Stable twisted cohomology of the mapping class groups in the unit tangent bundle homology

TL;DR

This paper computes the stable twisted cohomology of mapping class groups with coefficients from the unit tangent bundle's homology, revealing complex module structures beyond traditional stability frameworks.

Contribution

It introduces the computation of stable twisted cohomology with non-traditional covariant coefficients, expanding understanding of mapping class group cohomology.

Findings

Stable twisted cohomology is not free over the constant coefficient algebra.

The twisted coefficients define a covariant functor, unlike traditional contravariant frameworks.

Comparison with classical cohomology shows different structural properties.

Abstract

We compute the stable cohomology groups of the mapping class groups of compact orientable surfaces with one boundary, with twisted coefficients given by the homology of the unit tangent bundle of the surface. This stable twisted cohomology is not free as a module over the stable cohomology algebra with constant coefficients. In fact, it is out of the scope of the traditional framework for twisted cohomological stability, since these twisted coefficients define a covariant functor over the classical category associated to mapping class groups to study homological stability, rather than a contravariant one. For comparison, we also compute the stable cohomology group with coefficients in the first cohomology of the unit tangent bundle of the surface, which fits into the traditional framework.