The burnside ring-valued morse formula for vector fields on manifolds with boundary

TLDR: In this paper, an equivariant analog of the Morse formula for a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v was shown.

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 $$ {\rm Ind}^G(v) = \sum_{k = 0}^{n} (-1)^k \chi^G(\partial_{k}^{+}X) $$ which takes its values in A(G). Here IndG(v) denotes the equivariant index of the field v, $\{\partial_{k}^{+}X\}$ the v-induced Morse stratification (see [10]) of the boundary ∂X, and $\chi^G(\partial_{k}^{+}X)$ the class of the (n - k)-manifold $\partial_{k}^{+}X$ in A(G). We examine some applications of this formula to the equivariant real algebraic fields v in compact domains X ⊂ ℝn 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 ([3, 4]).