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

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 ${\mathcal{T}}_g\mathrm{/}[{\mathcal{T}}_g,{\mathcal{M}}_g]$ 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 ofSp_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$factorin$H_2($\mathbf{M}$_3;$\mathbf{Z}$)$.