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

TL;DR

This paper investigates the 2-torsion component in the second homology group of the genus 3 mapping class group, clarifying its structure and relation to existing results in algebraic topology.

Contribution

It provides a detailed analysis of the 2-torsion in H_2 of the genus 3 mapping class group, connecting it with prior work and existing algebraic structures.

Findings

Identifies the Z_2 torsion in H_2 of the genus 3 mapping class group.

Clarifies the relationship between the second homology and known algebraic invariants.

Links the torsion component to existing results on symplectic groups and Torelli groups.

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