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Surjectivity for Hamiltonian $ G$-spaces in $ K$-theory

Authors: Megumi Harada and Gregory D. Landweber
Journal: Trans. Amer. Math. Soc. 359 (2007), 6001-6025
MSC (2000): Primary 53D20; Secondary 19L47
Published electronically: June 4, 2007
MathSciNet review: 2336314
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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})$.

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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

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

Keywords: Equivariant $K$-theory, Kirwan surjectivity, Morse-Kirwan function, symplectic quotient, Atiyah-Bott lemma, equivariant formality
Received by editor(s): August 25, 2005
Received by editor(s) in revised form: September 8, 2005
Published electronically: June 4, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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