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The Quarterly Journal of Mathematics Advance Access published online on December 5, 2008
The Quarterly Journal of Mathematics, doi:10.1093/qmath/han031
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© 2008. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

KIRWAN SURJECTIVITY IN K-THEORY FOR HAMILTONIAN LOOP GROUP QUOTIENTS

Megumi Harada{dagger}

Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada, L8S 4K1

Paul Selick {ddagger}

Department of Mathematics, University of Toronto, 40 St.George Street, Toronto, Ontario, Canada, M5S 2E4

{dagger} Corresponding author. E-mail: Megumi.Harada{at}math.mcmaster.ca

Received 25 April 2008; revised 7 October 2008
  Abstract

Let G be a compact Lie group and LG be its associated loop group. The main result of this article is a surjectivity theorem from the equivariant K-theory of a Hamiltonian LG-space onto the integral K-theory of its Hamiltonian LG-quotient. Our result is a K-theoretic analogue of previous work in rational Borel-equivariant cohomology by R. Bott, S. Tolman and J. Weitsman, Surjectivity for Hamiltonian loop group spaces, Invent. Math. 155 (2004), 225–251, math.DG/0210036. Our proof techniques differ from that of Bott et al. in that they explicitly use the Borel construction, which we do not have at our disposal in equivariant K-theory; we instead directly construct G-equivariant homotopy equivalences to obtain the necessary isomorphisms in equivariant K-theory. The main theorem should also be viewed as a first step towards a similar theorem in K-theory for quasi-Hamiltonian G-spaces and their associated quasi-Hamiltonian quotients.

{ddagger} E-mail: selick{at}math.toronto.edu


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