On the structure of the top homology group of the Johnson kernel

The Johnson kernel is the subgroup \(\mathcal{K}_g\) of the mapping class group \({\rm Mod}(\Sigma_{g})\) of a genus \(g\) oriented closed surface \(\Sigma_{g}\) generated by all Dehn twists about separating curves. In this paper we study the structure of the top homology group \({\rm H}_{2g-3}(\mat...

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Bibliographic Details
Published in:arXiv.org 2023-04
Main Author: Spiridonov, Igor A
Format: Article
Language:English
Subjects:
Homology
Kernels
Subgroups
Online Access:Get full text
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Summary:The Johnson kernel is the subgroup \(\mathcal{K}_g\) of the mapping class group \({\rm Mod}(\Sigma_{g})\) of a genus \(g\) oriented closed surface \(\Sigma_{g}\) generated by all Dehn twists about separating curves. In this paper we study the structure of the top homology group \({\rm H}_{2g-3}(\mathcal{K}_g, \mathbb{Z})\). For any collection of \(2g-3\) disjoint separating curves on \(\Sigma_{g}\) one can construct the corresponding abelian cycle in the group \({\rm H}_{2g-3}(\mathcal{K}_g, \mathbb{Z})\); such abelian cycles will be called simplest. In this paper we describe the structure of \(\mathbb{Z}[{\rm Mod}(\Sigma_{g})/ \mathcal{K}_g]\)-module on the subgroup of \({\rm H}_{2g-3}(\mathcal{K}_g, \mathbb{Z})\) generated by all simplest abelian cycles and find all relations between them.
ISSN:2331-8422