Lagrangian mapping class groups from a group homological point of view
arXiv:0910.5262 · 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 by-product of this investigation, we determine the second homology of the mapping class group of a surface of genus 3.
20 pages. The proof of Lemma 4.2 is corrected. To appear in Algebraic & Geometric Topology