  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.
