Borsuk-Ulam theorem for filtered spaces

Abstract

Let X and Y be pathwise connected and paracompact Hausdorff spaces equipped with free involutions T:X→X and S:Y→Y, respectively. Suppose that there exists a sequence (Xi,Ti)→ hi (Xi+1,Ti+1) for 1≤i≤k, where, for each i, Xi is a pathwise connected and paracompact Hausdorff space equipped with a free involution Ti, such that Xk+1=X, and hi:Xi→Xi+1 is an equivariant map, for all 1≤i≤k. To achieve Borsuk-Ulam-type theorems, in several results that appear in the literature, the involved spaces X in the statements are assumed to be cohomological n-acyclic spaces. In this paper, by considering a more wide class of topological spaces X (which are not necessarily cohomological n-acyclic spaces), we prove that there is no equivariant map f:(X,T)→(Y,S) and we present some interesting examples to illustrate our results.