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

TL;DR

The paper investigates when equivariant maps between representation spheres exist for abelian compact Lie groups, focusing on $G= S^1\times C_p$. It constructs explicit $G$-maps between representation spheres for growing dimension using explicit algebraic maps and equivariant obstruction theory, showing $G$-maps $S(V_n)\to S(W_m)$ exist with $m\ge n$; consequently $G$ does not have the weak Borsuk-Ulam property. As a corollary, among abelian compact Lie groups, the weak Borsuk-Ulam property holds iff the group is a finite abelian $p$-group or a $k$-torus. The work combines representation-sphere maps, obstruction theory on lens spaces, and group-extension considerations to classify weak Borsuk-Ulam phenomena in the abelian setting.

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.

Theorems & Definitions (16)

• Theorem 1.1
• Corollary 1.2
• Theorem 2.1
• Remark 1
• Proposition 2.2
• proof
• Lemma 2.3
• proof : Proof of Lemma \ref{['l2-3']}
• proof : Proof of Corollary \ref{['c1-2']}
• Proposition 3.1