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 singular chain complex of a point

Example

Let $*$ be a one-point space. For each $n \geq 0$ there is exactly one singular $n$ -simplex $c_{n} : Δ^{n} \to *$ , so $C_{n} (*; Z) ≅ Z [c_{n}] .$ Moreover $\partial_{0} (c_{0}) = 0$ , and for $n \geq 1$ $\partial_{n} (c_{n}) = (\sum_{i = 0}^{n} (- 1)^{i}) c_{n - 1} = {\begin{cases} 0, & n odd, \\ c_{n - 1}, & n even . \end{cases}$

Hence $H_{0}^{s i n g} (*; Z) ≅ Z$ , all higher singular homology groups vanish, and the reduced singular homology groups are zero in every degree.

Facts & Assumptions

Given: The one-point space $*$ .

[L1]

Reduced singular homology is defined from the augmentation kernel in degree $0$ (Augmentation at 0-simplices and reduced singular homology).

[L2]

The singular boundary is the alternating sum of the face restrictions (The singular boundary operator).

Verification

technique · direct

1.1L2givenalgebra

Every map $Δ^{n} \to *$ is the same constant map $c_{n}$ , so each chain group is free of rank one on $c_{n}$ . By [L2], $\partial_{0} = 0$ , and for $n \geq 1$ one has $\partial_{n} (c_{n}) = \sum_{i = 0}^{n} (- 1)^{i} c_{n - 1},$ which is $0$ for odd $n$ and $c_{n - 1}$ for even $n$ .

2.1L1step 1.1∎

Therefore $\ker \partial_{n} = 0$ for even $n \geq 2$ and $\ker \partial_{n} = Z [c_{n}] = im \partial_{n + 1}$ for odd $n \geq 1$ , so $H_{n}^{s i n g} (*; Z) = 0$ for all $n > 0$ . Also $H_{0}^{s i n g} (*; Z) ≅ Z [c_{0}] ≅ Z$ . Since the augmentation of [L1] sends $c_{0}$ to $1$ , its kernel is $0$ , so the reduced degree-zero group also vanishes.

Depends on

Used by

The homology of an interval from contractibility Example

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)