  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 $SL_n$.
