Equivariant vector fields and their indices

Abstract:
We introduce Γ-equivariant local and global (total) indices of connected components of the zero set of a smooth Γ-equivariant (co)tangent vector field on a compact n-dimensional smooth manifold (possibly with boundary) with a given smooth action of a compact Lie group Γ. These indices belong to Γ-equivariant homotopy groups of spheres. The basis of our construction is the Ize–Vignoli degree theory of Γ-equivariant maps. Two existence theorems of solutions for nonlinear problems of Equivariant Analysis are obtained.