On the abelianization of the Torelli group De abelianisatie van de Torelligroep (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Uni- versiteit Utrecht op gezag van de Rector Magnificus, Prof.dr.W.H.Gispen,ingevolgehetbesluitvanhetCollege voor Promoties in het openbaar te verdedigen op maandag 6 oktober 2003 des ochtends te 10.30 uur door Barbara van den Berg geboren op 22 februari 1971, te Groningen Promotor: Prof. Dr. E.J.N. Looijenga Faculteit der Wiskunde en Informatica, Universiteit Utrecht Leescommissie: Prof. Dr. G.B.M. van der Geer Prof. Dr. R.M. Hain Dr. W.L.J. van der Kallen Prof. Dr. I. Moerdijk Prof. Dr. D. Siersma Dit proefschrift werd mede mogelijk gemaakt met financi¨ele steun van de Nederlandse Organisatie voor Wetenschappelijk Onderzoek. 2000 Mathematics Subject Classification: 57M50 ISBN 90-393-3467-6 Contents List of Notations v Introduction ix Chapter 1. Symplectic modules 1 1.1. Introduction 1 1.2. Surface modules 1 1.3. Quadratic forms 4 1.4. Simplicial complexes and posets 5 1.5. Definitions of simplicial complexes associated to a surface module. 8 1.6. The Cohen-Macaulay property of I(H,J) and Io(H,J) 11 1.7. The Cohen-Macaulay property I(π) 13 ≤g−2 1.8. The Cohen-Macaulay property of I(π−1(1)) and Io(π−1(1)) 17 ≤g−2 ≤g−2 1.9. The Cohen-Macaulay property of I(π−1(1)) and Io(π−1(1)) when g =1,2,3 18 1.10. The connectedness of Ao(H) and Ao(H,π) 27 1.11. Simplicial complexes with a group action 29 1.12. Computation of H (Σ,F) 30 0 Chapter 2. Surfaces 35 2.1. Introduction 35 2.2. The Torelli group 35 2.3. The work of Johnson and others on the Torelli group 38 2.4. Closing a hole of a surface 42 2.5. The arc-complexes of Harer 45 Chapter 3. The abelianization of the Torelli group 49 3.1. Introduction 49 3.2. Genus zero 51 3.3. Genus one 59 3.4. Genus two 62 3.5. Genus three or more 64 Bibliography 73 iii iv CONTENTS Samenvatting 75 Dankwoord 83 Curriculum vitae 85 List of Notations We use the following notations. The number refers to the page number where you can find the definition. General notations H (X) H (X)istheith (group, singular, cellular)homologygroup i i of(agroup,atopologicalspace,aspacewithcellstructure) X with integer coefficients, C (X) a chain complex to compute H (X) is denoted by C (X), i i i Z[G] Z[G] denotes the group ring of a group G, I[G] I[G]:=Ker((cid:178):Z[G]→Z) is the augmentation ideal, [a,b] [a,b]:=aba−1b−1 is the commutator of a and b, G we write G for the abelianization G/[G,G] of a group G, ab ab Z(A) If A is a set, then Z(A) denotes the group of maps A → Z with finite support. Notations introduced in Chapter 1 Rad(H) Rad(H) is the radical of a quasi-unimodular symplectic p. 1 module H, H H :=H/Rad(H), p. 1 g(H) g(H):= 1rk(H) is the genus of H, p. 1 2 (H(cid:101),H,∆) (H(cid:101),H,∆) denotes an extended surface module, p. 2 (S,P) (S,P) denotes a surface with boundary marking, p. 2 Sp(H(cid:101),H) Sp(H(cid:101),H) is the group of automorphisms of an extended p. 3 surface module (H(cid:101),H,∆), Sp(H) when H(cid:101) =H the group Sp(H(cid:101),H) is denoted by Sp(H), p. 3 δ foreveryv ∈H wehaveδ (x):=x+(x·v)v,thesymplectic p. 3 v v transvection determined by v, K(H(cid:101),H) K(H(cid:101),H):=Ker(Sp(H(cid:101),H)→Sp(H)), p. 3 S2V S2V is the submodule of V ⊗ V of invariants under the p. 3 involution determined by a⊗b(cid:55)→b⊗a, v vi LIST OF NOTATIONS V ◦V/U V ◦V/U :=S2V/S2U for a submodule U of V, p. 3 Ω if H is a symplectic module over Z/2 then Ω denotes the p. 5 H H affine space of associated quadratic forms on H, Ψ Ψ is the set of quadratic forms of Arf-invariant zero, p. 5 H H B (Ω ) B (Ω ) is the space of all polynomial functions on Ω of p. 5 r H r H H degree ≤r, |Σ| |Σ| is the topological realization of a simplicial complex Σ, p. 5 H (Σ,F) the pth homology group of Σ with values in the system of p. 6 p coefficients F is denoted by H (Σ,F), p f/y f/y :={x∈X :f(x)≤y}, p. 6 f\y f\y :={x∈X :f(x)≥y}, p. 6 Link−(y) Link−(y):=X ={x∈X :x<y}, p. 6 X X <y Link+(y) Link+(y):=X ={x∈X :x>y}, p. 6 X X >y Star (y) Star (y):=Link (y)∪{y}, p. 6 X X X CM CM is an abbreviation for Cohen-Macaulay of dimension p. 7 d d d, O(S) O(S) is the poset of nonempty finite subsets of S, p. 8 T(V,W) T(V,W) is the poset of nonzero proper subspaces U of V p. 9 such that U ⊕W →V is a primitive embedding, P(V,W) P(V,W) is the poset of partial bases E of V such that its p. 9 span (cid:104)E(cid:105) is in T(V,W) Ao(H) Ao(H) is the poset of arc-sequences in H, p. 9 Ao(H,π) if π : H → Z is an epimorpism that factorizes of Rad(H), p. 9 then Ao(H,π) is the poset of arc-sequences Eo in π−1(1) such that Eo ∈P(H,Rad(π−1(0))), I(H,I) I(H,I) is the poset of U ∈ T(H,I) such that U + I is p. 10 isotropic, I(H,I) I(H,I) is the poset of E ∈P(H,I) such that its span (cid:104)E(cid:105) p. 10 is in I(H,I), T(π) T(π) is the poset of U ∈ T(V) such that U is in general p. 13 position relative to π, T(π/ρ) T(π/ρ) is the poset of U ∈ T(π) such that U is primitive p. 13 relative to ρ, I(π) I(π):=I(H)∩T(π), p. 13 Notations introduced in Chapter 2 Sn Sn denotesacompact,oriented,connectedtopologicalsur- p. 35 g,r g,r faceofgenusg,withrboundarycomponentsandndistinct fixed points on the interior of Sn , g,r FSn FSn denotes the group of orientation preserving homeo- p. 35 g,r g,r morphismsof Sn thatare the identityon the boundaryof g,r S and fix the n distinct points pointwise, g,r LIST OF NOTATIONS vii Γn Γn is the mapping class group of Sn , that means, the p. 35 g,r g,r g,r group of isotopy classes of FSn , g,r D if γ is an embedded circle on the interior of Sn disjoint p. 35 γ g,r from the n fixed points, we denote by D the left Dehn γ twist around γ, D D D D means first apply D then D , β α β α α β T T :=Ker(Γ →Sp(H (S,P),H (S))) is the Torelli group p. 36 S S S 1 1 of S T(cid:101) T(cid:101) :=Ker(Γ →Sp(H (S))) is the big Torelli group of S p. 36 S S S 1 SCC simple closed curve, p. 35 BSCC bounding simple closed curve, p. 38 BP bounding pair, p. 38 T T is the set of BSCC-maps that bound a subsurface of p. 38 k k genus k, W W isthesetofBP-mapsthatboundasubsurfaceofgenus p. 38 k k k, τ for every m≥1 the Johnson homomorphism τ :Γ(m)→ p. 40 m m Hom(H,π /π ) is defined, [m] [m+1] t for an oriented loop α without self intersection such that p. 42 α ∂S ∩α = {p} on a component ∂ of ∂S, we define t := α D−1D . Here α ,α are the boundary components of α+ α− + − the regular neighborhood of α∪∂ such that α is on the − left of α and α on the right, + BX(Λ,Λ0) BX(Λ,Λ0) denotes the simplicial complex of (Λ,Λ0)-arc p. 45 systems defined by Harer, BX(p,q) BX(p,q) := T \BX(p,q), sometimes also abbreviated by p. 46 S BX, Notations introduced in Chapter 3 Fn(S) if S is a surface then Fn(S) is the configuration space of n p. 51 pairwise distinct points of S, Pn(S) Pn(S):=π(Fn(S))thepurebraidgroupofS onnstrings, p. 51 H if H is a free module over Z then H :=H ⊗ Z/2, p. 59 2 2 Z M M is the set of unimodular symplectic subspaces of H of p. 59 H H genus 1, N N is the set of unimodular symplectic subspaces of H of p. 59 H H genus 2, R R isthesetofelementsU⊕U(cid:48)−U−U(cid:48),whereU,U(cid:48) ∈M , p. 59 H H H U ⊥U(cid:48) and U ⊕U(cid:48) ∈N , H G(cid:101) G(cid:101) := Z/2(NH)⊕Z(MH), p. 59 H H RH G G := Z/2(NH2)⊕Z/2(MH2). p. 59 H H RH2 Introduction Let S be a surface of genus g, possibly with boundary (in this thesis by surface is meant a connected, orientable and compact topological surface). The mapping class group Γ of S is the group of isotopy classes of the orientation preserving S homeomorphisms of S that are the identity on the boundary ∂S. If we choose on each boundary component of S a point and denote this set of points by P, then Γ S actsonH (S,P)(relativehomologywithintegercoefficients),leavingthesubmodule 1 H (S)invariantandpreservingtheintersectionproductH (S,P)×H (S)→Zthat 1 1 1 can be defined. The kernel of this action is by definition the Torelli group T of S. S This means that if ∂S is empty or connected, then T is the subgroup of mapping S classes that act trivially on the homology of S. If ∂S has more than one component weneedthisrefineddefinitioninordertomakeT functorialforinclusionofsurfaces. S In this thesis we study the abelianization of the Torelli group of a surface S withanarbitrary(butfinite)numberofboundarycomponents. Thisstudytakesup and continues the work of Johnson from around 1980. He computes, among other things, that for a surface of genus at least three and with one boundary component, H (T ) ∼= ∧3H (S) modulo 2-torsion, and the torsion is also completely described 1 S 1 in terms of the homology of the surface, see [Johnson8]. Here we prove that this resultholdsforsurfacesofg ≥3withanarbitrarynumberofboundarycomponents. Our method of proof differs from his, is inductive in nature and may open the way to calculate the higher homology of these groups. We study how H (T ) changes 1 S comparedtoH (T ),whereS isobtainedfromS(cid:48) bygluingapairofpantstoit,by 1 S(cid:48) using the action of T on Harer’s arc-complexes. Furthermore we study the Torelli S group of a surface of low genus, the results of which are also needed to start the induction. We finish this introduction with an overview of the content of all chapters in this thesis, but let us first be more explicit about the method of proof we use. Let p,q ∈ ∂S. The basic tool in our study of T is its action on the highly S connected arc-complexes BX(p,q) that were introduced by Harer in [Harer]. The k-simplices of this simplicial complex are (k +1)-tuples of isotopy classes of arcs from p to q that can be represented by a (k+1)-tuple of embedded arcs which are disjoint away from p,q and whose complement in S is connected. They have the ix x INTRODUCTION property that the stabilizers under the action of Γ , respectively T , are mapping S S class groups, respectively Torelli groups, of surfaces of lower genus or with fewer boundary components. Harer proves in that BX(p,q) is spherical and, using the action of the mapping class group on BX(p,q), he establishes the stability of the homology of the mapping class groups induced by the inclusion of surfaces. His approach is analogous to the proof of the stabilization of homology of various arith- meticgroups. Seealso[Ivanov]onthestabilityofthehomologyofthemappingclass group. We use, just as Foisy does in [Foisy], the induced action of T on this arc- S complex. Here the quotient space is a simplicial complex that is closely related to the complexes of (isotropic) partial bases of a (symplectic) lattice, studied by Maazen, Van der Kallen, Charney, Vogtmann and others to prove the homology stabilityforcertainlineargroups. Seeforexample[Charney], [VdKallen], [Maazen], [Vogtmann]. Most of these complexes are known to be spherical and we show that this quotient space is connected up to a certain dimension. When the genus of the surface is ≥ 4 and we choose p,q on the same boundary component, this quotient is at least 2-connected and we get by a spectral sequence argument a description of H (T ) as an amalgam of the abelianization of the stabilizers of the vertices. When 1 S g = 3 the spectral sequence shows that H (T ) is a quotient of this amalgam, the 1 S Johnson and Birman-Craggs homomorphisms show that the kernel is trivial. Forlowgenera,(g ≤2),wedonothaveauniformdescriptionoftheabelianized Torelli group. Mess shows that for a closed surface of genus two, T is infinitely S freely generated by a set of Dehn twists around separating curves, see [Mess]. We give a description of H (T ) in some other cases. 1 S Thethesisconsistsofthreechapters,culminatingintheproofofJohnsonsresult for all surfaces of g ≥3. The outline of this thesis is as follows. Thefirstchapterisaboutsymplecticmodulesandsimplicialcomplexesdescribed in terms of such modules. We introduce the notion of an extended surface module which formalizes the situation of H (S,P) and the structures it has. It will appear 1 in the second chapter that the quotient of BX(p,q) by the action of the Torelli group, T \BX(p,q), is again a simplicial complex and can be described in terms of S symplectic modules. The main goal of this chapter is to prove that T \BX(p,q) S is (g−2)-connected when p and q are on the same boundary component; when p and q are on different components we show that it is 1-connected if g ≥ 2. We will do this in Sections 1.6 up to 1.10 and for this purpose we introduce some other complexes,someofthemareknowntobespherical,ofothersweproveconnectedness properties here. We finish this chapter with a discussion about simply connected simplicial complexes with a group action. Using a spectral sequence we derive an exact sequence that relates the abelianization of the group to the low-dimensional homology of the quotient complex and the abelianization of the stabilizers.