On the equivariant Hopf theorem

Let G be a finite group, X a compact locally smooth G-manifold and S an orthogonal G-sphere. The purpose of the paper is to compute, given a G-map f :X → S and under suitable assumptions, the number of distinct G-homotopy classes of maps f' :X → S such that deg(f'H) = deg(fH) for every subgroup H ⊂ G, i.e. to count the number of G-homotopy classes in [X,S]G with the same stable equivariant degree dG. To achieve this result, an unstable equivariant degree d̃G is introduced, with the property that, under the same assumptions, the number of G-homotopy classes of G-maps f :X → S having the same degree d̃G(f) is finite, and computable in terms of the orientation behavior of the Weyl groups WGH of the isotropy groups of X. This gives an equivariant analogue of the Hopf classification theorem. As a consequence, we find conditions under which the stable degree dG classifies G-maps X → S up to G-homotopy and we give some counter-examples. © 2002 Elsevier Science Ltd. All rights reserved.