CounterexampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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A cohomology class obstructs a global symplectomorphism

Counterexample

Let $ω$ be the standard positive area form on $S^{2}$ . The symplectic manifolds $(S^{2}, ω)$ and $(S^{2}, 2 ω)$ are not symplectomorphic.

Facts & Assumptions

Given: The oriented sphere and its positive area form.

[F1]

A symplectomorphism $f$ must satisfy $f^{*} ω_{1} = ω_{0}$ . Symplectomorphisms, local symplectomorphisms, and symplectic embeddings.

[F2]

On a compact connected oriented surface, integration identifies top de Rham cohomology with $R$ . Integration is an isomorphism on top compactly supported de Rham cohomology.

[F3]

Positivity of the oriented integral gives a strictly positive choice-free finite-chart integral for a nonzero positive top form on compact $S^{2}$ .

[F4]

Change of variables on oriented manifolds gives the signed choice-free finite-chart integral identity for a diffeomorphism and a compactly supported top form, including the orientation-preserving case used here.

Verification

technique · direct

1.1F2F3givenalgebra

Both forms are closed and positive, hence symplectic. By [F3] the integral $A = \int_{S^{2}} ω$ is strictly positive; by [F2], their cohomology classes are distinct because their integrals are $A$ and $2 A$ .

2.1F1F4step 1.1∎

If $f^{*} (2 ω) = ω$ , then $f$ is orientation preserving because both forms are positive. The choice-free change-of-variables formula [F4] gives $A = \int_{S^{2}} f^{*} (2 ω) = 2 A$ , impossible. Thus [F1] fails for every diffeomorphism, exhibiting the global cohomology obstruction.

Depends on

Used by

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Sources

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