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On spectral sequence for the action of genus 3 Torelli group on the complex of cycles

Alexander A. Gaifullin

Abstract

The Torelli group of a genus $g$ oriented surface $S_g$ is the subgroup $\mathcal{I}_g$ of the mapping class group $\mathrm{Mod}(S_g)$ consisting of all mapping classes that act trivially on the homology of $S_g$. One of the most intriguing open problems concerning Torelli groups is the question of whether the group $\mathcal{I}_3$ is finitely presented or not. A possible approach to this problem relies upon the study of the second homology group of $\mathcal{I}_3$ using the spectral sequence $E^r_{p,q}$ for the action of $\mathcal{I}_3$ on the complex of cycles. In this paper we obtain a partial result towards the conjecture that $H_2(\mathcal{I}_3;\mathbb{Z})$ is not finitely generated and hence $\mathcal{I}_3$ is not finitely presented. Namely, we prove that the term $E^3_{0,2}$ of the spectral sequence is infinitely generated, that is, the group $E^1_{0,2}$ remains infinitely generated after taking quotients by images of the differentials $d^1$ and $d^2$. If one proceeded with the proof that it also remains infinitely generated after taking quotient by the image of $d^3$, he would complete the proof of the fact that $\mathcal{I}_3$ is not finitely presented.

On spectral sequence for the action of genus 3 Torelli group on the complex of cycles

Abstract

The Torelli group of a genus oriented surface is the subgroup of the mapping class group consisting of all mapping classes that act trivially on the homology of . One of the most intriguing open problems concerning Torelli groups is the question of whether the group is finitely presented or not. A possible approach to this problem relies upon the study of the second homology group of using the spectral sequence for the action of on the complex of cycles. In this paper we obtain a partial result towards the conjecture that is not finitely generated and hence is not finitely presented. Namely, we prove that the term of the spectral sequence is infinitely generated, that is, the group remains infinitely generated after taking quotients by images of the differentials and . If one proceeded with the proof that it also remains infinitely generated after taking quotient by the image of , he would complete the proof of the fact that is not finitely presented.

Paper Structure

This paper contains 25 sections, 56 theorems, 204 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Scheme of the proof of Theorem \ref{['theorem_main']}
  3. Preliminaries
  4. Complex of cycles
  5. Spectral sequence
  6. Stabilizers of multicurves
  7. Birman--Craggs homomorphisms
  8. Fundamental exact sequence.
  9. Cohomology classes $\theta_{\mathbf{A}}$ and homomorphisms $\vartheta_A$
  10. Cells of $\mathcal{B}$ and their orientations
  11. Homomorphisms $\nu_{\gamma}$, $\nu_{\gamma,\mathcal{W}}$, and $\mu_{\gamma,\gamma'}$
  12. Stabilizers of four- and five-component multicurves without separating components and bounding pairs
  13. Type 1 four-component multicurves
  14. Five-component multicurves
  15. Type 2 four-component multicurves
  16. ...and 10 more sections

Key Result

Theorem \oldthetheorem

Suppose that $E^r_{p,q}$ is spectral sequence eq_SpSeqIntro for the action of $\mathcal{I}_3$ on $\mathcal{B}_3$. Then the group $E^3_{0,2}$ is not finitely generated.

Figures (10)

  • Figure 1: Lantern relation
  • Figure 2: Curves $\alpha_1,\ldots,\alpha_g,\beta_1,\ldots,\beta_g$ and hyperelliptic involution $\iota$
  • Figure 3: Abelian cycle $\mathcal{A}(T_{\delta_1},T_{\delta_2})$
  • Figure 4: Multicurves in the sets (a) $\mathcal{M}_0'$, (b) $\mathcal{M}_1^{(1)}$, (c) $\mathcal{M}_1^{(2)}$, and (d) $\mathcal{M}_2'$
  • Figure 5: Type 1 multicurve
  • ...and 5 more figures

Theorems & Definitions (107)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem: Putman Put07
  • ...and 97 more