# The Lusternik-Schniremann-category and the fundamental group

## TL;DR

This paper establishes an upper bound on the Lusternik-Schniremann category of a CW complex based on the cohomological dimension of its fundamental group and its dimension.

## Contribution

It provides a new inequality linking the Lusternik-Schniremann category with fundamental group cohomological dimension and the complex's dimension.

## Key findings

- Proved that at X d(\u03c0_1(X)) + igg\u2191(rac{ ext{dim} X - 1}{2})or CW complexes.
- Connected algebraic properties of fundamental groups with topological complexity.
- Extended understanding of Lusternik-Schniremann category bounds in algebraic topology.

## Abstract

We prove that $$ \cat X\le cd(\pi_1(X))+\bigg\lceil\frac{\dim X-1}{2}\bigg\rceil$$ for every CW complex $X$ where $cd(\pi_1(X))$ denotes the cohomological dimension of the fundamental group of $X$.