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

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 /: S{V) -> S(V), VG = {0}, of the unit sphere we have deg/ Φ 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 WG = VG = {0} and every G-equivariant map /: S(V) —> S{W) of the unit spheres we have deg/ Φ 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). 1. The main theorems. In order to formulate our main results we introduce the following notation. Let G be a compact Lie group. We denote by Go the component of identity of and by Γ the quotient group G/GQ . We use standard notation of the theory of compact transformation groups (see for instance [4] or [5]). In particular, for every subgroup H c G, the fixed point set of H on a G-space X is denoted by XH. Also, for a G-equivariant map f:X—>Y between two G-spaces, we denote by fH its restriction to the space XH . The symbol (n, m) stands for the greatest common divisor of the integers n, m with the notation (0, 1) = 0 and \G for the rank of the (finite) group G. We will work with the following definition of the BorsukUlam property (cf. [11]). DEFINITION I. (A) We say that G has the Borsuk-Ulam property in the weak sense A if for every orthogonal representation V of G and every G-equivariant map /: S(V) -> S(V) if (deg/ G, |Γ|) = 1 then