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arxiv: 2404.07953 · v3 · pith:ZY77UHMTnew · submitted 2024-04-11 · 🧮 math.SG · math.AT· math.DG

Floer Homology with DG Coefficients. Applications to cotangent bundles

Jean-Fran\c{c}ois Barraud , Mihai Damian , Vincent Humili\`ere , Alexandru Oancea This is my paper

Pith reviewed 2026-05-24 02:36 UTC · model grok-4.3

classification 🧮 math.SG math.ATmath.DG
keywords Floer homologyDG coefficientscotangent bundlesViterbo isomorphismsymplectic homologyalmost existenceperiodic orbitsaspherical manifolds
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The pith

Floer homology with differential graded local coefficients yields a Viterbo isomorphism for cotangent bundles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines Hamiltonian Floer homology with differential graded local coefficients on symplectically aspherical manifolds by letting the differential incorporate chain representatives of the fundamental classes of moduli spaces of Floer trajectories. This construction supports symplectic homology groups with the same coefficients and allows coefficients in chains on fibers of fibrations over the free loop space. For cotangent bundles the authors establish a Viterbo isomorphism theorem with these coefficients. The isomorphism then supplies criteria for almost existence of contractible periodic orbits and closed characteristics on hypersurfaces, giving the first access to a distinction between aspherical and non-aspherical base manifolds.

Core claim

The authors define Floer homology with DG local coefficients for symplectically aspherical manifolds, develop continuation maps, homotopies and spectral invariants, and prove that the resulting groups for cotangent bundles are isomorphic to the singular chains on the based loop space of the base manifold with DG coefficients. This isomorphism furnishes general almost-existence statements for contractible closed characteristics on regular energy levels of Hamiltonians in Liouville domains and, when specialized to hypersurfaces in cotangent bundles, distinguishes the behavior on aspherical versus non-aspherical manifolds.

What carries the argument

Differential graded local coefficients, with the Floer differential built from chain representatives of the fundamental classes of moduli spaces of trajectories of arbitrary dimension.

If this is right

  • Spectral invariants are defined and satisfy the usual properties with DG coefficients.
  • General criteria for almost existence of contractible periodic orbits hold inside Liouville domains.
  • Almost existence of contractible closed characteristics holds for closed smooth hypersurfaces in cotangent bundles.
  • The existence statements distinguish closed base manifolds that are aspherical from those that are not.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same DG-coefficient construction may apply to other Liouville domains whose loop-space fibrations admit suitable chain-level data.
  • The aspherical/non-aspherical distinction could be used to produce new symplectic invariants that detect the fundamental group or higher homotopy of the base manifold.
  • Continuation maps and homotopies developed here might allow comparison of DG Floer homology with other coefficient systems such as Novikov rings.

Load-bearing premise

The underlying manifolds must be symplectically aspherical so that the Floer differential can be defined using those chain representatives.

What would settle it

An explicit cotangent bundle and choice of DG coefficients for which the Viterbo isomorphism fails to hold, or a closed hypersurface in such a bundle on which contractible characteristics exist or fail to exist contrary to the predicted dichotomy.

Figures

Figures reproduced from arXiv: 2404.07953 by Alexandru Oancea, Jean-Fran\c{c}ois Barraud, Mihai Damian, Vincent Humili\`ere.

Figure 1
Figure 1. Figure 1: Inclusion of spheres into the (contractible) space of cones. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: We define: c ◦ HZ(V, Z) = sup{− min H : H is (V, Z)-admissible and admits no nontrivial contractible closed orbit of period ≤ 1}. Remark 5.4. If (V ′ , Z′ ) is any other pair consisting of an open set and a compact subset, such that Z ⊆ Z ′ ⊂ V ′ ⊆ V , then it follows immediately from the definition that c ◦ HZ(V ′ , Z′ ) ≤ c ◦ HZ(V, Z). 87 [PITH_FULL_IMAGE:figures/full_fig_p087_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: A Hamiltonian which is (V, Z)-admissible. We will be particularly interested in the case where V is the whole ambient manifold and Z is the closure of a small retraction of the domain bounded by Σ, i.e., (V, Z) = (W,Uint). Now consider a nonvanishing class α ∈ H∗(W, ∂W;i ∗ W F) that belongs to keriW∗ and define its spectral number c(α) = inf{c ∈ R : iW∗(α) = 0 in SH<c ∗ (W; F)}. (51) The fact that α is non… view at source ↗
Figure 3
Figure 3. Figure 3: Deformation. Proof. Let H be a (W,U)-admissible Hamiltonian such that − min H > c(α). We assume that H does not admit any nontrivial contractible closed orbit of period at most 1. Given R ≥ 1 we let WR = W ⊔ [1, R] × ∂W. Equivalently, WR is the image of W under the time log R flow of the Liouville vector field in the symplectic completion Wˆ . Note that we have an identification Wˆ = WR ⊔ [R, +∞) × ∂W. Rec… view at source ↗
read the original abstract

We define Hamiltonian Floer homology with differential graded (DG) local coefficients for symplectically aspherical manifolds. The differential of the underlying complex involves chain representatives of the fundamental classes of the moduli spaces of Floer trajectories of arbitrary dimension. This setup allows in particular to define and compute Floer homology with coefficients in chains on fibers of fibrations over the free loop space of the underlying symplectic manifold. We develop the DG Floer toolset, including continuation maps and homotopies, and we also define and study symplectic homology groups with DG local coefficients. We define spectral invariants and establish general criteria for almost existence of contractible periodic orbits on regular energy levels of Hamiltonian systems inside Liouville domains. In the case of cotangent bundles, we prove a Viterbo isomorphism theorem with DG local coefficients. This serves as a stepping stone for applications to the almost existence of contractible closed characteristics on closed smooth hypersurfaces. In this context, our methods allow to access for the first time the dichotomy between closed manifolds that are aspherical and those that are not.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper defines Hamiltonian Floer homology with DG local coefficients for symplectically aspherical manifolds, where the differential uses chain representatives of fundamental classes of moduli spaces of Floer trajectories of arbitrary dimension. It develops continuation maps, homotopies, symplectic homology with DG coefficients, and spectral invariants. For cotangent bundles it proves a Viterbo isomorphism with DG coefficients, which is used to obtain almost-existence results for contractible periodic orbits on hypersurfaces and to distinguish aspherical from non-aspherical manifolds.

Significance. If the central construction is valid, the work supplies a systematic extension of Floer theory to DG coefficients that permits new computations involving chains on loop-space fibrations. The Viterbo isomorphism and the resulting dichotomy for almost existence constitute a concrete advance in symplectic geometry. The full development of the algebraic toolkit (continuations, homotopies, spectral invariants) adds technical value beyond the main theorems.

major comments (1)
  1. [Abstract (Floer differential construction)] Abstract (definition of the Floer differential): the differential is defined by means of chain representatives of the fundamental classes of moduli spaces of trajectories of arbitrary dimension. For this to yield a chain complex (d² = 0), the chosen representatives must be compatible under the gluing maps that appear in the compactifications, so that the boundary operator reproduces the algebraic count of broken trajectories with the correct signs and DG coefficients. The manuscript gives no indication of how such a coherent system of representatives is constructed or verified; this compatibility is load-bearing for every subsequent statement.
minor comments (1)
  1. [Abstract] The abstract is unusually dense; separating the definition of the complex, the statement of the Viterbo isomorphism, and the almost-existence applications into distinct paragraphs would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying this foundational point about the Floer differential. We address the comment directly below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: Abstract (Floer differential construction)] Abstract (definition of the Floer differential): the differential is defined by means of chain representatives of the fundamental classes of the moduli spaces of trajectories of arbitrary dimension. For this to yield a chain complex (d² = 0), the chosen representatives must be compatible under the gluing maps that appear in the compactifications, so that the boundary operator reproduces the algebraic count of broken trajectories with the correct signs and DG coefficients. The manuscript gives no indication of how such a coherent system of representatives is constructed or verified; this compatibility is load-bearing for every subsequent statement.

    Authors: We agree that a coherent choice of chain representatives compatible with the gluing maps is required to guarantee d² = 0, including the correct signs and DG coefficients. The present manuscript states the definition but does not supply an explicit construction or verification of this compatibility. In the revised version we will add a dedicated subsection (most likely in Section 3) that constructs the system of representatives using the DG-module structure on the coefficient chains, verifies compatibility under the gluing maps of the compactified moduli spaces, and confirms that the resulting boundary operator satisfies d² = 0. Sign conventions will be fixed consistently with the orientation data already introduced for the moduli spaces. revision: yes

Circularity Check

0 steps flagged

No circularity; construction extends standard Floer theory independently

full rationale

The paper presents a definitional extension of Hamiltonian Floer homology to DG local coefficients on symplectically aspherical manifolds, using chain representatives of fundamental classes of moduli spaces to define the differential. The Viterbo isomorphism with DG coefficients is stated as a proved result in the cotangent bundle case, serving as a stepping stone rather than reducing to any fitted input, self-citation chain, or ansatz smuggled from prior work. No equation or step in the abstract or described derivation equates a claimed output to its inputs by construction; the setup is self-contained against the external benchmark of classical Floer theory and symplectic homology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the domain assumption of symplectically aspherical manifolds and the standard background of Floer theory; no free parameters or invented entities are introduced beyond the definitional extension to DG coefficients.

axioms (1)
  • domain assumption The manifold is symplectically aspherical.
    Explicitly required in the first sentence for the definition of the homology.

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