Lagrangian mapping class groups from a group homological point of view

Sakasai, Takuya¹

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

Algebraic and Geometric Topology, Tome 12 (2012) no. 1, pp. 267-291

Résumé

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.

Détail

DOI : 10.2140/agt.2012.12.267

Classification : 55R40, 32G15, 57R20
Keywords: mapping class group, Torelli group, Lagrangian filtration, Miller–Morita–Mumford class

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