An almost classification of compact Lie groups with Borsuk-Ulam properties

Authors:

• Wacław Bolesław Marzantowicz

Abstract

We say that a compact Lie group G has the Borsuk-Ulam property in the weak sense if for every orthogonal representation V of G and every G-equivariant map f: S(V) → S(V), V^G = {0}, of the unit sphere we have deg f ≠ 0. We say that G has the Borsuk-Ulam property in the strong sense if for any two orthogonal representations V, W of G with dim W = dim V and W^G = V^G = {0} and every G-equivariant map f: S(V) → S(W) of the unit spheres we have deg f ≠ 0. In this paper a complete classification, up to isomorphism, of group with the weak Borsuk-Ulam property is given. A classification of groups with the strong Borsuk-Ulam property does not cover nonabelian p-groups with all elements of the order p. In fact we deal with a more general definition admitting a nonempty fixed point set of G on the sphere S(V). © 1990 by Pacific Journal of Mathematics.