Spectral Floer theory and tangential structures
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866915427534241792 |
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| author | Porcelli, Noah Smith, Ivan |
| author_facet | Porcelli, Noah Smith, Ivan |
| contents | In \cite{PS}, for a stably framed Liouville manifold $X$ we defined a Donaldson-Fukaya category $\mathcal{F}(X;\mathbb{S})$ over the sphere spectrum, and developed an obstruction theory for lifting quasi-isomorphisms from $\mathcal{F}(X;\mathbb{Z})$ to $\mathcal{F}(X;\mathbb{S})$. Here, we define a spectral Donaldson-Fukaya category for any `graded tangential pair' $Θ\to Φ$ of spaces living over $BO \to BU$, whose objects are Lagrangians $L\to X$ for which the classifying maps of their tangent bundles lift to $Θ\to Φ$. The previous case corresponded to $Θ= Φ= \{\mathrm{pt}\}$. We extend our obstruction theory to this setting. The flexibility to `tune' the choice of $Θ$ and $Φ$ increases the range of cases in which one can kill the obstructions, with applications to bordism classes of Lagrangian embeddings in the corresponding bordism theory $Ω^{(Θ,Φ),\circ}_*$. We include a self-contained discussion of when (exact) spectral Floer theory over a ring spectrum $R$ should exist, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03257 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral Floer theory and tangential structures Porcelli, Noah Smith, Ivan Symplectic Geometry Algebraic Topology K-Theory and Homology In \cite{PS}, for a stably framed Liouville manifold $X$ we defined a Donaldson-Fukaya category $\mathcal{F}(X;\mathbb{S})$ over the sphere spectrum, and developed an obstruction theory for lifting quasi-isomorphisms from $\mathcal{F}(X;\mathbb{Z})$ to $\mathcal{F}(X;\mathbb{S})$. Here, we define a spectral Donaldson-Fukaya category for any `graded tangential pair' $Θ\to Φ$ of spaces living over $BO \to BU$, whose objects are Lagrangians $L\to X$ for which the classifying maps of their tangent bundles lift to $Θ\to Φ$. The previous case corresponded to $Θ= Φ= \{\mathrm{pt}\}$. We extend our obstruction theory to this setting. The flexibility to `tune' the choice of $Θ$ and $Φ$ increases the range of cases in which one can kill the obstructions, with applications to bordism classes of Lagrangian embeddings in the corresponding bordism theory $Ω^{(Θ,Φ),\circ}_*$. We include a self-contained discussion of when (exact) spectral Floer theory over a ring spectrum $R$ should exist, which may be of independent interest. |
| title | Spectral Floer theory and tangential structures |
| topic | Symplectic Geometry Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2411.03257 |