Determination of compact Lie groups with the Borsuk-Ulam property

A compact Lie group G is said to have the Borsuk-Ulam property if the Borsuk-Ulam theorem holds for G -maps between representation spheres. It is well-known that an elementary abelian p-group C-p(n)(p any prime) and an n-torus Tn have the Borsuk-Ulam property. 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 an 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 C(p)(n )and T-n, which is a final answer to the question. (C) 2022 Elsevier B.V. All rights reserved.