Surjectivity for Hamiltonian G-spaces in K-theory

Let G be a compact connected Lie group, and (M,ω) a Hamiltonian G-space with proper moment map μ. 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 μ. 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;\Z) admits an equivariant extension in H_{G}^{2}(M;\Z).