Cohomology at Infinity and the Well-Tempered Complex

TL;DR

This paper introduces a new framework for computing the action of Hecke operators on the cohomology of arithmetic groups using the well-tempered complex, extending previous boundary and retract results.

Contribution

It generalizes existing cohomology results to the well-tempered complex, providing a finite method for Hecke operator computations on arithmetic groups.

Findings

Established a sequence of commutative diagrams for the well-tempered complex.

Provided a finite computational method for Hecke operators on equivariant cohomology.

Extended cohomology techniques to new complex structures.

Abstract

We prove the existence of a sequence of commutative diagrams generalizing existing results on the cohomology of the Borel-Serre boundary and well-rounded retract to the context of the well-tempered complex. Our main theorem provides a method for computing in finite terms the action of Hecke operators on the equivariant cohomology of an arithmetic subgroup $\Gamma$ of the special linear group $SL_n$.