TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 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.

Cellular homology computes singular homology

Statement

For every CW complex $X$ , abelian group $G$ , and $n \geq 0$ , the cellular complex has an isomorphism $H_{n}^{c e l l} (X; G) ≅ H_{n} (X; G)$ , natural with respect to cellular maps.

Facts & Assumptions

Given: A CW complex $X$ , an abelian group $G$ , and $n \geq 0$ . Write $X^{- 1} = \emptyset$ , suppress $G$ in homology notation, and write $C_{m} = H_{m} (X^{m}, X^{m - 1}; G)$ .

[F1]

Consecutive relative skeletal homology is concentrated in the cell dimension (Relative homology of consecutive CW skeleta).

[F2]

Each skeletal pair has its long exact homology sequence (Long exact sequence of a pair).

[F3]

Skeletal inclusions induce isomorphisms in degrees below the old skeleton dimension (Skeletal homology stabilizes away from the cell dimension).

[F4]

The natural colimit of skeletal homology is singular homology of $X$ , also for infinite CW complexes (Homology of an infinite CW complex is the colimit of skeletal homology).

[F5]

Cellular homology is $\ker d_{n} / im d_{n + 1}$ (Cellular homology); $d_{m} = i_{m - 1} \partial_{m}$ , where $\partial_{m} : C_{m} \to H_{m - 1} (X^{m - 1})$ is the pair connecting map and $i_{m - 1} : H_{m - 1} (X^{m - 1}) \to C_{m - 1}$ is the relative quotient map (Cellular boundary from three consecutive skeleta).

Proof

technique · direct

1.1

For every finite $m \geq 0$ , $H_{k} (X^{m}) = 0$ when $k > m$ : begin with the discrete $X^{0}$ case of [F1], and use [F2] successively with [F1] to pass from $X^{m - 1}$ to $X^{m}$ . The empty skeleton also has zero homology.

F1F2given

2.1

For $n \geq 1$ , the exact sequence of $(X^{n}, X^{n - 1})$ and step 1.1 give an injection $i_{n} : H_{n} (X^{n}) \to C_{n}$ with image $\ker \partial_{n}$ . The same argument one degree lower makes $i_{n - 1}$ injective, including $i_{0} : H_{0} (X^{0}) \to H_{0} (X^{0}, \emptyset)$ , which is an isomorphism. Since $d_{n} = i_{n - 1} \partial_{n}$ , we obtain $\ker d_{n} = \ker \partial_{n} = i_{n} H_{n} (X^{n})$ . For $n = 0$ this identity follows directly from $i_{0}$ being an isomorphism and $d_{0} = 0$ .

F2F5step 1.1algebra

3.1

For $(X^{n + 1}, X^{n})$ , exactness and $H_{n} (X^{n + 1}, X^{n}) = 0$ give the exact tail $C_{n + 1} \overset{\partial_{n + 1}}{\to} H_{n} (X^{n}) ⟶ H_{n} (X^{n + 1}) ⟶ 0.$ Under the injection $i_{n}$ of step 2.1, $im \partial_{n + 1}$ corresponds exactly to $im d_{n + 1}$ by [F5]. Taking the quotient therefore yields $H_{n}^{c e l l} (X; G) ≅ H_{n} (X^{n}) / im \partial_{n + 1} ≅ H_{n} (X^{n + 1}; G) .$

F1F2F5step 2.1algebra

4.1

By [F3], all inclusions after $X^{n + 1}$ induce isomorphisms on $H_{n}$ . By [F4], their colimit is $H_{n} (X; G)$ . Combining with step 3.1 proves the comparison for arbitrary, possibly infinite-dimensional, CW complexes.

F3F4step 3.1

5.1

A cellular map preserves all skeleta. Its induced chain maps commute with inclusions, quotient maps and connecting homomorphisms: the latter send a relative cycle represented by $c$ to the class of $\partial c$ , and a chain map commutes with $\partial$ . Consequently every injection, quotient identification and colimit map in steps 2.1--4.1 commutes with cellular maps. The comparison is therefore natural in exactly the stated sense.

step 2.1step 3.1step 4.1given∎

Depends on

Used by

A CW complex with no cells in adjacent dimensions has zero cellular boundary Corollary
Cellular boundary matrix of a closed orientable surface Example
Cellular homology and ring-independent groups of complex projective space Example
Cellular homology of a lens space Example
Cellular homology of real projective space Example
Cellular maps induce cellular chain maps Proposition
Euler–Poincare formula for finite CW complexes Theorem
Relative cellular homology computes relative singular homology Theorem

Dependency tree · two levels

18 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 2.35 (standard reference, not scraped)