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Equivariant Floer homotopy via Morse-Bott theory

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arxiv 2309.15089 v2 pith:ZGX2DTHO submitted 2023-09-26 math.AT math.GTmath.SG

Laurent Côté , Yusuf Bar{i}ş Kartal This is my paper · ORCID
classification math.ATmath.GTmath.SG
keywords equivarianthomotopymorse-botttheorystabletypeclosedcompact
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We generalize the Cohen-Jones-Segal construction to the Morse-Bott setting. In other words, we define framings for Morse-Bott analogues of flow categories and associate a stable homotopy type to this data. We use this to recover the stable homotopy type of a closed manifold from Morse-Bott theory, and the stable equivariant homotopy type of a closed manifold with the action of a compact Lie group from Morse theory. We use this machinery in Floer theory to construct a genuine circle equivariant model for symplectic cohomology with coefficients in the sphere spectrum. Using the formalism of relative modules, we define equivariant maps to (Thom spectra over) the free loop space of exact, compact Lagrangians. We prove that this map is an equivalence of Borel equivariant spectra when the Lagrangian is the zero section of a cotangent bundle -- an equivariant Viterbo isomorphism theorem over the sphere spectrum.

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Cited by 3 Pith papers

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  1. 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.

  2. Parameterized Lagrangian Floer homotopy

    math.SG 2025-06 conditional novelty 7.0 of 10

    A parameterized Lagrangian Floer homotopy type over the moduli space of Maslov data is constructed, yielding a two-point distinct-action lower bound for degenerate Lagrangian intersections in plumbings of cotangent bundles.

  3. Floer homotopy theory for monotone Lagrangians

    math.SG 2025-06 conditional novelty 6.0 of 10

    An N-truncated flow category with a U-brane produces a Steenrod algebra action on monotone Lagrangian Floer cohomology, giving new clean-intersection restrictions for RP^n in CP^n.

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