Homotopy invariants of covers and KKM-type lemmas

Musin, Oleg¹

¹Department 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

Algebraic and Geometric Topology, Tome 16 (2016) no. 3, pp. 1799-1812

Résumé

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.

Détail

DOI : 10.2140/agt.2016.16.1799

Classification : 55M20, 55M25, 55P05
Keywords: KKM lemma, Sperner lemma, homotopy class, degree of mappings

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