The local structure of tangent G-vector fields

We introduce a notion of equivariant index in order to describe the behavior of tangent G-vector fields on smooth G-manifolds near isolated zeros. Our methods result in a calculation of the monoid of G-homotopy classes of self-maps of the unit sphere S(V) in a real orthogonal (finite dimensional) G-module V, this being the unstable analogue of a classical result of Segal. During the course of our calculation, we prove general position results on tangent G-vector fields and obtain canonical local structures for such fields.