Homotopy invariants of covers and KKM-type lemmas

Musin, Oleg

doi:10.2140/agt.2016.16.1799

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Homotopy invariants of covers and KKM-type lemmas

Oleg R Musin

Algebraic & Geometric Topology 16 (2016) 1799–1812

DOI: 10.2140/agt.2016.16.1799

Abstract

Given any (open or closed) cover of a space $T$ , we associate certain homotopy classes of maps from $T$ to $n$ –spheres. These homotopy invariants can then be considered as obstructions for extending covers of a subspace $A \subset X$ to a cover of all of $X$ . We use these obstructions to obtain generalizations of the classic KKM (Knaster–Kuratowski–Mazurkiewicz) and Sperner lemmas. In particular, we show that in the case when $A$ is a $k$ –sphere and $X$ is a $(k + 1)$ –disk there exist KKM-type lemmas for covers by $n + 2$ sets if and only if the homotopy group $π_{k} (S^{n})$ is nontrivial.

Keywords

KKM lemma, Sperner lemma, homotopy class, degree of mappings

Mathematical Subject Classification 2010

Primary: 55M20, 55M25

Secondary: 55P05

References

Publication

Received: 21 June 2015

Revised: 7 September 2015

Accepted: 22 September 2015

Published: 1 July 2016

Authors

Oleg R Musin
Department of Mathematics University of Texas Rio Grande Valley One West University Boulevard Brownsville, TX 78520 United States	Institute for Information Transmission Problems Russian Academy of Sciences Bolshoy Karetny per. 19 Moscow 127994 Russia

Volume 16, issue 3 (2016)