ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Relative homology of a disk and its boundary

Example

For $n \geq 1$ , $H_{k} (D^{n}, S^{n - 1}; G)$ is $G$ for $k = n$ and zero otherwise. If $c_{g}$ is the oriented relative fundamental cycle with coefficient $g \in G$ , our convention has $δ [c_{g}] = [\partial c_{g}]$ , the corresponding reduced fundamental class of $S^{n - 1}$ .

Facts & Assumptions

Given: The pair $(D^{n}, S^{n - 1})$ with $n \geq 1$ .

Verification

technique · direct

1.1givenconstruct

The disk is contractible, and the sphere has reduced homology only in degree $n - 1$ .

2.1step 1.1algebra∎

For $n > 1$ , exactness makes $δ : H_{n} (D^{n}, S^{n - 1}; G) \to H_{n - 1} (S^{n - 1}; G)$ an isomorphism. For $n = 1$ , it identifies the relative group with the augmentation kernel ${\tilde{H}}_{0} (S^{0}; G) \subset H_{0} (S^{0}; G)$ rather than with all of $H_{0} (S^{0}; G)$ . Exactness forces every other relative group to vanish, and the cycle formula gives $δ [c_{g}] = [\partial c_{g}]$ with the stated sign.

Depends on

Used by

Relative homology is not homology of the set difference Counterexample

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

Allen Hatcher, Algebraic Topology, Example 2.17 (standard reference, not scraped)