A simple proof of the Borsuk-Ulam theorem for Z_p-actions

TL;DR

This paper provides a straightforward proof of the Borsuk-Ulam theorem specifically for free actions of the cyclic group Z_p, establishing a key dimension inequality for equivariant maps between spheres.

Contribution

It introduces a simplified proof of the Borsuk-Ulam theorem for Z_p-actions, enhancing understanding of equivariant topology for prime cyclic groups.

Findings

If S^n and S^m have free Z_p-actions and a Z_p-equivariant map exists, then n ≤ m.

The proof clarifies the relationship between sphere dimensions under Z_p symmetries.

The result extends classical Borsuk-Ulam theorem to group actions of prime order.

Abstract

In this note, we give a simple proof of the Borsuk-Ulam theorem for $Z_p$-actions. We prove that, if $S^n$ and $S^m$ are equipped with free $Z_p$-actions (p prime) and $f: S^n \to S^m$ is a $Z_p$-equivariant map, then $n \leq m$.