Torsion at the Threshold for Mapping Class Groups

The mapping class group \({\Gamma}_g^ 1\) of a closed orientable surface of genus \(g \geq 1\) with one marked point can be identified, by the Nielsen action, with a subgroup of the group of orientation preserving homeomorphims of the circle. This inclusion pulls back the powers of the discrete univ...

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Bibliographic Details
Published in:arXiv.org 2024-09
Main Authors: Jekel, Solomon, Rita Jiménez Rolland
Format: Article
Language:English
Subjects:
Mapping
Subgroups
Online Access:Get full text
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Summary:The mapping class group \({\Gamma}_g^ 1\) of a closed orientable surface of genus \(g \geq 1\) with one marked point can be identified, by the Nielsen action, with a subgroup of the group of orientation preserving homeomorphims of the circle. This inclusion pulls back the powers of the discrete universal Euler class producing classes \(\text{E}^n \in H^{2n}({\Gamma}_g^1;\mathbb{Z})\) for all \(n\geq 1\). In this paper we study the power \(n=g,\) and prove: \(\text{E}^g\) is a torsion class which generates a cyclic subgroup of \(H^{2g}({\Gamma}_g^1;\mathbb{Z})\) whose order is a positive integer multiple of \(4g(2g+1)(2g-1)\).
ISSN:2331-8422