  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 $H_2(\mathbf{M}_3;\mathbf{Z})$.
