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Foundation of Floer homotopy theory I: Flow categories

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arxiv 2404.03193 v2 pith:LXYPVL4V submitted 2024-04-04 math.SG math.AT

Mohammed Abouzaid , Andrew J. Blumberg This is my paper · ORCID
classification math.SGmath.AT
keywords floerbordismcategoryhomotopytheoryflowconstructionspectra
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We construct a stable infinity category with objects flow categories and morphisms flow bimodules; our construction has many flavors, related to a choice of bordism theory, and we discuss in particular framed bordism and the bordism theory of complex oriented derived orbifolds. In this setup, the construction of homotopy types associated to Floer-theoretic data is immediate: the moduli spaces of solutions to Floer's equation assemble into a flow category with respect to the appropriate bordism theory, and the associated Floer homotopy types arise as suitable mapping spectra in this category. The definition of these mapping spectra is sufficiently explicit to allow a direct interpretation of the Floer homotopy groups as Floer bordism groups. In the setting of framed bordism, we show that the category we construct is a model for the category of spectra. We implement the construction of Floer homotopy types in this new formalism for the case of Hamiltonian Floer theory.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A homotopy coherent Pontryagin-Thom isomorphism

    math.AT 2026-07 unverdicted novelty 7.0 of 10

    There is a presentably symmetric monoidal stable infinity-category of homotopy-invariant Gysin sheaves whose unit is geometric cobordism and whose endomorphism E-infinity ring is the associated Thom spectrum.

  2. Ample divisor complements, Floer spectra, and relative Gromov-Witten theory

    math.SG 2026-01 conditional novelty 7.0 of 10

    The associated graded of the Floer homotopy type of an ample smooth divisor complement is computed, with the splitting obstruction encoded in a stable homotopy class from genus-0 relative Gromov–Witten moduli.

  3. Transport functions for principal bundles and Morse homology with differential graded coefficients

    math.AT 2026-06 unverdicted novelty 6.0 of 10

    Transport functions on Morse flow categories classify principal bundles, and their DG Morse complexes compute associated-bundle homology.

  4. Toric Mirror Symmetry for Homotopy Theorists

    math.AG 2025-01 conditional novelty 6.0 of 10

    A fully faithful symmetric monoidal coherent-constructible correspondence is constructed for smooth projective toric schemes over the sphere spectrum.

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