Integral Kirwan Surjectivity

TL;DR

This paper refines key theorems related to Hamiltonian group actions on symplectic manifolds, establishing surjectivity and splitting results for the Kirwan map, and introduces a stable version of equivariant formality using advanced cohomology theories.

Contribution

It provides new surjectivity and splitting results for the Kirwan map, including integral surjectivity for free quotients, and extends these results to MU-module spectra and stable equivariant formality.

Findings

Kirwan map is surjective after inverting stabilizer orders

Integral surjectivity for free quotients

Stable version of equivariant formality established

Abstract

We refine Kirwan's surjectivity and formality theorems for a Hamiltonian G-action on a compact symplectic manifold M. For a regular value of the moment map, we show that the Kirwan map is surjective and additively split after inverting the orders of stabilizers in the reduction. In particular, for a free quotient, it is surjective integrally. We generalize this to a splitting of MU-module spectra. We also give a stable version of Kirwan's equivariant formality theorem. The novel idea is to exploit the Atiyah-Bott argument in Morava K-theory, then return to bordism and cohomology.