TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-11-04 (gpt-6.1-sol)

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Regular aspherical Hamiltonian Floer differential and continuation contract

Cylinder and action conventions

Write $S^{1} = R / Z$ . A Floer cylinder from $x_{-}$ to $x_{+}$ is a smooth map $u : R \times S^{1} \to M$ satisfying $u_{s} + J_{t} (u) (u_{t} - X_{H_{t}} (u)) = 0$ and converging smoothly to $x_{\pm}$ as $s \to \pm \infty$ . Use the compatible metric $g_{t} (v, w) = ω (v, J_{t} w)$ and energy $E (u) = \int_{R \times S^{1}} ∣ u_{s} ∣_{g_{t}}^{2} d s d t$ . For a contractible loop $x$ and oriented cap $w : D^{2} \to M$ with $w (e^{2 π i t}) = x (t)$ , the capped action is $A_{H} ([x, w]) = \int_{0}^{1} H_{t} (x (t)) d t - \int_{D^{2}} w^{*} ω$ . Capping successive slices by adjoining the cylinder segment makes this a smooth function of $s$ ; its cap choice is irrelevant under spherical asphericity. These are the cap conventions of the orbit-module suppliers. Continuation cylinders use the same equation with smooth $s$ -dependent $H_{s, t}, J_{s, t}$ constant at their two ends; the parameter families below refer to these equations.

Statement

Let $(M, ω)$ be closed and satisfy $ω ∣_{π_{2} (M)} = c_{1} (T M) ∣_{π_{2} (M)} = 0$ . Assume every one-periodic orbit of $H$ is nondegenerate, use the Hamiltonian sign $ι_{X_{H}} ω = - d H$ , and take the aspherical orbit module over $F_{2}$ . For regular Floer data $(H, J_{t})$ , assume each zero-dimensional space $M_{1} (x, y)$ of index-one finite-energy Floer cylinders modulo $s$ -translation is compact, and define $d$ by its mod- $2$ count. Regular means that the linearized Floer operator at every relevant solution is surjective, with the stated Fredholm index, so these quotients are smooth manifolds of the expected dimension. If the index-two cylinder compactifications are compact one-manifolds with boundary exactly the broken-cylinder ends, then the boundary count gives $d^{2} = 0$ . For regular homotopies of admissible Hamiltonians, if all zero-dimensional continuation and parameterized count spaces used below are compact and the one-dimensional compactifications have the stated gluing and boundary descriptions, continuation counts are chain maps and induce isomorphisms on homology; reverse and concatenated homotopies give chain-homotopy inverse maps. With the action convention inherited from cappings and $ι_{X_{H}} ω = - d H$ , cylinder energy equals action drop. No compactness or gluing assertion is made without the stated hypotheses.

Statement references

Classical aspherical Hamiltonian Floer data

Tame J-holomorphic area equals metric Dirichlet energy

Facts & Assumptions

Given: Closed symplectic $(M, ω)$ ; $ω ∣_{π_{2}} = c_{1} ∣_{π_{2}} = 0$ ; a Hamiltonian with every one-periodic orbit nondegenerate; the $F_{2}$ orbit module and action convention; regular Floer and continuation data; compactness and gluing up to breaking; and the exact codimension-one boundary descriptions stated above.

[F1]

The aspherical chain module has finite-support generators given by nondegenerate contractible one-periodic Hamiltonian orbits, with cap-independent action and Conley–Zehnder grading. Classical aspherical Hamiltonian Floer data

[F2]

The symplectic-area/Dirichlet-energy identity is positive for every nonconstant compatible $J$ -holomorphic sphere, using a domain metric compatible with the sphere complex structure. Tame J-holomorphic area equals metric Dirichlet energy

[A1]

Each regular zero-dimensional space $M_{1} (x, y)$ is compact. The regularity, compactness-up-to-breaking, and exact index-two boundary/gluing description are explicit hypotheses of this conditional theorem; they are not supplied by [F1] or [F2].

[A2]

For each continuation homotopy and homotopy of homotopies used, assume compactness of every zero-dimensional space being counted and the stated compact one-dimensional manifold-with-boundary descriptions. Their faces identify with the indicated finite products, parameter-end faces, and broken-cylinder faces; for the inverse construction these parameter ends are the concatenation count and the identity count.

Proof

Given: The data, cylinder/action conventions and assumptions above.

1.1F1F2givenalgebra

For a cylinder $u$ set $v = \partial_{t} u - X_{H_{t}} (u)$ . The equation gives $\partial_{s} u = - J_{t} v$ , so $∣ \partial_{s} u ∣^{2} = ω (\partial_{s} u, v)$ . Differentiating the capped action along $x_{s} (t) = u (s, t)$ gives $\frac{d}{d s} A_{H} ([x_{s}, w_{s}]) = - \int_{0}^{1} ω (\partial_{s} u, v) d t = - \int_{0}^{1} ∣ \partial_{s} u ∣^{2} d t$ , because the loop has no boundary term. Integrating over $s$ proves $E (u) = A_{H} (x_{-}) - A_{H} (x_{+})$ . Moreover, any nonconstant sphere bubble would have positive symplectic area by [F2], while its class has zero $ω$ -area by the asphericity hypothesis; hence no such sphere bubble can occur in this branch.

2.1A1F1step 1.1algebra

Define $d (x) = \sum_{y} #_{2} M_{1} (x, y) y$ , where $M_{1}$ is the zero-dimensional index-one moduli space modulo translation. The full fixed-point set of the time-one map is closed in compact $M$ and hence compact; because every one-periodic orbit is nondegenerate, each fixed point is isolated. Thus the fixed-point set, and therefore its contractible subset, is finite. By [A1], each $M_{1} (x, y)$ is also finite: a compact zero-dimensional manifold is discrete and its singleton cover has a finite subcover. Thus each coefficient is defined and the sum is finite. For fixed generators $x, z$ , the coefficient of $z$ in $d^{2} x$ is the mod- $2$ count of ordered broken pairs through intermediate generators. By [A1], these pairs are exactly the boundary points of the compactified index-two one-manifold from $x$ to $z$ . Every compact one-manifold is a finite disjoint union of circles and closed intervals; circles have no boundary and each interval has two endpoints. Its total boundary count is therefore zero mod $2$ , so every coefficient of $d^{2}$ vanishes.

3.1A2step 2.1

For a regular homotopy, [A2] makes every zero-dimensional continuation count finite, by the compact-discrete argument of step 2.1; finitely many generators then define $K_{01}$ . The boundary of each compactified one-dimensional continuation moduli space has two types of ends: a Floer break before the continuation solution, contributing $K_{01} d_{0}$ , or a break after it, contributing $d_{1} K_{01}$ . The assumed boundary formula [A2] contains no other codimension-one faces, so the same even-boundary count gives $d_{1} K_{01} = K_{01} d_{0}$ .

4.1A2step 3.1algebra∎

Apply [A2] to a one-parameter family interpolating between the concatenated reverse/forward continuation homotopy and the constant homotopy. The resulting compactified one-dimensional parameterized moduli have parameter-end faces equal to $K_{10} K_{01}$ and the identity, while their broken-cylinder faces are precisely $d_{0} Q + Q d_{0}$ for the mod- $2$ count $Q$ of zero-dimensional parameterized solution spaces in that one-parameter family. These spaces are finite by [A2] and the same compact-discrete argument; the total one-dimensional spaces provide the boundary relation. The zero boundary count yields $K_{10} K_{01} - I = d_{0} Q + Q d_{0}$ (minus equals plus over $F_{2}$ ). Reversing the roles gives the other composition. Thus continuation maps are chain-homotopy inverses and induce inverse maps on homology.

Proof status

This is a complete conditional chain-level argument. The hypotheses [A1]–[A2] are part of the statement and are not supplied by the restricted source treatments; the item does not claim a construction of regular data or compactifications from them.

Depends on

Dependency tree · 2 levels

Sources

Dietmar Salamon, Lectures on Floer Homology