Loading…
On spectral sequence for the action of genus 3 Torelli group on the complex of cycles
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 qu...
Saved in:
Published in: | arXiv.org 2021-02 |
---|---|
Main Author: | |
Format: | Article |
Language: | English |
Subjects: | |
Online Access: | Get full text |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Summary: | 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. |
---|---|
ISSN: | 2331-8422 |
DOI: | 10.48550/arxiv.2011.00295 |