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.
"Homotopy invariants of covers and KKM-type lemmas." Algebr. Geom. Topol. 16 (3) 1799 - 1812, 2016. https://doi.org/10.2140/agt.2016.16.1799