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2016 Homotopy invariants of covers and KKM-type lemmas
Oleg Musin
Algebr. Geom. Topol. 16(3): 1799-1812 (2016). 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 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.

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Oleg Musin. "Homotopy invariants of covers and KKM-type lemmas." Algebr. Geom. Topol. 16 (3) 1799 - 1812, 2016. https://doi.org/10.2140/agt.2016.16.1799

Information

Received: 21 June 2015; Revised: 7 September 2015; Accepted: 22 September 2015; Published: 2016
First available in Project Euclid: 28 November 2017

zbMATH: 1350.55006
MathSciNet: MR3523055
Digital Object Identifier: 10.2140/agt.2016.16.1799

Subjects:
Primary: ‎55M20, 55M25
Secondary: 55P05

Rights: Copyright © 2016 Mathematical Sciences Publishers

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Vol.16 • No. 3 • 2016
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