Determination of compact Lie groups with the Borsuk-Ulam property

TL;DR

This paper characterizes which compact Lie groups possess the Borsuk-Ulam property, concluding that only elementary abelian p-groups and n-tori have this property.

Contribution

It provides a complete classification of compact Lie groups with the Borsuk-Ulam property, resolving a classical question in the field.

Findings

Only elementary abelian p-groups and n-tori have the Borsuk-Ulam property.

Groups extending n-tori by cyclic groups of prime order lack the Borsuk-Ulam property.

The classification answers a longstanding open problem.

Abstract

In this paper, we shall discuss the classical question of which compact Lie groups have the Borsuk-Ulam property and in particular we shall show that every extension group of a n-torus by a cyclic group of prime order does not have the Borsuk-Ulam property. This leads us that the only compact Lie groups with the Borsuk-Ulam property are an elementary abelian p-group and an n-torus, which is a final answer to the question.