Equivariant maps between representation spheres

Abstract

Let G be a finite group, V and W be finite representations of G, S(V) and S(W) be the unit spheres in V and W. Suppose dim VH ^dim WH for every subgroup H of G. We seek to classify the G- equivariant homotopy classes of 6-equivariant maps from S(V) to S(W). Introduction. We wish to consider the following problem: Let G be a finite group, and V and W be finite dimensional orthogonal representations of G. Let 5(V) and S(W) denote the unit spheres of V and W respectively. Then S(V) and S(W) inherit G-actions. Classify the equivariant homotopy classes of G-maps from S(V) to S(W). The case where G = Zp and S(V) and S(W) have free Zp actions was done by Olum (6) and was used to give a classification of lens spaces up to homotopy equivalence. In this paper we generalize Olum's result. Our approach is to consider the behavior of an equivariant map restricted to the various fixed point sets. Explicity, if X is a space with a left action of the group G, and H is a subgroup of G, we denote by XH the set of points in X left fixed by each element of H. If S(V)-*S(W) is a G- equivariant map (i.e., f(gυ) = gf(v) for all g G G and veS(V)), then / induces maps /: S(VH->S(W)H for each subgroup H. Since V and W are linear representations, these fixed point sets S(V)H and S(W)H are again spheres, and we may choose an orientation for each S(V)H and S(W)H. If X is a manifold, denote by dimX the (real) dimension of X. When dimS( V)H = dimS(HOH, fH has a well-determine d degree, denoted by deg/ H. Our major theorem asserts that, under suitable hypotheses, the homotopy classes of the maps fH for all H determine the equivariant homotopy class of DEFINITION. Let G be a finite group. If H is a subgroup of G, denote by N(H) the normalizer of H in G. An orthogonal representa- tion V of G is completely orientable if for every subgroup H of G, the induced action of N(H) on S(V)H is orientation-p reserving.