Lusternik–Schnirelmann category of spaces with free fundamental group

Abstract

We find upper bounds for the Lusternik–Schnirelmann category of spaces with free fundamental groups. We prove, in particular, that if $X$ is a finite-dimensional CW complex with a free fundamental group, then $cat\left(X\right)\le \frac{2}{3}dim\left(X\right)+1$. Our results specialize to resolve a question about the L–S category of manifolds posed by Dranishnikov, Katz and Rudyak.