Surjectivity for Hamiltonian $G$-spaces in $K$-theory
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- by Megumi Harada and Gregory D. Landweber PDF
- Trans. Amer. Math. Soc. 359 (2007), 6001-6025 Request permission
Abstract:
Let $G$ be a compact connected Lie group, and $(M,\omega )$ a Hamiltonian $G$-space with proper moment map $\mu$. We give a surjectivity result which expresses the $K$-theory of the symplectic quotient $M /\!\!/G$ in terms of the equivariant $K$-theory of the original manifold $M$, under certain technical conditions on $\mu$. This result is a natural $K$-theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the $K$-theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian $G$-spaces. We discuss this lemma in detail and highlight the differences between the $K$-theory and rational cohomology versions of this lemma. We also introduce a $K$-theoretic version of equivariant formality and prove that when the fundamental group of $G$ is torsion-free, every compact Hamiltonian $G$-space is equivariantly formal. Under these conditions, the forgetful map $K_{G}^{*}(M)\to K^{*}(M)$ is surjective, and thus every complex vector bundle admits a stable equivariant structure. Furthermore, by considering complex line bundles, we show that every integral cohomology class in $H^{2}(M;\mathbb {Z})$ admits an equivariant extension in $H_{G}^{2}(M;\mathbb {Z})$.References
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Additional Information
- Megumi Harada
- Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
- Address at time of publication: Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4K1
- Email: megumi@math.toronto.edu, megumi.harada@math.mcmaster.ca
- Gregory D. Landweber
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
- Address at time of publication: Department of Mathematics, Bard College, Annandale-on-Hudson, New York 12504
- Email: greg@math.uoregon.edu, landweber@bard.edu
- Received by editor(s): August 25, 2005
- Received by editor(s) in revised form: September 8, 2005
- Published electronically: June 4, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 6001-6025
- MSC (2000): Primary 53D20; Secondary 19L47
- DOI: https://doi.org/10.1090/S0002-9947-07-04164-5
- MathSciNet review: 2336314