{"paper":{"title":"Spectral Floer theory and tangential structures","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AT","math.KT"],"primary_cat":"math.SG","authors_text":"Ivan Smith, Noah Porcelli","submitted_at":"2024-11-05T16:55:08Z","abstract_excerpt":"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' $\\Theta \\to \\Phi$ 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 $\\Theta \\to \\Phi$. The previous case corresponded to $\\Theta = \\Phi = \\{\\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.03257","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.03257/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}