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This work is NOT to be used as reference. First, because asC.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 orientablesurfaces.} Math. Proc. Camb. Phil. Soc., was shown to exist by Skasai, see hiTheorem 4.9 and Corollary 4.10 in {\em Lagrangian mapping class groups from agroup homological point of view.} Algebr. Geom. Topol. 12 (2012), no. 1,267--291. Second, because one could obtain this result by gathering old resultsin the literature, first by noticing as Korkmaz kindly reminded me, thatD.~Johnson, in \emph{Homeomorphisms of a surface which act trivially onhomology} Porc. AMS Volume 75, Number 1, 1979. proved that the quotient of theTorelli 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 $H_2(\mathbf{M}_3;\mathbf{Z})$.

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