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arXiv · 2405.05821

Equivariant formality in complex-oriented theories

Shaoyun Bai·Daniel Pomerleano
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Abstract

Let $G$ be a product of unitary groups and let $(M,ω)$ be a compact symplectic manifold with Hamiltonian $G$-action. We prove an equivariant formality result for any complex-oriented cohomology theory $\mathbb{E}^*$ (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 for Hamiltonian fibrations over $\mathbb{CP}^1.$ Moreover, we establish analogues of the "localization" and "injectivity to fixed points" theorems for certain cohomology theories studied by Hopkins-Kuhn-Ravenel. As an application of these results, we establish a Goresky-Kottwitz-MacPherson theorem with Morava $K$-theory coefficients for Hamiltonian $T$-manifolds.

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Shaoyun Bai, Daniel Pomerleano. 2024-05-22. Equivariant formality in complex-oriented theories. https://arxiv.org/abs/2405.05821

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