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

Dranishnikov, Alexander

doi:10.2140/agt.2019.19.3601

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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

DOI: 10.2140/agt.2019.19.3601

Abstract

We prove that

{cat}_{LS} X \leq \frac{1}{2} (cd (π_{1} (X)) + dim X)

for every CW complex $X$ , where $cd (π_{1} (X))$ denotes the cohomological dimension of the fundamental group of $X$ . We obtain this as a corollary of the inequality

{cat}_{LS} X \leq \frac{1}{2} ({cat}_{LS} (u_{X}) + dim X),

where $u_{X} : X \to B π_{1} (X)$ is a classifying map for the universal covering of $X$ .

Keywords

Lusternik–Schnirelmann category, cohomological dimension of groups

Mathematical Subject Classification 2010

Primary: 55M30

References

Publication

Received: 12 June 2018

Revised: 18 February 2019

Accepted: 13 April 2019

Published: 17 December 2019

Authors

Alexander Dranishnikov
Department of Mathematics University of Florida Gainesville, FL United States

Volume 19, issue 7 (2019)