Surjectivity for Hamiltonian 𝐺-spaces in 𝐾-theory

Abstract

Let G G be a compact connected Lie group, and ( M , ω ) (M,\omega ) a Hamiltonian G G -space with proper moment map μ \mu . We give a surjectivity result which expresses the K K -theory of the symplectic quotient M / / G M /\!\!/G in terms of the equivariant K K -theory of the original manifold M M , under certain technical conditions on μ \mu . This result is a natural K K -theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the K K -theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian G G -spaces. We discuss this lemma in detail and highlight the differences between the K K -theory and rational cohomology versions of this lemma. We also introduce a K K -theoretic version of equivariant formality and prove that when the fundamental group of G G is torsion-free, every compact Hamiltonian G G -space is equivariantly formal. Under these conditions, the forgetful map K G ∗ ( M ) → K ∗ ( M ) 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 ) H^{2}(M;\mathbb {Z}) admits an equivariant extension in H G 2 ( M ; Z ) H_{G}^{2}(M;\mathbb {Z}) .