Uniform Versions of Index for Uniform Spaces with Free Involutions

TL;DR

This paper introduces uniform versions of index for uniform spaces with free involutions, extending classical concepts and exploring their properties, examples, and connections to coloring within the uniform setting.

Contribution

It develops uniform versions of the index for spaces with free involutions, linking them to existing indices, examples, and coloring concepts, advancing the theory in uniform spaces.

Findings

Examples of spaces with finite B-index but infinite uniform index.

Dense T-invariant subspaces determine the uniform index.

Connections established between uniform coloring and uniform index.

Abstract

In this paper, uniform versions of index for uniform spaces equipped with free involutions are introduced. They are mainly based on B-index defined and studied by C.-T. Yang in 1955, index studied by Conner and Floyd in 1960 and further development well collected by Matou$\check{s}$ek in his book on using the Borsuk-Ulam theorem in 2003. Examples of uniform spaces with finite B-index but infinite uniform version of index are given. It is also seen that for a uniform space $X$ with a free involution $T$, a dense $T$-invariant subspace is capable of determining the uniform version of index of $(X,T)$. In the end, the concept of coloring is carried over to uniform set up and, to a certain extent, connection between uniform versions of coloring and uniform versions of index is also established.