ExampleConstruction: Literature-sourcedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-04

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 simplicial homology of a point and an edge

Example

For the one-vertex complex ${\emptyset, {v}}$ , one has $C_{0} ≅ Z [v]$ and $C_{n} = 0$ for $n \geq 1$ . Hence $H_{0}^{s i m p} ≅ Z$ and $H_{n}^{s i m p} = 0$ for $n > 0$ . The augmentation $ε ([v]) = 1$ is an isomorphism, so ${\tilde{H}}_{n}^{s i m p} = 0$ for every $n$ .

For the one-edge complex with vertices $v_{0}, v_{1}$ and edge $[v_{0}, v_{1}]$ , one has $C_{1} ≅ Z [v_{0}, v_{1}]$ , $C_{0} ≅ Z [v_{0}] \oplus Z [v_{1}]$ , and $\partial [v_{0}, v_{1}] = [v_{1}] - [v_{0}]$ . The kernel of $\partial_{1}$ is $0$ , so $H_{1}^{s i m p} = 0$ . The quotient of $C_{0}$ by the subgroup generated by $[v_{1}] - [v_{0}]$ is $Z$ , so $H_{0}^{s i m p} ≅ Z$ . The augmented complex is exact, hence the reduced homology also vanishes.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)
Vidit Nanda, Computational Algebraic Topology, Lecture 03: Homology (standard reference, not scraped)