A Note on the Existence of G-maps between Spheres

Let G be a finite group, and let V and W be finite-dimensional real orthogonal G-modules with V 3 W, and with unit spheres S(V) and S( W) respectively. The purpose of this note is to give necessary sufficient conditions for the existence of a C-map /: S( V) -» S( W) in terms of the Burnside ring of G and its relationship with V and W. Note that if W has a nonzero fixed point, such a G-map always exists, so for nontriviality, we assume this not the case. Existence of G-maps. Let F be a finite-dimensional orthogonal G-module and let W c V be an invariant sub-G-module. Denote the unit spheres of V and W by S(V) and S(W) respectively. Here we obtain an algebraic criterion for the existence of a G-map /: S(V) -* S(W). Thus, for nontriviality, we assume WG = {0}. The case V = W has been studied in [3], and we first recall pertinent facts. Let A(G) be the Burnside ring of G. Thus, A(G) is the Grothendieck group of equivalence classes of finite G-sets with addition given by disjoint union. Its elements are thus represented by virtual G-sets, and A(G) is additively the free abelian group with basis {G/77}, where 77 runs through representatives of conjugacy classes of subgroups of G. The multiplicative structure is given by cartesian product. One has a natural isomorphism 0: A(G) = (G) the set of conjugacy classes of subgroups of G, and let d:A(G)-> fi Z=C denote its integral closure. Thus d[s t](H) = \s\H \t\H for a virtual G-set s — t. It is well known that d is a monomorphism [1]. Denote by A(W) the monoid of (free) G-homotopy classes of G-maps S(W) -+ S(W), and let v(W): A(W) -> A(G) denote the natural monoid homomorphisms obtained by suspending and applying 9~l. The results of [3] give a characterization of the image of v(W), which we now state. (The constructions there of G-maps S(W) -> S(W) representing suitable elements in A(G) are given in terms of appropriate tangent G-vector fields on S(W).) Received by the editors September 6, 1985 and, in revised form, December 19, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 54H15. ©1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page