Lagrangian mapping class groups from a group homological point of view (Q422096)

The author studies certain infinite index subgroups of the mapping class group associated to Lagrangian submodules of the first homology group of a surface. The point of view is that of group cohomology. Let \(H = H_1(\Sigma)\) be the first homology group of a closed surface. Let \(L\) be a Lagrangian submodule of \(H\) with respect to the intersection bilinear form. Let NEWLINE\[NEWLINE{\mathcal L}_g = \{ f \in {\mathcal M}_g : f_\ast (L)= L \}.NEWLINE\]NEWLINE NEWLINE\[NEWLINE{\mathcal{I L}}_g = \{ f \in {\mathcal M}_g : f_\ast|_L= id_L \}.NEWLINE\]NEWLINENEWLINENEWLINEIn the paper under review the author computes the following:NEWLINENEWLINE1) \(H_1({\mathcal{I L}}_g)\)NEWLINENEWLINE2) \(H_1({\mathcal{L}}_g/{\mathcal{I}}_g)\)NEWLINENEWLINE3) \(H_2({\mathcal{L}}_g/{\mathcal{I}}_g)\)NEWLINENEWLINE4) \(H_1({\mathcal{L}}_g)\) NEWLINENEWLINE5) a lower bound on \(H_2({\mathcal{L}}_g)\).