On generalization of Homotopy Axiom

TL;DR

This paper investigates the limitations of the homotopy axiom in Alexander-Spanier-Kolmogoroff cohomology, demonstrating that the parameter space cannot generally be replaced by locally compact connected spaces.

Contribution

It extends previous results by showing the homotopy axiom's parameter space restriction does not hold for all locally compact connected spaces.

Findings

Parameter space $T$ cannot be replaced by arbitrary locally compact connected spaces.

The homotopy axiom holds for compact connected spaces but not in general.

Provides counterexamples or proofs illustrating the limitations.

Abstract

In the paper \emph{K.~Sigmon} A strong homotopy axiom for Alexander cohomology, Proc. Amer. Math. Soc. \textbf{31:1} (1972), 271--275 it was proven that if $G$ is compact topological group or field then in the Homotopy Axiom for Alexander-Spanier-Kolmogoroff cohomology the parameter segment $[0, 1]$ can be replaced by any compact connected space $T$. The purpose of the paper is to shows that the parameter space $T$ can not be replaced in general by locally compact connected space.