The 2-torsion in the second homology of the genus $3$ mapping class group

Abstract

This work is NOT to be used as reference. First, because as C.F.~B\"odigheimer and M.~Korkmaz pointed to us the computation of the $\mathbf{Z}_2$ factor that remained undecided in M.~Korkmaz and A. Stipsicz, {\em The second homology groups of mapping class groups of orientable surfaces.} Math. Proc. Camb. Phil. Soc., was shown to exist by Skasai, see hi Theorem 4.9 and Corollary 4.10 in {\em Lagrangian mapping class groups from a group homological point of view.} Algebr. Geom. Topol. 12 (2012), no. 1, 267--291. Second, because one could obtain this result by gathering old results in the literature, first by noticing as Korkmaz kindly reminded me, that D.~Johnson, in \emph{Homeomorphisms of a surface which act trivially on homology} Porc. AMS Volume 75, Number 1, 1979. proved that the quotient of the Torelli group $\mathcal{T}_g/[\mathcal{T}_g,\mathcal{M}_g]$ is trivial for $g\geq 3$, the five term exact sequence then implies that the $\mathbf{Z}_2$ factor in Stein's computation of $H_2(Sp(6,\mathbf{Z});\mathbf{Z}) = \mathbf{Z}\oplus\mathbf{Z}_2$ (see his {\em The Schur Multipliers of $Sp_6(\mathbf{Z}), Spin_8(\mathbf{Z}), Spin_7(\mathbf{Z}),$ and $F_4(\mathbf{Z})$.} Math. Ann. 215 (1975), 173--193. ), detects the undecided $\mathbf{Z}_2$ factor in $H_2(\mathbf{M}_3;\mathbf{Z})$.