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

TL;DR

This paper establishes an upper bound on the Lusternik-Schnirelmann category of a CW complex based on the cohomological dimension of its fundamental group and the dimension of the space, linking algebraic and topological invariants.

Contribution

It introduces a new upper bound for the LS-category involving the fundamental group's cohomological dimension, extending previous inequalities.

Findings

Proves $ ext{cat} X \u2264 ( ext{cd}(\pi_1(X)) + ext{dim} X)/2$.

Derives the inequality from a more general bound involving the classifying map.

Connects algebraic properties of the fundamental group with topological complexity.

Abstract

We prove that $$ \cat X\le \frac{\cd(\pi_1(X))+\dim X}{2}$$ for every CW complex $X$ where $\cd(\pi_1(X))$ denotes the cohomological dimension of the fundamental group of $X$. We obtain this as a corollary of the inequality $$ \cat X\le\frac{\cat (u_X)+\dim X}{2} $$ where $u_X:X\to B\pi_1(X)$ is a classifying map for the universal covering of $X$.