Homology Computations

for Mapping Class Groups,

0in particular for Γ3,1

Dissertation

zur

Erlangung des Doktorgrades (Dr. rer. nat.)

der

Mathematisch-Naturwissenschaftlichen Fakult¨at

der

Rheinischen Friedrich-Wilhelms-Universit¨at Bonn

vorgelegt von

Rui Wang

aus

Beijing, China

Bonn, Februar 2011ii

Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at

der Rheinischen Friedrich-Wilhelms-Universita¨t Bonn

1. Gutachter: Prof. Dr. Carl-Friedrich B¨odigheimer

2. Gutachter: Prof. Dr. Jens Franke

Tag der Promotion: 1. Juli 2011

Erscheinungsjahr: 2011iii

Summary

In this thesis we compute the homology of mapping class groups of

orientableandnon-orientablesurfaces. Thesurfacesweconsiderare

of genus g, have one boundary curve and m permutable punctures.

m mThe corresponding moduli spaces M in the orientable and Ng,1 g,1

in the non-orientable case are classifying spaces for the mapping

class groups.

We are able to compute the integral homology of the modulispaces

m mM forh=2g+m<6andofN forh =g+m+1<5(Notethatg,1 g,1

we give a non-orientable surface the genus g if it is the connected

sum of g +1 projective planes). For h = 6 in the orientable case

0and h = 5 in the non-orientable case (these are the cases M ,3,1

2 4 0 1 2 3M and M resp. N , N , N and N ) we can compute2,1 1,1 4,1 3,1 2,1 1,1

some p-torsion in the homology and the mod-p Betti numbers for

several primes. But this is enough evidence to conjecture that we

have indeed the entire integral homology in these cases, too.

Thecomputationsarebasedonacellstructureofthemodulispaces.

Thiscell structureisbi-simplicial andtheassociated chain complex

Q (h,m) resp. NQ (h,m) can be describedby parts of the classi-•• ••

fying spaces of symmetric groupsS ,...,S resp. by parts of the2 2h

classifying space of a category of pairings.

MotivatedbyB.Visy’sDissertation, weinvestigatewaystosimplify

m mthehomology computationforM andN . Ontheonehand,weg,1 g,1

extend the notion of factorable groups to factorable categories and

studythe homology of the norm complex associated to a factorable

category; moreover, similar to the fact that a symmetric group is

factorable,weprovethatthecategoryofpairingsisafactorablecat-

m megory. Onthe otherhand, fromthecell structuresofM andNg,1 g,1

with their orientation systems, we construct the double complexese gQ (h,m) andNQ (h,m) and study their homology.•• ••

For the actual computations, we implemented the new algorithms

in a C++ program.ivContents

Introduction 1

1 Factorable Normed Categories 7

1.1 Norm Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Factorability and Homology of the Norm Complex . . . . . . . . . . 10

1.3 Homology Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Application to Moduli Spaces of Riemann Surfaces 25

2.1 Moduli Spaces of Riemann Surfaces. . . . . . . . . . . . . . . . . . . 25

2.2 Symmetric Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Orientation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Application to Moduli Spaces of Kleinian Surfaces 45

3.1 Moduli Space of Kleinian Surfaces . . . . . . . . . . . . . . . . . . . 45

3.2 The Category of Pairings . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Orientation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Computational Results 67

4.1 The Riemann Surface Case . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 The Kleinian Surface Case . . . . . . . . . . . . . . . . . . . . . . . . 69

Appendix: The computer program 75

References 81

vviIntroduction

The main aim of this thesis is to do new computations on the homology of moduli

spaces of surfaces, or equivalently, of mapping class groups of surfaces.

mWe consider the moduli space M of conformal equivalence classes of Riemanng,1

msurfaces F = F of genus g ≥ 0 with one boundary curve and m≥ 0 permutableg,1

mpunctures. Denote by Γ the corresponding mapping class group, i.e. the isotropyg,1

classesoforientation-preservingdiﬀeomorphismsﬁxingtheboundarypoint-wiseand

permutingthepunctures. Sincethediﬀeomorphismsarerequiredtoﬁxtheboundary

m mcurve, Γ is torsion free. Therefore Γ acts freely on the contractible Teichmu¨llerg,1 g,1

m m m m mspace Teich and the manifold M =Teich /Γ is a classifying space of Γ .g,1 g,1 g,1 g,1 g,1

A non-orientable surface together with a dianalytic (i.e. the coordinate changes

are holomorphic or antiholomorphic) structure is called a Kleinian surface. Analo-

mgously to the case of a Riemann surface, let N be the moduli space of dianalyticg,1

m 1equivalence classes of Kleinian surfaces NF = NF of genus g ≥ 0 with oneg,1

mboundary curve and m ≥ 0 permutable punctures. Let NΓ denote the corre-g,1

msponding mapping class group. Again, NΓ is torsion free, acts freely on theg,1

m mm mcontractible Teichmu¨ller space NTeich and the manifold N =NTeich /NΓg,1 g,1 g,1 g,1

mis the classifying space of NΓ .g,1

Inordertocomputethehomologygroups,itishelpfultoﬁndasuitablecellstructure

ofthese modulispaces. In[B1], B¨odigheimer ﬁrstintroducedanaﬃnevector bundle

m m m mHarm overM . The ﬁberofHarm over a point F ∈M consists of harmonicg,1 g,1 g,1 g,1

functions on F with certain prescribed singularities. He then analysed the gradient

mﬂow of a point u in Harm and associated to u a so called stable critical graphK.g,1

The graph K in turn produces a parallel slit domain, which is the complex plane

with h = 2m + g pairs of parallel horizontal slits. This method is called Hilbert

uniformization. Inthis way, B¨odigheimer obtained the space of parallel slit domains

m ′Par = Par(h,m)r Par(h,m), which is a manifold. Here Par(h,m) is a ﬁniteg,1

′cell complex and Par(h,m) is a subcomplex (consisting of “degenerate” surfaces).

m mThe main result in [B1] is that Par is homeomorphic to Harm . On the otherg,1 g,1

m mhand, Harm is a ﬂat ﬁber bundle with contractible ﬁber over M and thereforeg,1 g,1

m m mhomotopy equivalent to M , thus Par has the some homotopy type as M :g,1 g,1 g,1

m m ′Harm Par =Par(h,m)rPar(h,m)g,1 g,1∼=

≃

mMg,1

1The genus of a non-orientable surface is g, if it is a connected sum of g+1 real projective planes;

thus we diﬀer from the usual convention by one.

12

The cell structure on Par(h,m) is bi-simplicial, therefore the cellular chain complex

′Q (h,m) oftherelative manifold(Par(h,m),Par(h,m)) isadoublechain complex.••

Q (h,m) also has a more combinatorial description using symmetric groups. We••

will only sketch this approach here, more details can be found in section 2.1.

Denote by P (h) the free abelian group generated by all (q + 1)-tuples Σ =p,q

(σ ,...,σ ) with σ in the symmetric group S such thatq 0 i p+1

−1 −1N(Σ):=N(σ σ )++N(σ σ )≤h.q 1q−1 0

Here N(α) is the word length norm of α with respect to the generating set of Sp+1

which consists of all transpositions. L

Deﬁne a double complex P (h) := P (h), 0 ≤ p ≤ 2h, q ≤ h, whose ver-•• p,q

′ ′′tical and horizontal face operators are ∂ (Σ) = (σ ,...,σb ,...,σ ) and ∂ (Σ) =q i 0i j

(D (σ ),...,D (σ )) respectively, where D is deﬁned on page 30. The subcomplexj q j 0 j

′P (h,m) of P (h) is generated by those cells of P (h) which violate one of the•• ••••

conditions on page 30, for example

(1) N(Σ) =h

(2) σ has m+1 cyclesq

(3) σ is the rotation (0 1...p)0

′∼ThenQ (h,m) P (h)/P (h,m) is the double complex we are looking for.=•• •• ••

TheHilbert uniformization methodcan also beappliedto modulispaces ofKleinian

surfaces, see [Z1] and [E] for more details. Like in the orientable case, there is

m m man aﬃne vector bundle Nharm over N whose ﬁber over a point NF ∈ Ng,1 g,1 g,1

consists of dianalytic functions on NF with certain prescribed singularities. Again

mby analysing the gradient ﬂows of the points in Nharm , the space of parallel slitsg,1

m ′domains NPar = NPar(h,m)r NPar(h,m), which is also a manifold, can beg,1

′obtained. Here NPar(h,m) is a ﬁnite cell complex and NPar(h,m) is a subcomplex

m(consisting of “degenerate” surfaces). The main result in [E] is that NPar isg,1

m mhomeomorphic to Nharm . Moreover, since Nharm is a ﬂat ﬁber bundle withg,1 g,1

mm mcontractible ﬁberover N and therefore homotopy equivalent toN ,NPar hasg,1g,1 g,1

mthe some homotopy type as N .g,1

ThecellstructureonNPar(h,m)isbi-simplicial,thereforethecellularchaincomplex

′NQ (h,m) of the relative manifold (NPar(h,m),NPar (h,m)) is a double chain••

complex. Again there is a more combinatorial description of NQ (h,m), using••

pairings. We will give a brief review of this here, more details can be found in

section 3.1.

Denote by Λ ⊂S the set of ﬁxed-point free involutions – so-called pairings – onp 2p

2p letters. Let NP (h) be the free abelian group generated by all (q +1)-tuplesp,q

Λ=(λ ,...,λ ) with λ ∈Λ , such thatq 0 i p

1 −1 −1N (Λ) = (N (λ λ )+...+N (λ λ ))≤h,ΛΛ S q S 1p 2p q−1 2p 02

where N (α) is the word length norm of α with respect to the generating set ofS2p

S which consists of all transpositions.2p3

L

Deﬁne a double complex NP (h) := NP (h), 0 ≤ p ≤ 2h, q ≤ h, whose ver-•• p,q

′ ′′btical and horizontal face operators are ∂ (Λ) = (λ ,...,λ ,...,λ ) and ∂ (Λ) =q i 0i j

(D (λ ),...,D (λ )) respectively, where D is deﬁned on page 51. The subcomplexj q j 0 j

′NP (h,m) of NP (h) is generated by the cells of NP (h) which violate any of•• ••••

conditions listed on page 51. Some of these conditions are:

(1) N (Λ) =hΛΛp

(2) λ ◦J has 2(m+1) cycles, where the special element J ∈Λ is deﬁned in (3.1.5)q p

(3) λ is given by (3.1.6)0

′∼ThenNQ (h,m) =NP (h)/NP (h,m) is the desired double complex.•• •• ••

mComputations on the homology of M already have been done using the spectralg,1 esequencesofthedoublecomplexesQ (h,m)andQ (h,m)–whichwillbedescribed•• ••

later – in the series of works [Eh], [A] and [ABE]. In [ABE], the results up to

mh =5 forM are obtained; this article gives also a good overview of the homologyg,1

computations that have been done by other authors at this time. A special feature

of the computational results in [ABE] is that they do not lie in the stable range,

and very little information is known in this situation as h becomes bigger. In the

mnon-orientable situation, in [Z2], mod-2 homology ofN was computed for h =2,3g,1

via the double complexNQ (h,m).••

During the computations using the double complex Q (h,m), Ehrenfried, Abhau••

and B¨odigheimer realized that the spectral sequence ofQ (h,m) has the property••

1 2that its E -term concentrates on the top degree and thus it converges at E . This

phenomenon later was fully explained by Visy in [V]. He introduced the concept

of factorable groups while studying the norm complexN (G)[h] of a normed group∗

G and proved the important result that the homology ofN (G)[h] concentrates on∗

the top degree h. The behavior of the spectral sequence of Q (h,m) can then••

be explained from the facts that every symmetric group S is factorable and thatp

Q (h,m) is isomorphic to a direct summand ofN (S )[h].p,∗ ∗ p

1The theory developed by Visy also allows one to construct the E -term of the spec-

1tral sequence of Q (h,m) directly. The E -term is equivalent to a chain complex••

(W (h,m),d). This motivated us to do more homology computations. For example,∗

0 1theoretically the homology of Γ and Γ can be computed using (W (h,m),d),∗g,1 g,1

0 1since the spaces M and M are orientable.g,1 g,1

Inthepresentwork,wehave extendedthetheoryoffactorable groupsintwoaspects

to make full use of the idea of factorability.

mThe ﬁrst one is, in order to simplify the homology computation about N in theg,1

msame manner as that of M , we generalize the notion of a factorable group to ag,1

factorable category. We do this, because the categories of pairings are involved in

the double complex NQ (h,m) in the same way as symmetric groups are involved••

inQ (h,m).••

The other aspect is, since the double complexes Q (h,m) and NQ (h,m) are the•• ••

cell structure of relative manifolds, Poincar´e duality is needed to connect their co-4

m mhomology to the homology of the moduli spaces M and N :g,1 g,1

∗ ′ m∼H (Par(h,m),Par(h,m);O) =H (M ;Z),3h−∗ g,1

∗ ′ m∼H (NPar(h,m),NPar (h,m);O) =H (N ;Z).3h−∗ g,1

mHence when the moduli space is non-orientable (which is the case for M wheng,1

mm ≥ 2 and for N for all m) the orientation system O on the relative manifoldg,1

is involved in the computation of its integral homology. Therefore, we constructe gthe double complexes Q (h,m) and NQ (h,m), which are the cell complexes of•• ••

the relative manifolds with orientation system. They have the same Z-modules as

Q (h,m) and NQ (h,m), but diﬀerent boundary operators. These double com-•• ••

1plexes also turn out to have the properties that the E -terms of their spectral se-

2quences concentrate on the top degree and converge at E . We obtain these results

by arguments very similar to those used in proving the corresponding properties of

factorable groups.

After these theoretical preparations, we were able to dothe homology computations

m mfor the mapping class groups Γ with h ≤ 6 and for NΓ with h ≤ 5. Theg,1 g,1

computations were carried out with the help of a computer program written in

m mC++. But for Γ with h = 6 and m = 0,2,4, and for NΓ when h = 5 andg,1 g,1

m=0,1,2,3, we only get partial information about their homology groups, because

some of matrices involved were so huge, that it was not possible to compute the

Smith normal form on standard computers. However, we conjecture that we have

actually obtained the full information.

0Among the mapping class groups we considered, Γ is a particularly interesting3,1

example. Based on our computation, we conjecture: Z n=0 0 n=1 Z⊕Z n=2 2 Z⊕Z ⊕Z ⊕Z ⊕Z n=3 2 3 4 7 2 2 Z ⊕Z n=4 2 3

0H (M ) = Z⊕Z ⊕Z n=5n 2 33,1 3Z⊕Z n=6 2 Z n=72 0 n=8 Z n=9 0 n≥10

For n=0,1,4,7,8,9 and n≥10 in this list, the homology groups have been veriﬁed

by the computational results and the universal coeﬃcient theorem. It is known that

0H (Γ )liesinthestablerangeandshouldbe0accordingtothetheoryofthestable1 3,1

homology of mapping class groups, which is consistent with our result. Moreover,

0H (Γ ) recently was computed with completely diﬀerent methods by Sakasai ([S])2 3,1

to beZ⊕Z , which is also consistent with our conjecture. If we take this result into2

0consideration, then H (Γ ) = Z⊕Z ⊕Z ⊕Z ⊕Z is also veriﬁed. Thus only3 2 3 4 73,1

0 0H (Γ ) and H (Γ ) are not completely determined and remain as conjecture.5 63,1 3,1

Apart from this, our computational results for h ≤ 5 coincide with the results in

m[ABE]. Comparing the mod-2 homology computations of NΓ in [Z2] for h = 2,3g,1