Morse theory on Hamiltonian G-spaces and equivariant K-theory

Abstract

Let G be a torus and M a compact Hamiltonian G-manifold with finite fixed point set M G . If T is a circle subgroup of G with M G =M T , the T-moment map is a Morse function. We will show that the associated Morse stratification of M by unstable manifolds gives one a canonical basis of K G (M). A key ingredient in our proof is the notion of local index I p (a) for a∈K G (M) and p∈M G . We will show that corresponding to this stratification there is a basis τ p , p∈M G , for K G (M) as a module over K G (pt) characterized by the property: I q τ p =δ q p . For M a GKM manifold we give an explicit construction of these τ p 's in terms of the associated GKM graph.