An upper bound on the LS category in presence of the fundamental group

Abstract

We prove that

$${cat}_{LS}X\le \frac{1}{2}\left(cd\left({\pi }_1\left(X\right)\right)+dimX\right)$$

for every CW complex $X$, where $cd\left({\pi }_1\left(X\right)\right)$ denotes the cohomological dimension of the fundamental group of $X$. We obtain this as a corollary of the inequality

$${cat}_{LS}X\le \frac{1}{2}\left({cat}_{LS}\left(u_X\right)+dimX\right),$$

where $u_X:X\to B{\pi }_1\left(X\right)$ is a classifying map for the universal covering of $X$.