#### Volume 16, issue 3 (2016)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Homotopy invariants of covers and KKM-type lemmas

### Oleg R Musin

Algebraic & Geometric Topology 16 (2016) 1799–1812
##### Abstract

Given any (open or closed) cover of a space $T\phantom{\rule{0.3em}{0ex}}$, 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 $A\subset X$ 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 $\left(k+1\right)$–disk there exist KKM-type lemmas for covers by $n+2$ sets if and only if the homotopy group ${\pi }_{k}\left({\mathbb{S}}^{n}\right)$ is nontrivial.

##### Keywords
KKM lemma, Sperner lemma, homotopy class, degree of mappings
##### Mathematical Subject Classification 2010
Primary: 55M20, 55M25
Secondary: 55P05