The Topological Trace Formula

The topological trace formula is a computation of the Lefschetz number of a Hecke correspondence C acting on the weighted cohomology groups, defined in [GHM], of a locally symmetric space X . It expresses this Lefschetz number as a sum of contributions from fixed point components of C on the reductive Borel Serre compactification of X . The proof uses the Lefschetz fixed point formula of [GM2]. When the L cohomology of X is finite dimensional (the equal rank case), the Lefschetz number of a Hecke correspondence acting on the L cohomology was computed by J. Arthur in [Ar1, Ar3] using the trace formula. In this cased the L cohomology coincides with the “middle” weighted cohomology (see [GHM]), so this paper gives an independent computation of this Lefschetz number. In [GKM], it was shown that these two computations agree. Consequently, this paper completes an independent proof of Arthur’s formula. We also describe some of the rich geometry of Hecke correspondences and their fixed points. While [Ar1] and [GKM], because of their adelic setting, treated congruence subgroups, we treat here the slightly more general case of an arbitrary neat arithmetic subgroup. Some of the results of this paper were announced in [GM1].