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

Wacław Marzantowicz

Vol. 144 (1990), No. 2, 299–311

DOI: 10.2140/pjm.1990.144.299

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 dimW = dimV 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 ).

Mathematical Subject Classification 2000

Primary: 57S15

Secondary: 55M35

Milestones

Received: 6 July 1987

Published: 1 August 1990