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.