• home

Abstract: Farrell and Jones proved that the automorphism groups of compact hyperbolic manifolds in dimensions n≥1 do not have the homotopy-type of finite CW complexes. Here "automorphism group" means groups of diffeomorphisms, PL-homeomorphisms, and homeomorphisms respectively, i.e. we are not discussing homotopy-equivalences or isometries. Farrell and Jones primarily used the machinery of "Higher Simple Homotopy Theory", which reduces the problem of computing homotopy groups of automorphism groups (in a range) to that of the group of automorphisms of S^1 x D^{n-1}, which was worked-out by Hatcher and Wagoner, plus a good amount of K-theory. With David Gabai we found an alternate route to such theorems, that avoids Higher Simple Homotopy Theory entirely. For us the results occur in dimensions n≥4. I will describe how these kinds of arguments work.