The Lusternik–Schnirelmann category and the fundamental group

Abstract

We prove that

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

for every CW–complex $X$ where $cd\left({\pi }_1\left(X\right)\right)$ denotes the cohomological dimension of the fundamental group of $X$.