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Recovering generalized homology from Floer homology: the complex oriented case

Laurent Côté, Yusuf Barış Kartal

TL;DR

This work constructs and analyzes the completed Tate cohomology $\widehat{R}_{S^1}^*(X)$ for filtered $S^1$-equivariant spectra with a complex oriented theory $R$, and proves a stability result for Liouville manifolds: the completed Tate cohomology of the filtered spectral symplectic cohomology $SH_S(M,\mathbb{S})$ coincides with that of the trivial filtration $\Sigma^\infty (M/\partial_\infty M)$, thus recovering the stable homotopy type of $M$. The authors provide explicit calculations for Eilenberg–MacLane spectra, Morava $K$-theories, and complex $K$-theory, showing that torsion and integral homology—and in favorable ranges even $KU$-theory—are determined from the filtered equivariant Floer data. A key input is a local Floer homotopy computation near autonomous Hamiltonian orbits, realized as Thom spectra of equivariant virtual bundles, which underpins the global Tate-cohomology calculations. Together, these results support Treumann’s conjecture-style intuition that filtered equivariant Floer data encodes substantial stable-homotopy information about the underlying manifold, with potential implications for understanding complex $K$-theory and higher chromatic information from Floer theory.

Abstract

We associate an invariant called the completed Tate cohomology to a filtered circle-equivariant spectrum and a complex oriented cohomology theory. We show that when the filtered spectrum is the spectral symplectic cohomology of a Liouville manifold, this invariant depends only on the stable homotopy type of the underlying manifold. We make explicit computations for several complex oriented cohomology theories, including Eilenberg-Maclane spectra, Morava K-theories, their integral counterparts, and complex K-theory. We show that the result for Eilenberg-Maclane spectra depends only on the rational homology, and we use the computations for Morava K-theory to recover the integral homology (as an ungraded group). In a different direction, we use the completed Tate cohomology computations for the complex K-theory to recover the complex K-theory of the underlying manifold from its equivariant filtered Floer homotopy type. A key Floer theoretic input is the computation of local equivariant Floer theory near the orbit of an autonomous Hamiltonian, which may be of independent interest.

Recovering generalized homology from Floer homology: the complex oriented case

TL;DR

This work constructs and analyzes the completed Tate cohomology for filtered -equivariant spectra with a complex oriented theory , and proves a stability result for Liouville manifolds: the completed Tate cohomology of the filtered spectral symplectic cohomology coincides with that of the trivial filtration , thus recovering the stable homotopy type of . The authors provide explicit calculations for Eilenberg–MacLane spectra, Morava -theories, and complex -theory, showing that torsion and integral homology—and in favorable ranges even -theory—are determined from the filtered equivariant Floer data. A key input is a local Floer homotopy computation near autonomous Hamiltonian orbits, realized as Thom spectra of equivariant virtual bundles, which underpins the global Tate-cohomology calculations. Together, these results support Treumann’s conjecture-style intuition that filtered equivariant Floer data encodes substantial stable-homotopy information about the underlying manifold, with potential implications for understanding complex -theory and higher chromatic information from Floer theory.

Abstract

We associate an invariant called the completed Tate cohomology to a filtered circle-equivariant spectrum and a complex oriented cohomology theory. We show that when the filtered spectrum is the spectral symplectic cohomology of a Liouville manifold, this invariant depends only on the stable homotopy type of the underlying manifold. We make explicit computations for several complex oriented cohomology theories, including Eilenberg-Maclane spectra, Morava K-theories, their integral counterparts, and complex K-theory. We show that the result for Eilenberg-Maclane spectra depends only on the rational homology, and we use the computations for Morava K-theory to recover the integral homology (as an ungraded group). In a different direction, we use the completed Tate cohomology computations for the complex K-theory to recover the complex K-theory of the underlying manifold from its equivariant filtered Floer homotopy type. A key Floer theoretic input is the computation of local equivariant Floer theory near the orbit of an autonomous Hamiltonian, which may be of independent interest.
Paper Structure (28 sections, 27 theorems, 46 equations, 1 figure)

This paper contains 28 sections, 27 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Context
  3. Extra structures on spectral symplectic cohomology
  4. Summary of results
  5. Filtered equivariant spectra and completed Tate cohomology
  6. Reminders in homotopy theory
  7. Conventions
  8. Borel $G$-spectra
  9. Complex orientations
  10. Filtered objects of stable $\infty$-categories
  11. Completed Tate cohomology
  12. The definition
  13. Completed Tate cohomology of the building blocks
  14. $S^1$-equivariant spectral symplectic cohomology
  15. Conventions
  16. ...and 13 more sections

Key Result

Theorem \ref{thm:tatecohforsh}

If $M$ is a stably framed Liouville manifold and $R$ is a complex oriented ring spectrum, where $\Sigma^\infty (M/ \partial_\infty M)$ is endowed with the trivial filtration and trivial $S^1$ action.

Figures (1)

  • Figure 5.1: A spiked disc "over $\eta$"

Theorems & Definitions (59)

  • Theorem \ref{thm:tatecohforsh}
  • Corollary \ref{cor:eilenbergmaclanetatesh}
  • Corollary \ref{cor:deformedmoravash}
  • Theorem \ref{thm:hlgrecovery}
  • Corollary \ref{cor:fullhlgrecovery}
  • Theorem \ref{thm:kutatesh}
  • Remark 1.2: Potential dynamical applications
  • Lemma 2.1
  • proof
  • Definition 2.2
  • ...and 49 more