TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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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]

Every continuous image of a compact space in a CW complex with its supplied characteristic maps lies in a finite CW subcomplex, without a choice premise (Compact CW images have finite cell support without choice).

[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

1.1F1F2given

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.

2.1F2F5step 1.1algebra

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 $d_{0} = 0$ .

3.1F1F2F5step 2.1algebra

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 $H_{n}^{c e l l} (X; G) ≅ H_{n} (X^{n}) / im \partial_{n + 1} ≅ H_{n} (X^{n + 1}; G) .$

4.1F3F4step 3.1

By [F3], all inclusions after $X^{n + 1}$ induce isomorphisms on $H_{n}$ . Each singular cycle is a finite sum of simplices; [F4] puts the image of each simplex in a finite subcomplex. Their finite union lies in some skeleton, so every class in $H_{n} (X; G)$ comes from a skeleton. If a skeletal cycle bounds in $X$ , its finite bounding chain likewise lies in some later skeleton, where that class already vanishes. Singular chain inclusions are injective, so these are respectively surjectivity and injectivity of the natural colimit map. Thus the colimit is $H_{n} (X; G)$ without any choice premise. Combining with step 3.1 proves the comparison for arbitrary, possibly infinite-dimensional, CW complexes.

5.1step 2.1step 3.1step 4.1given∎

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.

Depends on

Dependency tree · 2 levels

Sources

Allen Hatcher, Algebraic Topology, Theorem 2.35