On infinitely generated homology of Torelli groups

On infinitely generated homology of Torelli groups

Alexander A. Gaifullin Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, Moscow, 119991, Russia Moscow State University, Leninskie gory 1, Moscow, 119991, Russia Skolkovo Institute of Science and Technology, Nobel str. 3, Moscow, 121205, Russia Institute for Information Transmission Problems (Kharkevich Institute), Bolshoy Karetny per. 19, build. 1, Moscow, 127051, Russia
Abstract.

Let be the Torelli group of an oriented closed surface  of genus , that is, the kernel of the action of the mapping class group on the first integral homology group of . We prove that the th integral homology group of  contains a free Abelian subgroup of infinite rank, provided that and . Earlier the same property was known only for (Bestvina, Bux, Margalit, 2007) and in the special case (Johnson, Millson, 1992). Moreover, we show that the group  contains a free Abelian subgroup of infinite rank generated by Abelian cycles and we construct explicitly an infinite system of Abelian cycles generating such subgroup. As a consequence, we obtain that an Eilenberg–MacLane CW-complex of type  cannot have a finite -skeleton. The proofs are based on the study of the Cartan–Leray spectral sequence for the action of  on the complex of cycles constructed by Bestvina, Bux, and Margalit.

Key words and phrases:
Torelli group, homology of groups, complex of cycles, Abelian cycles, Cartan–Leray spectral sequence
2010 Mathematics Subject Classification:
20F34 (Primary); 57M07, 20J05 (Secondary)
The work is supported by a grant of the President of the Russian Federation (grant MD-2907.2017.1) and by the Russian Foundation for Basic Research (grant 18-51-50005).

1. Introduction

Let be an oriented closed surface of genus . Recall that the mapping class group of  is the group

 Mod(Sg)=π0Homeo+(Sg),

where is the group of orientation preserving homeomorphisms of  on itself. The action of the mapping class group on the first integral homology group of  preserves the intersection form and yields the surjective homomorphism

 Mod(Sg)→Sp(2g,Z).

The kernel of this homomorphism is called the Torelli group of  and is denoted by .

It is a classical result that , and so  is trivial, cf. [FaMa12, Theorem 2.5]. Mess [Mes92] proved that  is an infinitely generated free group. (The fact that is not finitely generated was earlier proved by McCullough and Miller [MCM86].) Johnson [Joh83] showed that for , the group  is finitely generated, and described explicitly a finite set of generators. Nevertheless, the structure of the Torelli groups , where , is still rather poorly understood.

An interesting and important problem is to study the homology of the Torelli groups , . For the sake of simplicity, we denote the homology groups  with integral coefficients simply by . The group , i. e., the Abelianization of  was computed by Johnson [Joh85b]. He showed that

 H1(Ig)≅Z(2g3)−2g⊕(Z/2Z)(2g2)+2g

and described explicitly the structure of an -module on this group. For , the problem of explicit computation of  seems to be extremely hard. So the reasonable questions are:

• Which groups  are trivial and which are not?

• Which groups  are finitely generated and which are not?

One of the first results towards the latter question was obtained by Akita [Aki01] who showed that for , the total rational homology group

 H∗(Ig;Q)=⨁k≥0Hk(Ig;Q)

is an infinitely-dimensional vector space. For , it is known that either of the groups  and  contains a free Abelian subgroup of infinite rank, see [Mes92] and [Hai02], respectively.

In 2007 Bestvina, Bux, and Margalit [BBM07] developed a new method for studying the homology of the Torelli groups based on the Cartan–Leray spectral sequence for the action of  on a special contractible cell complex  called the complex of cycles. They proved that the cohomological dimension of  is equal to  and the top-dimensional homology group  is not finitely generated. (Recall that the cohomological dimension  of a group  is the largest integer  for which there exists a -module  such that .) Moreover, the proof in [BBM07] actually yields that contains a free Abelian subgroup of infinite rank, see Remark 2.5 below. Until now, for no intermediate dimension , it has been known whether the group  is finitely generated or not, except for the above mentioned case due to Mess. The main result of the present paper is as follows.

Theorem 1.1.

Suppose that and . Then the group is not finitely generated. Moreover, contains a free Abelian subgroup of infinite rank.

Recall the definition of an Abelian cycle. Let be pairwise commuting elements of a group . Consider the homomorphism that sends the generator of the th factor  to  for every . We denote by  the image of the standard generator  of the group  under the homomorphism . Homology classes are called Abelian cycles.

Vautaw [Vau02] proved that the Torelli group  does not contain a free Abelian subgroup of rank greater than . Since is torsion-free, this result implies that the group  contains no non-trivial Abelian cycles, provided that . For , we can refine Theorem 1.1 in the following way.

Theorem 1.2.

Suppose that . Then the group contains a free Abelian subgroup of infinite rank whose generators are Abelian cycles.

Remark 1.3.

For , Theorem 1.1 exactly recovers the already mentioned result by Johnson and Millson (unpublished, see [Mes92]) that contains a free Abelian subgroup of infinite rank. Nevertheless, Theorem 1.2 seems to be a new result even for genus . For each pair such that and any , the assertion of Theorem 1.1 is a new result.

Remark 1.4.

Our proof of Theorem 1.2 is constructive, which means that we shall construct explicitly an infinite set of linearly independent Abelian cycles in . (Recall that elements of an Abelian group are linearly independent if they form a basis for a free Abelian subgroup of this group.) Notice also that this construction does not use the result of Bestvina, Bux, and Margalit [BBM07] claiming that the group is infinitely generated. On the contrary, our construction of an infinite set of linearly independent homology classes in , where , is in a sense a mixture of our construction of an infinite set of linearly independent Abelian cycles in  and the Bestvina–Bux–Margalit description of an infinite set of linearly independent homology classes in . It is a debatable question whether the latter description can be regarded as an explicit construction. Our proof of Theorem 1.1 is constructive modulo this description.

The proofs of Theorems 1.1 and 1.2 are based on a more delicate study of the Cartan–Leray spectral sequence  for the action of  on the contractible complex of cycles  constructed by Bestvina, Bux, and Margalit [BBM07]. Recall that this spectral sequence converges to the homology of , see Section LABEL:subsection_CL for details.

Remark 1.5.

Bestvina, Bux, and Margalit asked explicitly whether the groups are infinitely generated whenever (cf. Question 8.1 in [BBM07]), and suggested a possible approach towards obtaining the affirmative answer to this question. Though our main result (Theorem 1.1) is exactly the answer to this question, our proof does not follow the approach suggested in [BBM07]. Namely, Bestvina, Bux, and Margalit showed that the term  of the Cartan–Leray spectral sequence for the action of  on  is infinitely generated, provided that , and concluded that if one could prove that this group remains infinitely generated after taking consecutive quotients by the images of the differentials , then he would obtain that is infinitely generated, since injects into . This plan seems to be very hard to realize because it requires computation of higher differentials of the spectral sequence. Instead, we show that every group , where , contains a free Abelian subgroup of infinite rank. This also implies that contains a free Abelian subgroup of infinite rank for , since the groups , , are consecutive quotients for certain filtration in . The crucial point in our approach is that we avoid explicit computation of higher differentials by mapping the Cartan–Leray spectral sequence  to certain auxiliary spectral sequences  in which the corresponding differentials are trivial by dimension reasons.

Kirbi’s list of problems in low-dimensional topology [Kir97] contains the following Problem 2.9(B) attributed to Mess: Given , what is the largest  for which admits a classifying space with finite -skeleton. (This is also Problem 5.11 in [Far06].) The best previously known estimate was obtained by Bestvina, Bux, and Margalit [BBM07]. The following is a direct consequence of Theorem 1.1.

Corollary 1.6.

If admits a classifying space with finite -skeleton, then .

This paper is organized as follows. In Section 2 we describe explicitly an infinite linearly independent system of Abelian cycles in  (Theorem 2.1) and an infinite linearly independent system of homology classes in  for every (Theorem LABEL:theorem_explicit). The rest part of the paper contains the proof that these systems of homology classes are indeed linearly independent. The main tool is the Cartan–Leray spectral sequence for the action of  on the complex of cycles . Section LABEL:section_background contains necessary information on the complex of cycles  and on the Cartan–Leray spectral sequence. In the last three sections we study the Cartan–Leray spectral sequence  for the action of  on . In Section LABEL:section_aux_spectral we construct auxiliary spectral sequences  and  and morphisms of spectral sequences

 E∗∗,∗→E∗∗,∗(N)→ˆE∗∗,∗(N).

This construction is needed to avoid explicit computation of higher differentials of the spectral sequence . In Sections LABEL:section_Abelian_proof and LABEL:section_main_proof we prove Theorems 2.1 and LABEL:theorem_explicit, respectively.

The author is grateful to S. I. Adian and A. L. Talambutsa for fruitful discussions.

2. Systems of linearly independent homology classes

A multicurve on the surface  is a union of a finite number of pairwise disjoint simple closed curves such that no curve  is homotopic to a point and no two curves  and  are homotopic to each other. A multicurve  is said to be oriented if all components of  are endowed with orientations. (Notice that, by definition, an oriented multicurve is not allowed to contain a pair of components that are homotopic to each other with the homotopy either preserving or reversing the orientation.)

Most constructions in the present paper concerning curves and multicurves are determined up to isotopy. For the sake of simplicity, we shall later not distinguish between notation for a curve (or a multicurve) and its isotopy class. For instance, the subgroup of  consisting of all mapping classes  that stabilize the isotopy class of a multicurve  will be denoted by and will be called the stabilizer of the multicurve .

We denote by the homology class of an oriented simple closed curve  in . We denote by the algebraic intersection index of homology classes . We denote by the subgroup of  generated by homology classes . We denote by  the (left) Dehn twist about a simple closed curve .

We choose a primitive element and fix it throughout the whole paper.

2.1. System of linearly independent Abelian cycles

Consider a splitting

 H1(Sg)=U0⊕⋯⊕Ug−1, (2.1)

where every  has rank  and is orthogonal to  with respect to the intersection form unless . Then there is a unique decomposition

 x=x0+⋯+xg−1,xi∈Ui. (2.2)

Let be the set of all splittings of form (2.1) such that all summands  in the corresponding decomposition (2.2) are nonzero. Splittings obtained from each other by permutations of summands are regarded to be different, i. e., a splitting is an ordered -tuple . Obviously, the set  is infinite.

If a splitting belongs to , then every summand in decomposition (2.2) can be uniquely written as , where is primitive and . We denote by the set of all -tuples such that and for every . (Notice that we do not choose elements  and  in and .)

It is easy to see that for each pair  such that and , there exists a -component multicurve , where

 Δ =δ1∪⋯∪δg−1, L =β1∪β′1∪⋯∪βg−2∪β′g−2,

that satisfies the following conditions (see Fig. 1):

• are separating simple closed curves,

• consists of connected components such that

• and  are once-punctured tori adjacent to  and to , respectively,

• for , is a twice-punctured torus adjacent to  and ,

• the image of the homomorphism induced by the inclusion is ,

• and are oriented simple closed curve contained in ,

• ,

• the connected component of  that is adjacent to  lies on the right-hand side from .

It is easy to check that all multicurves satisfying these conditions for the fixed pair  lie in the same -orbit. Therefore the homology class

 AU,b=A(Tβ1T−1β′1,…,Tβg−2T−1β′g−2,Tδ1,…,Tδg−1)∈H2g−3(Ig)

is well defined and independent of the choice of .

For and , we denote by and the tuples obtained from  and  by reversing the orders of ’s and ’s, respectively, i. e., , . It is easy to see that

 A¯¯¯¯U,¯¯¯b=AU,b.
Theorem 2.1.

Let be a subset containing exactly one splitting in every pair . Choose any representatives , where . Then the Abelian cycles , where runs over , are linearly independent in .

Since the set is infinite, Theorem 1.2 follows from Theorem 2.1.

2.2. On the group H3g−2(Ig,1)

Consider an oriented compact surface  of genus  with one boundary component. Let be the mapping class group of , where is the group of all homeomorphisms that are identical on the boundary. Let be the corresponding Torelli group, that is, the kernel of the natural surjective homomorphism .

We shall need the following proposition, which is a reformulation of a result by Bestvina, Bux, and Margalit [BBM07].

Proposition 2.2.

Suppose that . Then and the top homology group  contains a free Abelian subgroup of infinite rank.

Proof.

Bestvina, Bux, and Margalit considered the Torelli group  for the oriented surface of genus  with one puncture rather than one boundary component. (However, they denoted this group by  rather than by .) Their result is precisely as follows: and the top homology group  is infinitely generated, see Theorem C and Lemma 8.8 in [BBM07]. However, in fact, their proof yields a stronger result, namely, that  contains a free Abelian subgroup of infinite rank. Indeed, their proof is by induction on  and is based on the following two facts:

for .

Fact 2.4.

injects into for .

The former fact is contained in the proof of Lemma 8.8 in [BBM07]. The latter fact can be extracted from [BBM07] in the following way. First, by Lemma 8.10 in [BBM07], the group , where is a non-separating simple closed curve on , injects into . Second, in the proof of Theorem C in [BBM07], a torsion free Abelian group  is constructed such that the group injects into . Finally, by the construction, is a subgroup of the first homology of a free group, hence, is free Abelian. Therefore injects into .

By Fact 2.3, . Since is an infinitely generated free group (cf. [Mes92]), we obtain that is a free Abelian group of infinite rank. Using Facts 2.3 and 2.4, we obtain by induction that contains a free Abelian subgroup of infinite rank.

Now, consider the short exact sequence

 1→Z→Ig,1→Ig,∗→1. (2.3)

Here the subgroup  is generated by the Dehn twist about a simple curve isotopic to the boundary of , hence, is central in . Since , the Hochschild–Serre spectral sequence for central extention (2.3) yields the natural isomorphism , which completes the proof of the proposition. ∎

Remark 2.5.

The above proof also yields that the group , where , contains a free Abelian subgroup of infinite rank.

In the sequel, we shall need the following proposition, which is equivalent to Fact 2.4.

Proposition 2.6.

Let be a non-separating simple closed curve on , where . Then any embedding induces an injection .

Proof.

We have the commutative diagram with exact rows:

 1−−−−→Z−−−−→Ig−1,1−−−−→Ig−1,∗−−−−→1i⏐⏐↓φ∗⏐⏐↓∥∥1−−−−→K−−−−→StabIg(v)−−−−→Ig−1,∗−−−−→1 (2.4)

Here the first row is exact sequence (2.3), the second row is the restriction of the Birman exact sequence obtained in [BBM07, Lemma 8.3], and is the commutator subgroup of , where is a once-punctured surface of genus  obtained from  by forgetting one of the two punctures. Obviously, is a free group. The group  in the first row is generated by the Dehn twist , where is a simple curve isotopic to the boundary of . The element  represents a non-trivial homology class in , see [BBM07, Lemma 8.6]. Since is torsion-free, we obtain that the homomorphism is an injection. This injection induces the homomorphism

 i∗:H3g−6(Ig−1,∗)=H3g−6(Ig−1,∗,H1(Z))→H3g−6(Ig−1,∗,H1(K)), (2.5)

where is endowed with the structure of an -module corresponding to the second row of (2.4). (Since lies in the centre of , the action of  on the coefficient group is trivial.) It is proved in [BBM07, proof of Theorem C] that (2.5) is an injection. (Notice that the submodule  that is constructed in [BBM07, Lemma 8.7] and used in the proof of Theorem C in [BBM07] is exactly .)

Finally, consider the Hochschild–Serre spectral sequences for the rows of (2.4). Since and , these spectral sequences yield the commutative diagram

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