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

TL;DR

This paper extends Morse theory to G-manifolds with boundary by formulating an equivariant Morse formula valued in the Burnside ring, with applications to algebraic fields and Gauss map degrees.

Contribution

It introduces an equivariant Morse formula for vector fields on G-manifolds with boundary, generalizing classical results to the Burnside ring context.

Findings

Derived an equivariant Morse formula in the Burnside ring for G-manifolds with boundary.

Applied the formula to real algebraic vector fields defined by polynomial inequalities.

Connected the formula with equivariant degrees of Gauss maps, generalizing Gottlieb's results.

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.