Partial Torelli groups and homological stability
Andrew Putman
TL;DR
This paper establishes homological stability results for certain subgroups of the mapping class group, extending known theorems to new contexts involving fixed homology data and finite group actions.
Contribution
It introduces new homological stability theorems for subgroups of the mapping class group related to fixed homology and finite group maps, generalizing prior work on braid groups.
Findings
Proves stability for subgroups fixing part of the homology.
Establishes stability for subgroups preserving a map to a finite group.
Uses complexes of subsurfaces to generalize previous results.
Abstract
We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping class group preserving a fixed map from the fundamental group to a finite group, which can be viewed as a mapping class group version of a theorem of Ellenberg-Venkatesh-Westerland about braid groups. These results require studying various simplicial complexes formed by subsurfaces of the surface, generalizing work of Hatcher-Vogtmann.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models