Homotopy invariants of covers and KKM type lemmas

With any (open or closed) cover of a space T we associate certain homotopy classes of maps T into n-spheres. These homotopy invariants can be considered as obstructions for extensions of covers of a subspace A to a space X. We using these obstructions for 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 k-homotopy group of n-sphere is not zero.