Steve Ferry, Rutgers University

Higson's Compactification and the Novikov Conjecture

In 1990, Nigel Higson introduced a very general compactification, reminiscent of the Stone-Cech compactification, for locally compact metric spaces. By 1993, it was known that if the Higson compactification of the universal cover of a closed aspherical manifold is acyclic, then the rational Novikov conjecture follows for the fundamental group of that manifold.

Unfortunately, Dranishnikov and Ferry showed that even for euclidean space, the rational cohomology of the Higson compactification is enormous. In recent joint work with Dranishnikov and Weinberger, we have shown that the rational Novikov conjecture also follows from the mod 2 acyclicity of the Higson compactification. We have also shown that the Higson compactification of EG is mod p acyclic for all groups G such that BG is a finite complex and EG is asymptotically finite dimensional. This shows that Higson's compactification can be used to recover an integral Novikov result of Guoliang Yu.