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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The standard volume form generates top cohomology of a sphere

Example

Under countable choice, for $n \geq 1$ the form $ω = \sum_{i = 1}^{n + 1} (- 1)^{i - 1} x_{i} d x_{1} \land \dots \land \hat{d x_{i}} \land \dots \land d x_{n + 1} ∣_{S^{n}}$ generates $H^{n} (S^{n})$ .

Facts & Assumptions

Given: Assume countable choice. The outward orientation on the unit sphere.

[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.

[F2]

Nonzero total integral obstructs exactness on a closed manifold: Let $M^{n}$ be compact, oriented, and boundaryless, $n \geq 1$ . A smooth top form $ω$ with $\int_{M} ω \neq 0$ is not exact. In particular every positive smooth top form on a nonempty such $M$ is not exact.

Verification

technique · direct

1.1givenalgebra

For tangent vectors $v_{1}, \dots, v_{n}$ , expansion along the first column gives $ω_{x} (v_{1}, \dots, v_{n}) = \det (x, v_{1}, \dots, v_{n})$ . The outward orientation is precisely the convention that this determinant is positive on positive tangent bases. Since $x$ is a nonzero normal to the tangent space, the determinant is nonzero on every tangent basis; thus $ω$ is smooth, positive and nowhere zero.

2.1F1F2step 1.1∎

The sphere is nonempty, compact, oriented and boundaryless, so the positive-top-form clause of the integral obstruction theorem makes $ω$ nonexact. It is closed by top degree. The sphere computation gives a one-dimensional $H^{n}$ , and its nonzero class therefore generates it.

Source locator

Lee, Theorem 17.21, pp.450–451, and Proposition 16.28, p.422, positivity of volume integration; the proof verifies nonexactness by the stated Stokes supplier.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

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