Surjectivity for Hamiltonian $G$-spaces in $K$-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 $K_{G}^{*}(M)\to K^{*}(M)$ 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 $H_{G}^{2}(M;\mathbb{Z})$.