Equivariant formality in complex-oriented theories

Abstract Let G be a product of unitary groups and let ( M , ω ) $(M,\omega )$ left parenthesis upper M comma omega right parenthesis be a compact symplectic manifold with Hamiltonian G -action. We prove an equivariant formality result for any complex-oriented cohomology theory E ∗ $\mathbb {E}^*$ double struck upper E Superscript asterisk (in particular, integral cohomology). This generalizes the celebrated result of Atiyah–Bott–Kirwan for rational cohomology from the 1980s. The proof does not use classical ideas but instead relies on a recent cohomological splitting result of Abouzaid–McLean–Smith [AMS21] for Hamiltonian fibrations over CP 1 . $\mathbb {CP}^1.$ double struck upper C upper P Superscript 1 Baseline period Moreover, we establish analogues of the ‘localization’ and ‘injectivity to fixed points’ theorems for certain cohomology theories studied by Hopkins–Kuhn–Ravenel in [HKR00]. As an application of these results, we establish a Goresky–Kottwitz–MacPherson theorem with Morava K -theory coefficients for Hamiltonian T -manifolds.