The existence of $S^1\times C_p$-maps between representation spheres and its applications

TL;DR

This paper proves the existence of specific $S^1\times C_p$-maps between representation spheres and characterizes when abelian compact Lie groups have the weak Borsuk-Ulam property, linking group structure to topological properties.

Contribution

It establishes the existence of certain $S^1\times C_p$-maps and characterizes abelian compact Lie groups with the weak Borsuk-Ulam property.

Findings

Existence of $S^1\times C_p$-maps between representation spheres.

Characterization of abelian compact Lie groups with the weak Borsuk-Ulam property.

Identification of groups with the property as finite abelian $p$-groups or $k$-tori.

Abstract

We show the existence of $S^1\times C_p$-maps between certain representation spheres. As an application, we show that, in the family of abelian compact Lie groups, a group $G$ has the weak Borsuk-Ulam property (in the sense of Bartsch) if and only if $G$ is either a finite abelian $p$-group or a $k$-torus.