Characteristic classes of foliated surface bundles with area-preserving holonomy

0.22 MB·English

by  D. Kotschick

CHARACTERISTIC CLASSES OF FOLIATED SURFACE
BUNDLES WITH AREA-PRESERVING HOLONOMY
6
0
0 D.KOTSCHICKANDS.MORITA
2

n
ABSTRACT. Making use of the extended flux homomorphism defined
a
J in [13]onthe groupSympΣg ofsymplectomorphismsofa closedori-
6 entedsurfaceΣgofgenusg ≥2,weintroducenewcharacteristicclasses
2 offoliatedsurfacebundleswithsymplectic,equivalentlyarea-preserving,
totalholonomy. Thesecharacteristicclassesarestablewithrespecttog

]
and we show that they are highly non-trivial. We also prove that the
G
secondhomologyofthegroupHamΣ ofHamiltoniansymplectomor-
S g
phismsofΣ , equippedwiththediscretetopology,isverylargeforall
. g
h g ≥2.
t
a
m
[

2 1. INTRODUCTION
v
4 In this paper we study the homology of symplectomorphism groups of
1
surfaces considered as discrete groups. We shall prove that certain homol-
2
9 ogy groups are highly non-trivial by constructing characteristic classes of
0 foliated surface bundles with area-preserving holonomy, and proving non-
4
vanishingresultsfor them.
0
/ Let Σ be a closed oriented surface of genus g ≥ 2, and Diff Σ its
h g + g
t group of orientation preserving selfdiffeomorphisms. We fix an area form
a
m ω on Σ , which, for dimension reasons, we can also think of as a symplec-
g
tic form. We denote by SympΣ the subgroup of Diff Σ preserving the
: g + g
v
form ω. Theclassifyingspace BSympδΣ for thegroup SympΣ with the
i g g
X
discrete topology is an Eilenberg-MacLane space K(SympδΣ ,1) which
g
r
a classifiesfoliated Σg-bundleswitharea-preserving totalholonomygroups.
Ourconstructionofcharacteristicclassesproceedsasfollows. LetSymp Σ
0 g
betheidentitycomponentofSympΣ . Awell-knowntheoremofMoser[24]
g
concerning volume-preserving diffeomorphisms implies that the quotient
SympΣ /Symp Σ can be naturally identified with the mapping class
g 0 g
1991MathematicsSubjectClassification. Primary57R17,57R50, 57M99;secondary
57R50,58H10.
Keywordsandphrases. symplectomorphism,area-preservingdiffeomorphism,foliated
surfacebundle,Hamiltoniansymplectomorphism.
ThefirstauthorisgratefultotheDeutscheForschungsgemeinschaftforsupportofthis
work.ThesecondauthorispartiallysupportedbyJSPSGrant16204005.
1
2 D.KOTSCHICKANDS.MORITA
groupM , sothatwehavean extension
g
p
1−→Symp Σ −→SympΣ −→M −→1.
0 g g g
ThereisasurjectivehomomorphismFlux: Symp Σ →H1(Σ ;R),called
0 g g
thefluxhomomorphism. In[13]weprovedthatthishomomorphismcanbe
extendedtoacrossed homomorphism
Flux: SympΣ −→H1(Σ ;R) ,
g g
which we call the extended flux homomorphism. This extension is essen-
g
tiallyuniquein thesensethat theassociated cohomologyclass
[Flux] ∈ H1(SympδΣ ;H1(Σ ;R))
g g
withtwistedcoefficients isuniquelydefined. Nowweconsiderthepowers
g
[Flux]k ∈ Hk(SympδΣ ;H1(Σ ;R)⊗k) (k = 2,3,···) ,
g g
andapply M -invarianthomomorphisms
g
g
λ: H1(Σ ;R)⊗k−→R
g
toobtaincohomologyclasses
λ([Flux]k) ∈ Hk(BSympδΣ ;R)
g
withconstantcoefficients. Theusualcup-productpairingH1(Σ ;R)⊗2→R
g
g
isthemainexampleofsuchahomomorphismλ.
This method of constructingconstant cohomologyclasses out of twisted
oneswasalreadyusedin[23]inthecaseofthemappingclassgroup,where
theTorelligroup(respectivelytheJohnsonhomomorphism)playedtherole
of Symp Σ (respectively of the flux homomorphism) here. In that case,
0 g
it was proved in loc. cit. that all the Mumford–Morita–Miller classes can
be obtained in this way. The precise formulae were given in [11, 12], with
the important implicationthat no other classes appear. In our context here,
we can go further by enhancing the coefficients R to associated Q-vector
spaceswhichappearasthetargetsofvariousmultiplesofthediscontinuous
cup-productpairing
H1(Σ ;R)⊗ H1(Σ ;R)−→S2R ,
g Z g Q
whereS2RdenotesthesecondsymmetricpowerofRoverQ,seeSection2
Q
forthedetails. In thisway,weobtainmanynew characteristicclassesin
H∗(SympδΣ ;S∗(S2R)) ,
g Q
where
∞
S∗(S2R) = Sk(S2R)
Q Q
k=1
M
CHARACTERISTICCLASSESOFFOLIATEDSURFACEBUNDLES 3
denotes the symmetric algebra of S2R. On the other hand, we proved
Q
in[13]thatany power
ek ∈ H2k(SympδΣ ;Q)
1 g
of the first Mumford–Morita–Millerclass e is non-trivial for k ≤ g. Now
1 3
we can consider the cup products of ek with the new characteristic classes
1
defined above. Themainpurposeofthepresentpaper istoprovethat these
characteristicclassesare allnon-trivialin asuitablestablerange.
The contents of this paper is as follows. In Section 2 precise statements
of the main results are given. In Section 3 we study, in detail, the trans-
versesymplecticclassoffoliatedΣ -bundleswitharea-preservingtotalho-
g
lonomy groups. In Section 4 we construct two kinds of extended flux ho-
momorphismsforopensurfacesΣ0 = Σ \D2. Wecomparetheseextended
g g
flux homomorphisms with the one obtained in the case of closed surfaces.
This is used in Section 5 to generalize our result on the second homology
of the symplectomorphism group to the case of open surfaces. Sections 4
and 5 are the heart of this paper. Then in Section 6 we use the results of
the previous sections to show the non-t riviality of cup products of vari-
ouscharacteristicclasses,thusyieldingproofsofthemainresultsaboutthe
homology of symplectomorphism groups as discrete groups. In the final
Section 7 we give definitions of yet more characteristic classes, other than
the ones given in Section 2. We propose several conjectures and problems
aboutthem.
2. STATEMENT OF THE MAIN RESULTS
Considertheusualcup-productpairing
ι: H1(Σ ;R)⊗H1(Σ ;R)−→R
g g
incohomology,dualtotheintersectionpairinginhomology. Forsimplicity,
wedenoteι(u,v)byu·v,whereu,v ∈ H1(Σ ;R). Wefirstliftthispairing
g
asfollows.
As in Section 1, let S2R denote the second symmetric power of R over
Q
Q. In other words, this is a vectorspace overQ, consistingof thehomoge-
neouspolynomialsofdegreetwogeneratedbytheelementsofRconsidered
as a vector space overQ. For each element a ∈ R, we denote by aˆ the cor-
respondingelement inS1R. Thus any element inS2R can be expressedas
Q Q
afinitesum
ˆ ˆ
aˆ b +···+aˆ b
1 1 k k
witha ,b ∈ R. Wehaveanaturalprojection
i i
S2R−→R
Q
givenbythecorrespondenceaˆ 7→ a(a ∈ R).
4 D.KOTSCHICKANDS.MORITA
Withthisterminologywemakethefollowingdefinition:
Definition1 (Discontinuousintersectionpairing). Define apairing
˜ι: H1(Σ ;R)×H1(Σ ;R)−→S2R
g g Q
asfollows. Chooseabasisx ,··· ,x ofH1(Σ ;Q). Foranytwoelements,
1 2g g
u,v ∈ H1(Σ ;R),write
g
u = a x , v = b x (a ,b ∈ R) .
i i i i i i
i i
X X
Thenweset
˜ι(u,v) = ι(x ,x )aˆ ˆb ∈ S2R .
i j i j Q
i,j
X
Clearly ˜ι followed by the projection S2R−→R is nothing but the usual
Q
intersectionorcup-productpairingι. Henceforthwesimplywriteu⊙v for
˜ι(u,v).
Itiseasytoseethat˜ιiswelldefinedindependentlyofthechoiceofbasis
inH1(Σ ;Q). Wecanseefromthefollowingpropositionthat˜ιenumerates
g
all Z-multilinear skew-symmetric pairings on H1(Σ ;R) which are M -
g g
invariant. Let Λ2H1(Σ ;R) denote the second exterior power, over Z, of
Z g
H1(Σ ;R) considered as an abelian group, rather than as a vector space
g
overR. Alsolet(Λ2H1(Σ ;R)) denotetheabeliangroupofcoinvariants
Z g M
g
ofΛ2H1(Σ ;R) withrespect tothenaturalaction ofM .
Z g g
Proposition2. Thereexists acanonicalisomorphism
Λ2H1(Σ ;R) ∼= S2R
Z g M Q
g
givenbythecorrespond(cid:0)ence (cid:1)
ˆ
a u ∧ b v 7−→ ι(u ,v ) aˆ b ,
i i j j i j i j
! !
i j i,j
X X X
wherea ,b ∈ R, u ,v ∈ H1(Σ ;Q).
i j i j g
In ordernotto digress,werefer thereaderto theappendix foraproof.
BeforewecandefinesomenewcocyclesonthegroupSympΣ ,wehave
g
to recall some facts from [13]. The symplectomorphism group SympΣ
g
acts onits identitycomponentby conjugation,and acts on H1(Σ ;R) from
g
theleftviaϕ(w) = (ϕ−1)∗(w). ThefluxhomomorphismFlux: Symp Σ →
0 g
H1(Σ ;R)is equivariantwithrespect totheseactions byLemma6 of[13].
g
Its extension Flux: SympΣ → H1(Σ ;R) is a crossed homomorphism
g g
fortheaboveaction inthesensethat
(1) gFlux(ϕψ) = Flux(ϕ)+(ϕ−1)∗Flux(ψ) .
g g g
CHARACTERISTICCLASSESOFFOLIATEDSURFACEBUNDLES 5
Definition3. Let ϕ ,...,ϕ ∈ SympΣ , and
1 2k g
ξ = ((ϕ ...ϕ )−1)∗Flux(ϕ ) .
i 1 i−1 i
Definea2k-cocycleα˜(k) withvaluesin Sk(S2R)by
Q
g
α˜(k)(ϕ ,...,ϕ )
1 2k
1
= sgnσ (ξ ⊙ξ )...(ξ ⊙ξ ) ∈ Sk(S2R) ,
(2k)! σ(1) σ(2) σ(2k−1) σ(2k) Q
σX∈S2k
wherethesumisoverpermutationsinthesymmetricgroupS .
2k
That α˜(k) is indeed a cocycle is easy to check by a standard argument
in the theory of cohomology of groups using (1). Thus we have the corre-
spondingcohomologyclasses
α˜(k) ∈ H2k(SympδΣ ;Sk(S2R)) ,
g Q
denotedbythesameletters. IfweapplythecanonicalprojectionSk(S2R)→R
Q
totheseclasses,weobtainreal cohomologyclasses
αk ∈ H2k(SympδΣ ;R) ,
g
which are the usual cup products of the first one α ∈ H2(SympδΣ ;R).
g
Therefinedclassesα˜(k) canbeconsideredasatwistedversionofdiscontin-
uousinvariantsinthesenseof[18]arisingfrom thefluxhomomorphism.
Nowwecan stateourfirst mainresult.
Theorem 4. Foranyk ≥ 1 andg ≥ 3k, thecharacteristicclasses
ek,ek−1α˜,...,e α˜(k−1),α˜(k)
1 1 1
inducea surjectivehomomorphism
H (SympδΣ ;Z)−→Z⊕S2R⊕···⊕Sk(S2R) .
2k g Q Q
For k = 1 this is not hard to see, so we give the proof right away. For
k > 1theproofisgiveninSection6belowandrequiresthetechnicalresults
developedinthebodyofthispaper.
ConsiderthesubgroupHamΣ ofSymp Σ consistingofallHamilton-
g 0 g
iansymplectomorphismsofΣ . As iswellknown(see[1,16]), wehavean
g
extension
(2) 1−→HamΣ −→Symp Σ −Fl→ux H1(Σ ;R)−→1.
g 0 g g
Thisgivesrisetoa5-term exactsequencein cohomology:
0−→H1(H1(Σg;R)δ;Z) F−lu→x∗ H1(Sympδ0Σg;Z)−→H1(HamδΣg;Z)HR1
∗
−→H2(H1(Σ ;R)δ;Z) F−lu→x H2(SympδΣ ;Z) ,
g 0 g
6 D.KOTSCHICKANDS.MORITA
where we havewritten H1 for H1(Σ ;R). Now HamΣ is a perfect group
R g g
by a result of Thurston [25], see also Banyaga [1]. Therefore, Flux∗ in-
jects the second cohomology of H1(Σ ;R) as a discrete group into that of
g
SympδΣ . By definition, the class α˜ is the image of the class of ˜ι un-
0 g
der Flux∗. So α˜ is nontrivial on SympδΣ , and is defined on the whole
0 g
SympδΣ . Weconcludethat α˜ defines a surjectivehomomorphism
g
H (SympδΣ ;Z) → S2R,
2 g Q
for any g ≥ 2. We already proved in [13] that the first Mumford–Morita–
Miller class e defines a surjection H (SympδΣ ;Z) −→ Z for all g ≥ 3.
1 2 g
Clearly the two classes are linearly independent because α˜ is nonzero on
SympδΣ ,towhiche restrictstrivially. ThisprovesTheorem4intheeasy
0 g 1
casewhen k = 1.
Wecan restrictthehomomorphism
Flux∗: H∗(H1(Σ ;R)δ;R)−→H∗(SympδΣ ;R)
g 0 g
tothecontinuouscohomology
H∗(H1(Σ ;R)δ;R) ∼= H (T2g;R) ⊂ H∗(H1(Σ ;R)δ;R) ,
ct g ∗ g
see Section 3 for the precise definition. Thereby we obtain a ring homo-
morphism
Flux∗: H (T2g;R) −→ H∗(SympδΣ ;R),
∗ 0 g
whereT2g = K(π Σ ,1)istheJacobianmanifoldofΣ ,andtheringstruc-
1 g g
ture on the homology of T2g is induced by the Pontrjagin product. Let
ω ∈ H (T2g;R) be the homology class represented by the dual of the
0 2
standard symplectic form on T2g. We decompose the Sp(2g,R)-module
H (T2g;R) into irreducible components. For this, consider the homomor-
k
phism
ω ∧ : H (T2g;R) −→ H (T2g;R)
0 k−2 k
induced by the wedge product with ω . On the one hand, it is easy to see
0
using Poincare´ duality on T2g, that the above homomorphism is surjective
for any k ≥ g + 1. On the other hand, it is well-known (see [4]), that the
kernelofthecontractionhomomorphism
C: H (T2g;Q) −→ H (T2g;Q)
k k−2
induced by the intersection pairing H (T2g;Q) ∼= Λ2H (Σ ;Q)→Q is the
2 1 g
irreducible representation of the algebraic group Sp(2g,Q) corresponding
to the Young diagram [1k] for any k ≤ g. Let [1k] = [1k]⊗R denote the
R
real formofthisrepresentation. Then wehaveadirectsumdecomposition
(3) H (T2g;R) = [1k] ⊕ω ∧H (T2g;R) (k ≤ g).
k R 0 k−2
CHARACTERISTICCLASSESOFFOLIATEDSURFACEBUNDLES 7
Theorem 5. The kernel ofthehomomorphism
Flux∗: H (T2g;R)−→H∗(SympδΣ ;R)
∗ 0 g
induced by the flux homomorphism is the ideal generated by the subspace
ω ∧H (T2g;R) ⊂ H (T2g;R), and the image of this homomorphism can
0 1 3
bedescribedas
g
ImFlux∗ ∼= R⊕ [1k] ,
R
k=1
M
whereR denotestheimageof thesubspaceofH (T2g;R) spannedbyω .
2 0
ThegroupH1(Σ ;R)actsonHamΣ byouterautomorphisms. Inpartic-
g g
ular,itactsonthehomologyH (HamδΣ ;Z)ofthediscretegroupHamδΣ
∗ g g
sothatwecanconsiderthecoinvariantsH∗(HamδΣg;Z)H1,whereforsim-
R
plicity we have written H1 instead of H1(Σ ;R). The following result
R g
shows that the second homology group H (HamδΣ ;Z) is highly non-
2 g
trivial.
Theorem 6. Foranyg ≥ 2, thereexistsa naturalinjection
H1(Σg;R) ⊂ H2(HamδΣg;Z)H1 .
R
3. THE TRANSVERSE SYMPLECTIC CLASS
SinceΣ isan Eilenberg-MacLanespace, thetotalspaceoftheuniversal
g
foliatedΣ -bundleovertheclassifyingspaceBSympδΣ isagainaK(π,1)
g g
space. Hence if we denote by ESympδΣ the fundamental group of this
g
totalspace, thenweobtainashort exactsequence
(4) 1−→π Σ −→ESympδΣ −→SympδΣ −→1
1 g g g
andanycohomologyclassofthetotalspacecanbeconsideredasanelement
in the group cohomology of ESympδΣ . Now on the total space of any
g
foliated Σ -bundle with total holonomy group contained in SympΣ there
g g
isa closed2-form ω˜ which restrictsto thesymplecticform ω on each fiber.
At the universal space level, the de Rham cohomology class of ω˜ defines a
class v ∈ H2(ESympδ;R) which we call the transverse symplectic class.
We normalize the symplecticform ω on Σ so that its total area is equal to
g
2g−2. Itfollowsthattherestrictionofvtoafiberisthesameasthenegative
oftheEulerclasse ∈ H2(ESympδ;R)oftheverticaltangentbundle.
LetESymp Σ denotethesubgroupofESympδΣ obtainedbyrestrict-
0 g g
ing the extension (4) to Symp Σ ⊂ SympΣ . Since any foliated Σ -
0 g g g
bundle with total holonomy in Symp Σ is trivial as a differentiable Σ -
0 g g
bundle,thereexistsan isomorphism
∼
ESymp Σ = π Σ ×Symp Σ .
0 g 1 g 0 g
8 D.KOTSCHICKANDS.MORITA
Henceforth we identify theabovetwo groups. By theKu¨nneth decomposi-
tion,we havean isomorphism
H2(ESympδΣ ;R) ∼= H2(Σ ;R)⊕
0 g g
(5)
⊕ H1(Σ ;R)⊗H1(SympδΣ ;R) ⊕H2(SympδΣ ;R) ,
g 0 g 0 g
where we iden(cid:0)tify H∗(π1Σg;R) with H∗(Σg;R(cid:1)). Let µ ∈ H2(Σg;Z) be
the fundamental cohomology class of Σ . Clearly the Euler class e ∈
g
H2(ESympδΣ ;R) is equal to (2 − 2g)µ. The flux homomorphism gives
0 g
riseto an element
[Flux] ∈ Hom (H (SympδΣ ;Z),H1(Σ ;R))
Z 1 0 g g
∼=Hom (H (SympδΣ ;R),H1(Σ ;R))
R 1 0 g g
∼=H1(Σ ;R)⊗H1(SympδΣ ;R),
g 0 g
wherethelastisomorphismexistsbecauseH1(Σ ;R)isfinitedimensional.
g
Choose a symplectic basis x ,...,x , y ,...,y of H (Σ ;R) and denote
1 g 1 g 1 g
by x∗,...,x∗, y∗,...,y∗ thedual basis of H1(Σ ;R). Then Poincare´ dual-
1 g 1 g g
ityH (Σ ;R) ∼= H1(Σ ;R)isgivenbythecorrespondencex 7→ −y∗,y 7→
1 g g i i i
x∗. Theelement[Flux]can bedescribed explicitlyas
i
g
(6) [Flux] = (x∗ ⊗x˜ +y∗⊗y˜) ∈ H1(Σ ;R)⊗H1(SympδΣ ;R)
i i i i g 0 g
i=1
X
where x˜ ,y˜ ∈ H1(SympδΣ ;R) ∼= Hom(H (Symp Σ ;Z);R) is defined
i i 0 g 1 0 g
bytheequality
g
Flux(ϕ) = (x˜ (ϕ)x +y˜(ϕ)y ) (ϕ ∈ Symp Σ ).
i i i i 0 g
i=1
X
The elements x˜ ,y˜ can be also interpreted as follows. The flux homomor-
i i
phisminducesahomomorphismin cohomology
(7) Flux∗: H∗(H1(Σ ;R)δ;R)−→H∗(SympδΣ ;R)
g 0 g
wherethedomain
H∗(H1(Σ ;R)δ;R) ∼= Hom (Λ∗(H1(Σ ;R)),R)
g Z Z g
is the cohomology group of H1(Σ ;R) considered as a discrete abelian
g
group, rather than as a vector space over R, so that it is a very large group.
Itscontinuouspartis defined as
H∗(H1(Σ ;R)δ;R) =Hom (Λ∗(H1(Σ ;R)),R)
ct g R R g
⊂Hom (Λ∗(H1(Σ ;R)),R)
Z R g
⊂Hom (Λ∗(H1(Σ ;R)),R) ∼= H∗(H1(Σ ;R)δ;R) ,
Z Z g g
CHARACTERISTICCLASSESOFFOLIATEDSURFACEBUNDLES 9
wherethesecondinclusionis inducedby thenatural projection
Λ∗(H1(Σ ;R)) −→ Λ∗(H1(Σ ;R)) .
Z g R g
Denotingby T2g = K(π Σ ,1)theJacobian torusof Σ , thereis a canoni-
1 g g
cal isomorphism
Λ∗H1(Σ ;R) ∼= H∗(T2g;R),
R g
sothatwecan identify
H∗(H1(Σ ;R)δ;R) ∼= Hom (H∗(T2g;R);R) ∼= H (T2g;R).
ct g R ∗
Thus, by restricting the homomorphism Flux∗ in (7) to the continuous co-
homology,weobtainahomomorphism
Flux∗: H (T2g;R) −→ H∗(SympδΣ ;R).
∗ 0 g
Itis easy toseethat underthishomomorphismwehave
∗ ∗
x˜ = Flux (x ) , y˜ = Flux (y ) .
i i i i
Letω ∈ Λ2H (Σ ;R)bethesymplecticclassdefined by
0 R 1 g
g
ω = x ∧y
0 i i
i=1
X
andset
g
ω˜ = Flux∗(ω ) = x˜ y˜ ∈ H2(SympδΣ ;R) .
0 0 i i 0 g
i=1
X
Lemma 7. Wehavetheequality
[Flux]2 = −2µ⊗ω˜ ∈ H2(Σ ;R)⊗H2(SympδΣ ;R).
0 g 0 g
Proof. A direct calculationusingtheexpression(6)yields
g
[Flux]2 = − (x∗y∗ ⊗x˜ y˜ +y∗x∗ ⊗y˜x˜ ) .
i i i i i i i i
i=1
X
Sincex∗y∗ = −y∗x∗ = µ,weobtain
i i i i
g
[Flux]2 = −2µ⊗ x˜ y˜ = −2µ⊗ω˜
i i 0
i=1
X
asrequired. (cid:3)
Now we can completely determine the transverse symplectic class v of
foliatedΣ -bundleswhosetotalholonomygroupsarecontainedintheiden-
g
titycomponentSymp Σ ofSympΣ as follows.
0 g g
10 D.KOTSCHICKANDS.MORITA
Proposition8. OnthesubgroupESympδΣ thetransversesymplecticclass
0 g
v ∈ H2(ESympδΣ ;R) isgiven by
0 g
1
v = (2g −2)µ+[Flux]+ ω˜
0
2g −2
undertheisomorphism(5). Furthermore,thehomomorphism
ω˜ ⊗H1(SympδΣ ;R)−→H3(SympδΣ ;R)
0 ct 0 g 0 g
inducedbythecupproductistrivial,whereH1(SympδΣ ;R) ∼= H (Σ ;R)
ct 0 g 1 g
denotesthesubgroupofH1(SympδΣ ;R)generatedbythecontinuousco-
0 g
homologyclasses x˜ ,y˜. In particular,ω˜2 = 0.
i i 0
Proof. Sincetherestrictionofv toeachfiberisequaltothenegativeofthat
of the Euler class e by our normalization, v restricts to (2g − 2)µ on the
fiber. Thisgivesthefirstcomponentoftheformula. Thesecondcomponent
followsfromLemma8of[13]. Thuswecan write
v = (2g −2)µ+[Flux]+γ ∈ H∗(Σ ;R)⊗H∗(SympδΣ ;R)
g 0 g
forsomeγ ∈ H2(SympδΣ ;R). Nowobservethatv2 = 0becauseω˜2 = 0.
0 g
Also, because wehave restricted to Symp Σ , we haveµ2 = 0, µ[Flux] =
0 g
0. Hence weobtain
[Flux]2 +γ2 +2(2g −2)µγ +2[Flux]γ = 0 .
Itfollowsthat
[Flux]2 +2(2g −2)µγ = 0
[Flux]γ = 0
γ2 = 0
because thesethree elements belong to different summandsin theKu¨nneth
decomposition of H∗(Σ ;R) ⊗ H∗(Sympδ;R). If we combine the first
g 0
equalityaboveand Lemma7, thenwe can concludethat
1
(8) γ = ω˜ .
0
2g −2
This proves the first claim of the proposition. If we substitute (6) and (8)
in the second equality above, then we see that ω˜ x˜ = ω˜ y˜ = 0 for any i,
0 i 0 i
whence the second claim. Observe that the third equality γ2 = 0, which is
equivalent to ω˜2 = 0 by the above, is a consequence of the second claim.
0
(cid:3)
NowwecancalculatetherestrictionofthecocycleαdefinedinSection2
totheidentitycomponentSymp Σ .
0 g