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

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 $K_G^{\ast }(M)\to K^{\ast }(M)$ 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 $H_G^2(M;\mathbb{Z})$ .