Volume 12, issue 1 (2012)

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Lagrangian mapping class groups from a group homological point of view

Takuya Sakasai

Algebraic & Geometric Topology 12 (2012) 267–291

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.

mapping class group, Torelli group, Lagrangian filtration, Miller–Morita–Mumford class
Mathematical Subject Classification 2010
Primary: 55R40
Secondary: 32G15, 57R20
Received: 20 November 2010
Revised: 1 November 2011
Accepted: 10 November 2011
Published: 25 February 2012
Takuya Sakasai
Department of Mathematics
Tokyo Institute of Technology
2-12-1 Oh-okayama
152-8551 Tokyo