Floer homotopy theory for monotone Lagrangians
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916825246203904 |
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| author | Bonciocat, Ciprian Mircea |
| author_facet | Bonciocat, Ciprian Mircea |
| contents | We circumvent one of the roadblocks in associating Floer homotopy types to monotone Lagrangians, namely the curvature phenomena occurring in high dimensions. Given $N \ge 3$ and $R$ a connective $\mathbb E_1$-ring spectrum, there is a notion of an $N$-truncated, $R$-oriented flow category, to which we associate a module prospectrum over the Postnikov truncation $τ_{\le N - 3}R$. This endows ordinary Floer cohomology with an action of the Steenrod algebra over $τ_{\le N-3}R$, and also induces certain generalized cohomology theories. We give sufficient conditions for a closed embedded monotone Lagrangian to admit such well-defined invariants for $N = N_μ$ the minimal Maslov number, and $R = MU$ complex bordism. Finally, we formulate Oh-Pozniak type spectral sequences for these invariants, and show that in the case of $\mathbb{RP}^n \subset \mathbb{CP}^n$ they provide further restrictions on the topology of clean intersections with a Hamiltonian isotopy, not detected by ordinary Floer (co)homology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17431 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Floer homotopy theory for monotone Lagrangians Bonciocat, Ciprian Mircea Symplectic Geometry Algebraic Topology Geometric Topology 53D05 (Primary) 55P42, 37D15, 57R58, 57R19 (Secondary) We circumvent one of the roadblocks in associating Floer homotopy types to monotone Lagrangians, namely the curvature phenomena occurring in high dimensions. Given $N \ge 3$ and $R$ a connective $\mathbb E_1$-ring spectrum, there is a notion of an $N$-truncated, $R$-oriented flow category, to which we associate a module prospectrum over the Postnikov truncation $τ_{\le N - 3}R$. This endows ordinary Floer cohomology with an action of the Steenrod algebra over $τ_{\le N-3}R$, and also induces certain generalized cohomology theories. We give sufficient conditions for a closed embedded monotone Lagrangian to admit such well-defined invariants for $N = N_μ$ the minimal Maslov number, and $R = MU$ complex bordism. Finally, we formulate Oh-Pozniak type spectral sequences for these invariants, and show that in the case of $\mathbb{RP}^n \subset \mathbb{CP}^n$ they provide further restrictions on the topology of clean intersections with a Hamiltonian isotopy, not detected by ordinary Floer (co)homology. |
| title | Floer homotopy theory for monotone Lagrangians |
| topic | Symplectic Geometry Algebraic Topology Geometric Topology 53D05 (Primary) 55P42, 37D15, 57R58, 57R19 (Secondary) |
| url | https://arxiv.org/abs/2506.17431 |