Volume 16, issue 3 (2016)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Homotopy invariants of covers and KKM-type lemmas

Oleg R Musin

Algebraic & Geometric Topology 16 (2016) 1799–1812
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 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 πk(Sn) 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