LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Finite dimensional axiomatic homology has finite subcomplex support

Statement

For a finite-dimensional CW pair $(X, A)$ and ordinary $h$ , the canonical map $\underset{K \subset X finite subcomplex}{colim} h_{n} (K, K \cap A) ⟶ h_{n} (X, A)$ is an isomorphism for every integer $n$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

For every finite-dimensional CW pair $(X, A)$ and ordinary theory $h$ , there is a canonical isomorphism $h_{n} (X, A) ≅ H_{n} (C_{*}^{h} (X, A))$ for every integer $n$ , natural for cellular maps. It is the skeletal lift isomorphism described below and commutes with the homology connecting maps of pairs. The number of cells need not be finite. (Finite dimensional skeletal exactness computes axiomatic homology)

[F2]

For a CW pair $(X, A)$ , an ordinary theory $h$ with coefficient $G$ , and chosen cell orientations, the complex $C_{*}^{h} (X, A)$ is canonically $C_{*}^{c e l l} (X, A; Z) \otimes G .$ Its differential is the integral incidence matrix acting on $G$ . In degree one the entries are signed terminal-minus-initial endpoints. The direct-sum matrices have finite support in each column. (Axiomatic cellular boundaries are integral incidence matrices with coefficients)

[F3]

If $K$ is compact and $f : K \to X$ is continuous into a CW complex, then $f (K)$ lies in a finite CW subcomplex of $X$ . (The image of a compact space lies in a finite CW subcomplex)

Proof

1.1

By F1 and F2, compute each of these groups using its oriented cellular direct-sum complex with coefficients $G$ . Subcomplex inclusion preserves the basis cells and their incidence coefficients. The inclusion from a finite subcomplex therefore gives the actual inclusion of its relative cellular chains into those of $(X, A)$ .

F1F2

2.1

A cycle in the latter complex has finite support. Include the closures of its support cells in a finite CW subcomplex $K$ : each closed cell is the compact image of a disk, and F3 places it in a finite subcomplex; a finite union of these remains finite. The cycle equation is unchanged in this subcomplex, so its homology class comes from $K$ .

F3step 1.1

3.1

If a class from $K$ maps to zero, its representing cycle bounds a finite-support cellular chain in $X$ . Enlarge $K$ to a finite subcomplex containing the closures of that chain's support cells. There the same boundary equation already witnesses zero. This is exactly injectivity of the colimit map. Finite unions show the indexing collection is directed, with the empty subcomplex included; zero complexes and negative degrees cause no exception.

F3step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

9 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

May, A Concise Course in Algebraic Topology, 15§2, cellular calculation pp.119–120 (standard reference, not scraped)
Hatcher, Algebraic Topology, Lemma 2.34 p.138, finite support reasoning (standard reference, not scraped)