Volume 16, issue 3 (2016)

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Homotopy invariants of covers and KKM-type lemmas

Oleg R Musin

Algebraic & Geometric Topology 16 (2016) 1799–1812

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.

KKM lemma, Sperner lemma, homotopy class, degree of mappings
Mathematical Subject Classification 2010
Primary: 55M20, 55M25
Secondary: 55P05
Received: 21 June 2015
Revised: 7 September 2015
Accepted: 22 September 2015
Published: 1 July 2016
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
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Moscow 127994