2006/12/21 by Megumi Harada, Harada, Megumi, Gregory D. Landweber +1 · 1 citation
Mathematics · #19L47 #53D20 #FOS: Mathematics #K-Theory and Homology (math.KT) #Symplectic Geometry (math.SG) #math.KT #math.SG #msc:19L47 #msc:53D20
paper · pdf · doi:10.48550/arxiv.math/0612660
15 pages; typos corrected
arxiv created 2008/01/02 · arxiv updated 2009/12/01
Let T be a compact torus and (M,ω) a Hamiltonian T-space. In a previous paper, the authors showed that the T-equivariant K-theory of the manifold M surjects onto the ordinary integral K-theory of the symplectic quotient M \mod T of M by T, under certain technical conditions on the moment map. In this paper, we use equivariant Morse theory to give a method for computing the K-theory of the symplectic quotient by obtaining an explicit description of the kernel of the surjection κ: K^*T(M) \onto K^*(M \mod T). Our results are K-theoretic analogues of the work of Tolman and Weitsman for Borel equivariant cohomology. Further, we prove that under suitable technical conditions on the T-orbit stratification of M, there is an explicit Goresky-Kottwitz-MacPherson (``GKM'') type combinatorial description of the K-theory of a Hamiltonian T-space in terms of fixed point data. Finally, we illustrate our methods by computing the ordinary K-theory of compact symplectic toric manifolds, which arise as symplectic quotients of an affine space \CN by a linear torus action.