On a conjecture of Gottlieb

Abstract

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.