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.~Bödigheimer 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 $g\ge 3$, 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 $S{p}_6(\mathbf{Z}),Spi{n}_8(\mathbf{Z}),Spi{n}_7(\mathbf{Z}),$ and $F_4(\mathbf{Z})$.} Math. Ann. 215 (1975), 173--193. ), detects the undecided ${\mathbf{Z}}_2$ factor in $H_2({\mathbf{M}}_3;\mathbf{Z})$.

4 pages, 2 figures, some typos corrected, v3. Reference to previous work by Sakasai proving the result by different methods added, See the Abstract

References in corpus (1)

• An elementary approach to the mapping class group of a surface

Cited by in corpus (1)

• A twisted first homology group of the handlebody mapping class group