On a conjecture of Gottlieb

We give a counterexample to a conjecture of D H Gottlieb and prove a strengthened version of it. The conjecture says that a map from a finite CW–complex X to an aspherical CW–complex Y with non-zero Euler characteristic can have non-trivial degree (suitably defined) only if the centralizer of the image of the fundamental group of X is trivial. As a corollary we show that in the above situation all components of non-zero degree maps in the space of maps from X to Y are contractible. We use L 2 –Betti numbers and homological algebra over von Neumann algebras to prove the modified conjecture.