Topological fundamental groups can distinguish spaces with isomorphic homotopy groups

Paul Fabel

Published 2005-02-18, updated 2005-04-19Version 2

We exhibit a map f between aspherical spaces X and Y such that f induces an isomorphism on homotopy groups but, with natural topologies, X and Y fail to have homeomorphic fundamental groups. Thus the topological fundamental group has the capacity to distinguish homotopy type when the Whitehead theorem fails.