Volume 19, issue 7 (2019)

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

Alexander Dranishnikov

Algebraic & Geometric Topology 19 (2019) 3601–3614
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 $X\phantom{\rule{0.3em}{0ex}}$, where $cd\left({\pi }_{1}\left(X\right)\right)$ denotes the cohomological dimension of the fundamental group of $X\phantom{\rule{0.3em}{0ex}}$. 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 $X\phantom{\rule{0.3em}{0ex}}$.

Keywords
Lusternik–Schnirelmann category, cohomological dimension of groups
Primary: 55M30