Surjectivity for Hamiltonian -spaces in -theory

Let be a compact connected Lie group, and a Hamiltonian -space with proper moment map . We give a surjectivity result which expresses the -theory of the symplectic quotient in terms of the equivariant -theory of the original manifold , under certain technical conditions on . This result is a natural -theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the -theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian -spaces. We discuss this lemma in detail and highlight the differences between the -theory and rational cohomology versions of this lemma.

We also introduce a -theoretic version of equivariant formality and prove that when the fundamental group of is torsion-free, every compact Hamiltonian -space is equivariantly formal. Under these conditions, the forgetful map 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 admits an equivariant extension in .