Degree theory for equivariant maps. I

Abstract

A degree theory for equivariant maps is constructed in a simple geometrical way. This degree has all the basic properties of the usual degree theories and takes its values in the equivariant homotopy groups of spheres. For the case of a semifree S 1 {S^1} -action, a complete computation of these groups is given, the range of the equivariant degree is determined, and the general S 1 {S^1} -action is reduced to that special case. Among the applications one recovers and unifies both the degree for autonomous differential equations defined by Fuller [F] and the S 1 {S^1} -degree for gradient maps introduced by Dancer [Da]. Also, a simple but very useful formula of Nirenberg [N] is generalized (see Theorem 4.4(ii)).