LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Skeletal homology stabilizes away from the cell dimension

Statement

Let $X$ be a CW complex, let $G$ be an abelian group, and let $n \geq 0$ . The skeletal inclusion induces an isomorphism $H_{k} (X^{n}; G) \to H_{k} (X^{n + 1}; G)$ for $k < n$ and a surjection for $k = n$ .

Facts & Assumptions

Given: A CW complex $X$ , an abelian coefficient group $G$ , and the consecutive-skeleton pair $(X^{n + 1}, X^{n})$ .

Proof

technique · direct

1.1given

Its relative homology vanishes outside degree $n + 1$ by Relative homology of consecutive CW skeleta.

2.1step 1.1∎

In the pair long exact sequence, the two relative groups adjacent to $H_{k} (X^{n})$ vanish for $k < n$ , and only the outgoing one need vanish for $k = n$ ; exactness gives respectively an isomorphism and a surjection.

Depends on

Dependency tree · 2 levels

Sources

Allen Hatcher, Algebraic Topology, Section 2.2