Algebraic & Geometric Topology

Homotopy invariants of covers and KKM-type lemmas

Oleg Musin

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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.

Article information

Algebr. Geom. Topol., Volume 16, Number 3 (2016), 1799-1812.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M20: Fixed points and coincidences [See also 54H25] 55M25: Degree, winding number
Secondary: 55P05: Homotopy extension properties, cofibrations

KKM lemma Sperner lemma homotopy class degree of mappings


Musin, Oleg. Homotopy invariants of covers and KKM-type lemmas. Algebr. Geom. Topol. 16 (2016), no. 3, 1799--1812. doi:10.2140/agt.2016.16.1799.

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