The Burnside Ring-Valued Morse Formula for Vector Fields on Manifolds with Boundary

Abstract

Let G be a compact Lie group and A(G) its Burnside Ring. For a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v, we prove an equivariant analog of the Morse formula Ind^G(v) = \sum_{k = 0}^{n} (-1)^k \chi^G(\d_k^+X) which takes its values in A(G). Here Ind^G(v) denotes the equivariant index of the field v, {\d_k^+X\} the v-induced Morse stratification (see [M]) of the boundary \d X, and \chi^G(\d_k^+X) the class of the (n - k)-manifold \d_k^+X in $A(G)$. We examine some applications of this formula to the equivariant real algebraic fields v in compact domains X \subset \R^n defined via a generic polynomial inequality. Next, we link the above formula with the equivariant degrees of certain Gauss maps. This link is an equivariant generalization of Gottlieb's formulas.