ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The angular form generates the first de rham cohomology of the circle

Example

Under countable choice, $α = (x d y - y d x) / (2 π) ∣_{S^{1}}$ has period one and its class generates $H_{d R}^{1} (S^{1})$ .

Facts & Assumptions

Given: Assume countable choice. The counterclockwise oriented unit circle and the displayed one-form.

[F1]

De rham cohomology of spheres: Assume countable choice. For $n \geq 1$ , $H_{d R}^{k} (S^{n})$ is $R$ in degrees $0, n$ and zero otherwise. For $S^{0}$ it is $R^{2}$ in degree zero and zero otherwise.

[F3]

A nonzero period obstructs exactness and bounding: Let $S ↪ M$ be an oriented compact boundaryless embedded $k$ -submanifold, $k \geq 1$ , and let $ω$ be a closed smooth $k$ -form on $M$ . If $\int_{S} ω \neq 0$ , then $ω$ is not exact on $M$ , and $S$ cannot be the induced oriented boundary of a compact embedded $(k + 1)$ -submanifold of $M$ .

Verification

technique · direct

1.1

The form is smooth and closed because two-forms on a one-manifold vanish. For $γ (t) = (\cos t, \sin t)$ , $0 \leq t \leq 2 π$ , substitution gives $γ^{*} α = d t / (2 π)$ , hence $\int_{S^{1}} α = 1$ .

givenalgebra

2.1

The circle is compact, oriented, boundaryless and embedded, and the form is closed, so its nonzero period obstructs exactness. Since the sphere theorem gives $\dim H^{1} (S^{1}) = 1$ , this nonzero class is a basis.

F1F3step 1.1∎

Source locator

Lee, angular form (17.1), p.441, and Theorem 17.21, pp.450–451; the period is calculated explicitly.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

Nigel Hitchin, Differentiable Manifolds (2014) (standard reference, not scraped)
John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)