Homotopy invariants of covers and KKM-type lemmas

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 $A\subset X$ 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.