Equivariant formality in complex-oriented theories
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916257110949888 |
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| author | Bai, Shaoyun Pomerleano, Daniel |
| author_facet | Bai, Shaoyun Pomerleano, Daniel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05821 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivariant formality in complex-oriented theories Bai, Shaoyun Pomerleano, Daniel Symplectic Geometry Algebraic Geometry Algebraic Topology 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. |
| title | Equivariant formality in complex-oriented theories |
| topic | Symplectic Geometry Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2405.05821 |