False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Cohomologous symplectic forms on a noncompact manifold are always isotopic

Statement refuted

Cohomologous symplectic forms on a noncompact manifold are always related by a Moser isotopy.

Facts & Assumptions

Given: The proposed universal claim.

[F1]

Compact Moser stability requires compact $M$ ; its noncompact replacement requires primitives with one common compact support. Moser stability theorem, Compact-support Moser stability on a noncompact manifold.

Refutation

technique · direct

1.1

On $R^{2}$ let $ω_{0} = d x \land d y = d (x d y)$ and $ω_{1} = e^{- (x^{2} + y^{2})} d x \land d y = d (F d y)$ , where $F (x, y) = \int_{0}^{x} e^{- (s^{2} + y^{2})} d s$ . Both are symplectic and exact, hence cohomologous.

givenalgebra

2.1

Their total areas are respectively $+ \infty$ and $π$ . A diffeomorphism pulling $ω_{1}$ back to $ω_{0}$ would be orientation preserving and the change-of-variables formula would preserve total area, an impossibility. Thus no such symplectomorphism, and therefore no Moser isotopy, exists. The missing common-support/global-flow hypothesis in [F1] is substantive.

F1step 1.1∎

Depends on

Used by

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Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)