Homotopy invariants of covers and KKM type lemmas

TL;DR

This paper introduces homotopy invariants associated with covers of spaces, using them to generalize KKM and Sperner lemmas through homotopy obstructions, especially relating to spheres and disks.

Contribution

It develops a new homotopy-theoretic framework for covers, providing generalized KKM and Sperner lemmas based on homotopy invariants and obstructions.

Findings

Homotopy invariants serve as obstructions for cover extensions.

Generalized KKM lemmas depend on the non-vanishing of certain homotopy groups.

Existence of KKM-type lemmas relates to the non-triviality of the k-homotopy group of spheres.

Abstract

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.