2.
Applying this to φt and noting suppφt ⊂D2 ×D2, we obtain
Fi Fi + +
Graphφt ∆⊃∆ ∆
Fi D−2 D+2\D+2(1−δ)
forsomeδ >0forallt∈[0,1],in\dependently[ofsufficientlylargei’sbutdepending
onlyonF. (See[OM]orDefinition2.7ofthepresentpaperfortheprecisedefinition
ofapproximatingsequenceonopenmanifolds.) Thenweconsiderthenormalization
F of F on S2 and define Hamiltonian
i i
F (t,x):=χ(x)F (t,x), x=(x,y)
i i
on T∗∆ with a slight abuse of notation for F , where χ is a cut-off function such
i
that χ≡1 on a neighborhood V of ∆ with
∆
suppφ ⊂V ⊂V ⊂S2×S2\∆.
F ∆ ∆
Two kinds of the associated generating functions, denoted by hF and hF respec-
i i
tively, are given by
hFi(q)=AcFl zFqi , hFi(x)=AcFli zxFi , e (1.6)
where the Hamiltoniaen trajectorie(cid:0)s zFq(cid:1)i and zxFi are defin(cid:0)ed b(cid:1)y
zq(t) = φt (q), q∈o
F F ∆
i i
zxFi(t) = φtFi((φ1Fi)−1(x)), x∈φ1Fi(o∆).
We note that zFq(0) = q and zxFi(1) = x. Later we will review the definition from
i
[Oh1, Oh9] of the basic phase function fF and the Lagrangianselector σF . These
i i
maps have the properties that
fF =hF ◦σF (1.7)
i i i
and σF (q) = (q,dfF (q)) ∈ T∗∆ whenever dfF (q) exists. This ends the review of
i i i
construction of basic phase function.
The following theorem exhibits the relationship between the limit of Calabi in-
variants and that of the basic phase function.
Theorem 1.4 (Theorem 6.1). Let (M,ω) be an arbitrary closed symplectic man-
ifold. Let U = M \ B where B is a closed subset of nonempty interior. Let
λ=φ be any engulfed topological Hamiltonian loop in Pham(Sympeo (M,ω),id)
F U
with φt ≡id on B. Then
Fi
Cal (F)
U
lim fF (x)= (1.8)
i→∞ i volω(M)
uniformly over x∈M, for any approximating sequence F of F. In particular, the
i
limit function fF defined by fF(x):=limi→∞fFi(x) is constant.
6 YONG-GEUNOH
It is crucial for the equality (1.8) to hold in the generalcase that we are consid-
ering topologicalHamiltonian loop, not just a path. (We refer readers to the proof
of Theorem 6.1 to see how the loop property is used therein. We also refer to the
proof of Lemma 7.5 [Oh9] for a similar argument used for a similar purpose.)
The following is the main conjecture to beat which was previously proposed by
the present author in [Oh7].
Conjecture 1.5 (Main Conjecture). Let M = S2 be the 2 sphere with standard
symplectic structure. Let Λ = φt be a hameotopy contracting a
H(s)
(s,t)∈[0,1]2
topological Hamiltonian loop φ wnith F o=H(1) such that H(s)≡id on D2 where
F −
D−2 is the lower hemisphere of S2. Then fF =0.
It turns out that this conjecture itself is strong enough to directly give rise to
a proof of Conjecture 1.2 in a rather straightforward manner with little usage of
Floerhomologyargumentinitsoutsetexceptafewfunctorialpropertiesofthebasic
phase function that are automatically carried by the Floer theoretic construction
given in [Oh1].
Weindicatevalidityofthisconjecturebyprovingtheconjectureforthefollowing
special class consisting of weakly graphical topological Hamiltonian loops.
1.3. Graphical Hamiltonian diffeomorphism on D2 and its Calabi invari-
ant. We start with the following definition. We refer readers to Definition 4.2 for
the definition of engulfed diffeomorphisms.
Definition 1.6. Let Ψ : U → V be a Darboux-Weinstein chart of the diagonal
∆
∆⊂ M ×M and denote π = πΨ : U → ∆ to be the composition of Ψ followed
∆ ∆ ∆
by the canonical projection T∗∆→∆.
(1) We call an engulfed symplectic diffeomorphism φ : M → M Ψ-graphical
if the projection π | → ∆ is one-to-one, and an engulfed symplec-
∆ Graphφ
tic isotopy is {φt} Ψ-graphical if each element φt Ψ-graphical. We call a
Hamiltonian F =F(t,x) Ψ-graphicalif its associated Hamiltonian isotopy
φt is Ψ-graphical.
F
(2) We call a topological Hamiltonian loop F is strongly (resp. weakly) Ψ-
graphical,ifitadmitsanapproximatingsequenceF eachelementofwhich
i
is Ψ-graphical(resp. whose time-one map φ1 is Ψ-graphical).
Fi
DenotebyFathetime-dependentHamiltoniangeneratingthepatht7→φat. The
F
statement (2) of this definition is equivalent to saying that each Fa is Ψ-graphical
for a∈[0,1].
WeremarkthatanysymplecticdiffeomorphismssufficientlyC1-closetotheiden-
tityisgraphical,butnoteveryC0-closeone. Wealsoremarkthatπ | issur-
∆ Graphφ
jectiveandhenceadiffeomorphismifφisaΨ-graphicalsymplecticdiffeomorphism
isotopic to the identity via a Ψ-engulfed isotopy.
In 2 dimension, we prove the following interesting phenomenon. We doubt that
similar phenomenon occurs in high dimension. This theorem will not be used in
the proofs of main results of the present paper but has its own interest.
Theorem 1.7. Let M be a closed 2 dimensional surface. Suppose φ : M → M
is a Ψ-graphical symplectic diffeomorphism isotopic to the identity via Ψ-graphical
isotopy. and let Graphφ =Imageα for a closed one-form α . Then for any 0 ≤
φ φ
AREA-PRESERVING HOMEOMOPHISM GROUP OF D2 7
r≤1, the projection π :M×M →M restricts to a one-one map to Imagerα ⊂
2 φ
M ×M. In particular
Imagerα =Graphφ (1.9)
φ r
for some symplectic diffeomorphism φ :M →M for each 0≤r≤1.
r
FinallyweproveConjecture1.5fortheweaklygraphicaltopologicalHamiltonian
loop on S2 that arises as follows.
Theorem 1.8. Conjecture1.5 holds for any weakly graphical topological Hamilton-
ian loop on S2 arising from one on D2 as in subsection 1.2.
The proof of this theorem strongly relies on Theorem 1.3.
An immediate corollaryof Theorem1.4 and 1.8 is the following vanishing result
of Calabi invariant.
Corollary 1.9. Suppose λ=φ is a weakly graphical topological Hamiltonian loop
F
path
on D2. Then Cal (λ)=0.
Unraveling the definitions, this corollary establishes the main conjecture with
the additional graphicality hypothesis on φ .
i
Theorem 1.10. Consider a sequence φ ∈ Ham(D2,∂D2) that satisfies the fol-
i
lowing conditions:
(1) Each φ is graphical, and φ →id in C0-topology,
i i
(2) φ =φ1 with convergent H in L(1,∞)-topology.
i Hi i
Then lim Cal(φ )=0.
i→∞ i
WehopetostudyelsewheregeneralengulfedtopologicalHamiltonianloopdrop-
ping the graphicality condition.
Remark 1.11. Previously the author announced a ‘proof’ of the nonsimpleness
resultin[Oh6]modulotheproof ofConjecture1.5inwhichnonsimplenessisderived
out of the homotopy invariance of spectral invariants whose proof also strongly
reliedonthis vanishingresult. Unlikethe previouslyproposedschemeofthe proof,
thecurrentschemedoesnotrelyonthehomotopyinvarianceofspectralinvariantsof
topologicalHamiltonian paths but moredirectly follows fromthe abovementioned
vanishing result.
We thank M. Usher for his carefulreadingof the previousversionof the present
paper and useful discussions in relation to the proof of Theorem 10.1. We also
take this opportunity to thank A. Fathi for explaining to us, during his visit of
KIAS in the summer of year 2004, how the question of extendability of the Calabi
homomorphism on DiffΩ(D2,∂D2) to Hameo(D2,∂D2) is related to the non-
simpleness of the area-preserving homeomorphism group HomeoΩ(D2,∂D2). We
also thank much the anonymous referee for pointing out an error in the proof of
Theorem 10.1. We newly add section 9 in the present version to fix the error, and
revise the proof of Theorem 10.1 accordingly. We also thank her/him for making
manyusefulcommentswhichwebelievegreatlyimproveourexpositionofthepaper.
Part 1. Calabi invariant and basic phase function
path
2. Calabi homomorphism Cal on the path space
8 YONG-GEUNOH
2.1. Hamiltoniantopologyandhamiltonianhomotopy. In[OM],Mu¨llerand
the author introduced the notion of Hamiltonian topology on the space
Pham(Symp(M,ω),id)
of Hamiltonian paths λ : [0,1] → Symp(M,ω) with λ(t) = φt for some time-
H
dependent Hamiltonian H. We would like to emphasize that we do not assume
that H is normalized unless otherwise said explicitly. This is because we need to
consider both compactly supported and mean-normalized Hamiltonians and suit-
ably transform one to the other in the course of the proof of the main theorem of
this paper.
We first recall the definition of this Hamiltonian topology.
We start with the case of closed (M,ω). For a given continuous function h :
M →R, we denote
osc(h)=maxh−minh.
We define the C0-distance d on Homeo(M) by the symmetrized C0-distance
d(φ,ψ)=max d (φ,ψ),d (φ−1,ψ−1)
C0 C0
and the C0-distance, again denoted(cid:8)by d, on (cid:9)
Pham(Symp(M,ω),id)⊂P(Homeo(M),id)
by
d(λ,µ)= max d(λ(t),µ(t)).
t∈[0,1]
The Hofer length of Hamiltonian path λ=φ is defined by
H
1
leng(λ)= osc(H )dt=kHk.
t
Z0
Following the notations of [OM], we denote by φ the Hamiltonian path
H
φ :t7→φt ; [0,1]→Ham(M,ω)
H H
and by Dev(λ) the associated normalized Hamiltonian
Dev(λ):=H, λ=φ (2.1)
H
where H is defined by
1
H(t,x)=H(t,x)− H(t,x)ωn. (2.2)
vol (M)
ω ZM
Definition 2.1. Let (M,ω) be a closed symplectic manifold. Let λ, µ be smooth
Hamiltonian paths. The Hamiltonian topology is the metric topology induced by
the metric
d (λ,µ):=d(λ,µ)+leng(λ−1µ). (2.3)
ham
Now we recall the notion of topological Hamiltonian flows and Hamiltonian
homeomorphisms introduced in [OM].
Definition 2.2 (L(1,∞) topologicalHamiltonianflow). Acontinuous mapλ:R→
Homeo(M) is called a topological Hamiltonian flow if there exists a sequence of
smooth Hamiltonians H :R×M →R satisfying the following:
i
(1) φ →λ locally uniformly on R×M.
Hi
(2) the sequence H is Cauchy in the L(1,∞)-topology locally in time and so
i
has a limit H lying in L(1,∞) on any compact interval [a,b].
∞
AREA-PRESERVING HOMEOMOPHISM GROUP OF D2 9
We call any such φ or H an approximating sequence of λ. We call a continuous
Hi i
path λ:[a,b] →Homeo(M) a topological Hamiltonian path if it satisfies the same
conditions with R replaced by [a,b], and the limit L(1,∞)-function H called a
∞
L(1,∞) topological Hamiltonian or just a topological Hamiltonian.
Following the notations from [OM], we denote by Sympeo(M,ω) the closure of
Symp(M,ω) in Homeo(M) with respect to the C0-metric d, and by H ([0,1]×
m
M,R) the set of mean-normalized topological Hamiltonians, and by
ev :Pham(Sympeo(M,ω),id)→Sympeo(M,ω),id) (2.4)
1 [0,1]
theevaluationmapdefinedbyev (λ)=λ(1).BytheuniquenesstheoremofBuhovsky-
1
Seyfaddini [BS], we can extend the map Dev given in (2.1) to
Dev:Pham(Sympeo(M,ω),id)→H ([0,1]×M,R)
[0,1] m
inanobviousway. Following the notationof [OM, Oh5], we denote the topological
Hamiltonian path λ=φ when Dev(λ)=H in this general context.
H
Definition 2.3 (Hamiltonian homeomorphism group). We define
Hameo(M,ω)=ev Pham(Sympeo(M,ω),id)
1 [0,1]
and call any element therein a Hamilt(cid:16)onian homeomorphism. (cid:17)
The group property and its normality in Sympeo(M,ω) are proved in [OM].
Theorem2.4([OM]). Let(M,ω)beaclosedsymplecticmanifold. ThenHameo(M,ω)
is a normal subgroup of Sympeo(M,ω).
Especially when dimΣ=2, we have a smoothing result
Sympeo(Σ,ω)=HomeoΩ(Σ) (2.5)
ofarea-preservinghomeomorphismsbyarea-preservingdiffeomorphisms(see[Oh3],
[Si] for a proof). Therefore combining this with the above theorem, we obtain the
following corollary,which is the starting point of our research to apply continuous
Hamiltoniandynamicstothestudyofthesimplenessquestionofthearea-preserving
homeomorphism group of D2 (or S2).
Corollary 2.5. Let Σ be a compact surface with or without boundary and let Ω
be an area form of Σ, which we also regard as a symplectic form ω = Ω. Then
Hameo(M,ω) is a normal subgroup of HomeoΩ(Σ).
Bothresultshavetheircounterpartsevenwhen∂M 6=∅. We refertothe discus-
sion below at the end of this subsection.
Next we consider the notion of homotopy in this topological Hamiltonian cat-
egory. The following notion of hamiltonian homotopy, which we abbreviate as
hameotopy, of topological hamiltonian paths is introduced in [Oh6, Oh8]. The
guiding principle for the choice of this as the definition of homotopy in this topo-
logicalHamiltoniancategoryistoincludetheAlexanderisotopywedefineinsection
3 as a special case.
Definition2.6(Hameotopy). Letλ , λ ∈Pham(Sympeo(M,ω),id). Ahameotopy
0 1
Λ:[0,1]2 →Sympeo(M,ω)between λ andλ basedat the identity is a map such
0 1
that
Λ(0,t)=λ (t), Λ(1,t)=λ (t), (2.6)
0 1
10 YONG-GEUNOH
and Λ(s,0) ≡ id for all s ∈ [0,1], and which arises as follows: there is a sequence
of smooth maps Λ :[0,1]2 →Ham(M,ω) that satisfy
j
(1) Λ (s,0)=id,
j
(2) Λ →Λ in C0-topology,
j
(3) Any s-section Λ : {s}×[0,1] → Ham(M,ω) converges in hamiltonian
j,s
topology in the following sense: If we write
Dev Λ Λ−1 =:H (s),
j,s j,0 j
thenHj(s)convergesinh(cid:0)amiltonia(cid:1)ntopologyuniformlyovers∈[0,1]. We
call any such Λ an approximating sequence of Λ.
j
When λ (1) = λ (1) = ψ, a hameotopy relative to the ends is one that satisfies
0 1
Λ(s,0)=id, Λ(s,1)=ψ for all s∈[0,1] in addition.
We saythatλ , λ ∈Pham(Sympeo(M,ω),id)arehameotopic (resp. relativeto
0 1
the ends), if there exists a hameotopy (resp. a hameotopy relative to the ends).
We emphasize that by the requirement (3),
H (0)≡0 (2.7)
j
in this definition.
All the above definitions can be modified to handle the case of open mani-
folds,either noncompactorcompactwithboundary,by consideringH’s compactly
supported in the interior as done in section 6 [OM]. We recall the definitions of
topologicalHamiltonian paths and Hamiltonian homeomorphisms supported in an
open subset U ⊂M from [OM].
We first define Pham(Symp (M,ω),id) to be the set of smooth Hamiltonian
U
pathssupportedinU. ThefollowingdefinitionistakenfromDefinition6.2[OM]to
whichwereferreadersformoredetaileddiscussions. FirstforanyopensubsetV ⊂
U withcompactclosureV ⊂U,wecandefineacompletionofPham(Symp (M,ω),id)
V
using the same metric given above.
Definition 2.7. LetU ⊂M be anopensubset. Define Pham(Sympeo (M,ω),id)
U
to be the union
Pham(Sympeo (M,ω),id):= Pham(Sympeo (M,ω),id)
U K
K⊂U
[
with the direct limit topology, where K ⊂ U is a compact subset. We define
Hameo (U,ω) to be the image
c
Hameo (U,ω):=ev (Pham(Sympeo (M,ω),id)).
c 1 U
We wouldliketoemphasizethatthis setisnotnecessarilythe sameasthesetof
λ ∈ Pham(Sympeo(M,ω),id) with compact suppλ ⊂ U. The same definition can
be applied to general open manifolds or manifolds with boundary.
2.2. Calabi invariants of topological Hamiltonian paths in D2. Denote by
Pham(Symp(D2,∂D2);id) the group of Hamiltonian paths supported in Int(D2),
i.e.,
suppH ⊂Int(D2).
t
t∈[[0,1]
We denote by Pham(Sympeo(D2,∂D2),id) the L(1,∞) hamiltonian completion of
Pham(Symp(D2,∂D2);id).