Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 16, Number 3 (2016), 1799-1812.
Homotopy invariants of covers and KKM-type lemmas
Given any (open or closed) cover of a space , we associate certain homotopy classes of maps from to –spheres. These homotopy invariants can then be considered as obstructions for extending covers of a subspace to a cover of all of . 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 is a –sphere and is a –disk there exist KKM-type lemmas for covers by sets if and only if the homotopy group is nontrivial.
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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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. https://projecteuclid.org/euclid.agt/1511895862