Stable twisted cohomology of the mapping class groups in the exterior powers of the unit tangent bundle homology (2311.01791v1)

Abstract: We study the stable cohomology groups of the mapping class groups of surfaces with twisted coefficients given by the $d^{th}$ exterior powers of the first rational homology of the unit tangent bundles of the surfaces $\tilde{H}{\mathbb{Q}}$. These coefficients are outside of the traditional framework of cohomological stability. They form a module $H{\mathrm{st}}^{*}(\Lambda^{d}\tilde{H}_{\mathbb{Q}})$ over the stable cohomology algebra of the mapping class groups with trivial coefficients denoted by $\mathrm{Sym}{\mathbb{Q}}(\mathcal{E})$. If $d\neq 2$, the $\mathrm{Tor}$-group in each degree of $H{\mathrm{st}}^{*}(\Lambda^{d}\tilde{H}_{\mathbb{Q}})$ does not vanish, and we compute all these $\mathrm{Tor}$-groups explicitly for $d \leq 5$. In particular, for each $d\neq 2$, the module $H_{\mathrm{st}}^{*}(\Lambda^{d}\tilde{H}_{\mathbb{Q}})$ is not free over $\mathrm{Sym}_{\mathbb{Q}}(\mathcal{E})$, while it is free for $d=2$.For comparison, we also compute the stable cohomology group with coefficients in the $d^{th}$ exterior powers of the first rational cohomology of the unit tangent bundle of the surface, which fit into the classical framework of cohomological stability.