Journal of Differential Geometry

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

Victor Guillemin and Mikhail Kogan


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 aK G (M) and pM G . We will show that corresponding to this stratification there is a basis τ p , pM G , for K G (M) as a module over K G (pt) characterized by the property: I q τ p q p . For M a GKM manifold we give an explicit construction of these τ p 's in terms of the associated GKM graph.

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J. Differential Geom., Volume 66, Number 3 (2004), 345-375.

First available in Project Euclid: 18 October 2004

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Guillemin, Victor; Kogan, Mikhail. Morse theory on Hamiltonian G -spaces and equivariant K -theory. J. Differential Geom. 66 (2004), no. 3, 345--375. doi:10.4310/jdg/1098137837.

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