CLASSIFICATION OF HOMOTOPY CLASSES OF EQUIVARIANT GRADIENT MAPS E. N. DANCER, K. GE ¸BA, AND S. M. RYBICKI † Abstract. Let V be an orthogonal representation of a compact Lie group G and let S(V ),D(V ) be the unit sphere and disc of V, respec- tively. If F : V → R is a G-invariant C 1 -map then the G-equivariant gradient C 0 -map ∇F : V → V is said to be admissible provided that (∇F ) -1 (0) ∩ S(V )= ∅. We classify the homotopy classes of admissible G-equivariant gradient maps ∇F :(D(V ),S(V )) → (V,V \{0}). 1. Introduction The purpose of this paper is to understand the topological invariants associated with equivariant gradient maps. More precisely, we assume V is a real finite-dimensional orthogonal re- presentation of a compact Lie group G. In other words, V is a real finite- dimensional linear space with a scalar product and there is an orthogonal action of G on V denoted by gx for g ∈ G, x ∈ V. We are then interested in the homotopy classes of G-equivariant gradient mappings from V into V , which are non-zero on the unit sphere S (V ) of V. Here a map ∇F : V → V is said to be G-equivariant if ∇F (gx)= g∇F (x) for g ∈ G, x ∈ V. We completely classify the homotopy classes of such maps in terms of the G-equivariant gradient degree defined here. Our gradient degree is a natu- ral generalization of the classical Brouwer degree with the ring of integers replaced by a ring U (G) determined by the orbit types of G (see [4] for the definition of U (G)). Results of this type are of considerable interest because they show that there are no extra invariants to be found. Note that the study of the ho- motopy classes of G-equivariant gradient maps is a natural problem as these maps arise in many applications (see [3, 10, 11, 12] for example). The stronger homotopy invariance properties of our degree are reflected in the possibility to obtain connected sets of bifurcation solutions as in [3, 10, 11, 12]. The Conley index is not suitable for this. Date : July 15, 2005. 1991 Mathematics Subject Classification. 47H11, 37G40. Key words and phrases. equivariant degree theory, variational methods. † Research supported by the State Committee for Scientific Research, Poland, under grant No. 5 PO3A 026 20. 1