TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Simplicial and singular homology agree

Statement

For every simplicial complex $K$ , the natural simplicial-to-singular chain map induces $H_{n}^{s i m p} (K; G) ≅ H_{n} (∣ K ∣; G)$ for all $n$ .

Facts & Assumptions

Given: A simplicial complex $K$ and the natural simplicial-to-singular map.

Proof

technique · induction

1.1

Extend the simplicial chains, the characteristic-simplex map, and its boundary identity $G$ -linearly. For a finite-dimensional complex, filter by skeleta. The relative comparison for $(K^{r}, K^{r - 1})$ is an isomorphism: both relative theories are a direct sum of one copy of $G$ for each $r$ -simplex in degree $r$ and vanish in the other degrees.

givenbaseconstruct

2.1

The simplicial and singular pair long exact sequences commute with this comparison. Skeletal induction and the five lemma therefore give an isomorphism for every finite-dimensional $K$ .

step 1.1ihalgebra

3.1

A finite singular cycle, and likewise a finite chain witnessing a boundary, has compact image. In a simplicial complex this image meets only finitely many open simplices and is contained in a finite-dimensional skeleton. The finite-dimensional result gives respectively surjectivity and injectivity for arbitrary $K$ . The characteristic-simplex construction commutes with simplicial maps, so the isomorphism is natural.

step 2.1discharge-induction∎

Depends on

Used by

Homology of spheres Corollary

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

Allen Hatcher, Algebraic Topology, Theorem 2.27 (standard reference, not scraped)