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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...

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Bibliographic Details
Published in:arXiv.org 2021-02
Main Author: Gaifullin, Alexander A
Format: Article
Language:English
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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