Cohomology at Infinity and the Well-Tempered Complex

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 $\mathrm{\Gamma }$ of the special linear group $S{L}_n$.