TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07‡ 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.

Relative cellular homology computes relative singular homology

Statement

For a CW pair $(X, A)$ , the quotient cellular complex $C_{*}^{c e l l} (X; G) / C_{*}^{c e l l} (A; G)$ , whose degree- $n$ group is $⨁_{e^{n} \subseteq X ∖ A} G$ , computes $H_{*} (X, A; G)$ .

Facts & Assumptions

Given: A CW pair $(X, A)$ .

Proof

technique · direct

1.1

If $A = \emptyset$ , the assertion is the absolute cellular-homology theorem. Suppose $A \neq \emptyset$ . The quotient $X / A$ is a CW complex with one base vertex coming from $A$ and exactly the cells of $X ∖ A$ otherwise. Consequently its reduced cellular complex is canonically $C_{*}^{c e l l} (X; G) / C_{*}^{c e l l} (A; G)$ , including in degree zero.

given

2.1

A CW subcomplex is closed. The cellwise radial collar construction, assembled over the skeleta by the weak topology, gives an open neighborhood $V$ of $A$ that deformation retracts onto $A$ while fixing $A$ ; equivalently, every nonempty CW pair is a good pair. Therefore Good pairs and quotient reduced homology identifies $H_{*} (X, A; G)$ with ${\tilde{H}}_{*} (X / A; G)$ . Applying Cellular homology computes singular homology to $X / A$ and using step 1.1 proves the claim. Under this identification, the triple connecting maps for the relative skeleta are exactly the quotient cellular differential.

step 1.1construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

J. Peter May, A Concise Course in Algebraic Topology, Chapter 13 (standard reference, not scraped)
Allen Hatcher, Algebraic Topology, Proposition A.5 and Section 2.1 (standard reference, not scraped)