ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedaudited 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.

Relative homology of a handle by excision

Example

For a single $2$ -handle crossing in a $4$ -manifold satisfying the compact-band hypotheses (and ${A C}_{ω}$ ), collar excision reduces the relative homology to $(D^{2} \times D^{2}, S^{1} \times D^{2})$ and then to $(D^{2}, S^{1})$ . With any abelian coefficients $G$ , the result is $G$ in degree $2$ and zero in every other degree.

Facts & Assumptions

[F1]

Relative homology of a single handle pair: Assume ${A C}_{ω}$ and the one-critical-point compact-band hypotheses, with critical index $k$ . For every abelian group $G$ and $i \geq 0$ , $H_{i} (M^{b}, M^{a}; G) ≅ G$ if $i = k$ and zero otherwise. In particular this holds for the additive group of any coefficient ring. No orientation of $M$ is needed.

[F2]

Relative homology of the standard handle pair: For any abelian group $G$ , integers $0 \leq k \leq n$ , and $i \geq 0$ , the standard handle pair has $H_{i} (D^{k} \times D^{n - k}, S^{k - 1} \times D^{n - k}; G) ≅ G$ if $i = k$ and zero otherwise. Here $D^{0}$ is a point and $S^{- 1} = \emptyset$ .

Verification

Given: The objects and hypotheses in the example.

1.1

The collar-excision argument for a single critical point replaces the sublevel pair in relative homology by the standard handle pair with $k = 2, n = 4$ . It uses an open collar thickening of the lower sublevel before excision, so the closure-in-interior requirement is met.

2.1

The explicit pair homotopy $(u, v) \mapsto (u, (1 - t) v)$ contracts the second disk factor. The standard-pair calculation then identifies $H_{2} (D^{2}, S^{1}; G)$ with $H_{1} (S^{1}; G) = G$ . In degree zero the map from the circle to the disk is the identity on $G$ , so the relative degree-zero group and degree-one group vanish; all higher groups except degree two vanish as well. This includes $G = 0$ .

F2step 1.1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Audin–Damian, Morse Theory and Floer Homology (standard reference, not scraped)