Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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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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Skeletal homology stabilizes away from the cell dimension

Statement

Let X be a CW complex, let G be an abelian group, and let n≥0. The skeletal inclusion induces an isomorphism Hk(Xn;G)→Hk(Xn+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 (Xn+1,Xn).

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 Hk(Xn) vanish for k<n, and only the outgoing one need vanish for k=n; exactness gives respectively an isomorphism and a surjection.

Depends on

Used by

Dependency tree · two levels

11 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