ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

The homology of punctured Euclidean space by deformation retraction

Example

For every $n \geq 1$ and every abelian group $G$ , the punctured space $R^{n} ∖ {0}$ has the same singular homology groups as $S^{n - 1}$ : $H_{k}^{s i n g} (R^{n} ∖ {0}; G) ≅ H_{k}^{s i n g} (S^{n - 1}; G) (k \geq 0) .$

Facts & Assumptions

Given: An integer $n \geq 1$ and an abelian group $G$ .

[L1]

$R^{n} ∖ {0}$ deformation retracts onto $S^{n - 1}$ (For $n \geq 1$ , radial normalisation is a deformation retraction of $R^{n} ∖ {0}$ onto $S^{n - 1}$ ).

[L2]

A deformation retract inclusion induces an isomorphism on singular homology (Singular homology is invariant under deformation retracts).

Verification

technique · direct

1.1L1given

By [L1], the sphere inclusion $S^{n - 1} ↪ R^{n} ∖ {0}$ is a deformation retract inclusion.

2.1L2step 1.1∎

Applying [L2] to that inclusion gives isomorphisms on singular homology in every degree, which is exactly the displayed statement.

Depends on

Used by

Nothing in the library uses this result yet.

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 (standard reference, not scraped)