Lagrangian mapping class groups from a group homological point of view

Sakasai, Takuya

doi:10.2140/agt.2012.12.267

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Lagrangian mapping class groups from a group homological point of view

Takuya Sakasai

Algebraic & Geometric Topology 12 (2012) 267–291

DOI: 10.2140/agt.2012.12.267

Abstract

We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-type invariants of 3–manifolds. In this paper, we discuss these groups from a group (co)homological point of view. The results include the determination of their abelianizations, lower bounds of the second homology and remarks on the (co)homology of higher degrees. As a byproduct of this investigation, we determine the second homology of the mapping class group of a surface of genus $3$ .

Keywords

mapping class group, Torelli group, Lagrangian filtration, Miller–Morita–Mumford class

Mathematical Subject Classification 2010

Primary: 55R40

Secondary: 32G15, 57R20

References

Publication

Received: 20 November 2010

Revised: 1 November 2011

Accepted: 10 November 2011

Published: 25 February 2012

Authors

Takuya Sakasai
Department of Mathematics Tokyo Institute of Technology 2-12-1 Oh-okayama Meguro-ku 152-8551 Tokyo Japan

Volume 12, issue 1 (2012)