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

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Additivity and compact cell support control the infinite cw colimit

Statement

For every ordinary theory $h$ with arbitrary additivity and every CW pair $(X, A)$ , the canonical map ${colim}_{i \geq 0} h_{n} (X^{i}, A^{i}) ⟶ h_{n} (X, A)$ is an isomorphism. So is the canonical colimit over finite subcomplex pairs $(K, K \cap A)$ of $X$ . The latter identification is natural for every continuous map of CW pairs.

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

Let $X = U \cup V$ be a cover by CW subcomplexes and $W = U \cap V$ . Every ordinary homology theory has a natural exact sequence $\dots \to h_{n} (W) \overset{(i_{*}, - j_{*})}{\to} h_{n} (U) \oplus h_{n} (V) \overset{a_{*} + b_{*}}{\to} h_{n} (X) \overset{Δ}{\to} h_{n - 1} (W) \to \dots,$ where all four maps $i, j, a, b$ are inclusions. The same sequence holds for a CW pair $(X, C)$ covered by $(U, C \cap U)$ and $(V, C \cap V)$ , with the corresponding relative groups. (Ordinary homology theories have mayer vietoris for cw covers)

[F2]

For every CW pair $(X, A)$ , the skeletal telescope projection $p : (T_{X}, T_{A}) \to (X, A)$ is a homotopy equivalence of pairs. (The skeletal telescope projects by a homotopy equivalence of pairs)

[F3]

For any sequence of abelian groups $G_{0} \overset{u_{0}}{\to} G_{1} \overset{u_{1}}{\to} \dots$ , let $D = ⨁_{i \geq 0} G_{i}$ and let $s : D \to D$ send the $i$ th coordinate by $u_{i}$ into coordinate $i + 1$ . Then $0 ⟶ D \overset{1 - s}{\to} D ⟶ {colim}_{i} G_{i} ⟶ 0$ is exact, where the last map sums the canonical maps to the colimit. The maps $u_{i}$ need not be injective. (A sequential abelian colimit is the cokernel of one minus shift)

[F4]

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 $n$ . (Finite dimensional axiomatic homology has finite subcomplex support)

Proof

1.1

Use the telescope from F2. Subdivide its height intervals at half-integers. Let $B_{0} = X^{0} \times [0, 1]$ , and for $i \geq 1$ let $B_{i} = (X^{i - 1} \times [i - 1 / 2, i]) \cup (X^{i} \times [i, i + 1])$ . Even $B_{i}$ form a disjoint-union subcomplex $U$ , and odd $B_{i}$ form a disjoint-union subcomplex $V$ ; they cover the telescope. Their intersection is the disjoint union $C_{i} = X^{i} \times [i + 1 / 2, i + 1]$ . Each $B_{i}$ retracts onto $X^{i}$ at height $i$ , and each $C_{i}$ onto $X^{i}$ , with all retractions preserving the corresponding pieces of $T_{A}$ .

F2given

2.1

Apply relative Mayer–Vietoris F1 and arbitrary additivity. After the retractions, the overlap map has from the $i$ th summand the identity into stage $i$ and the inclusion-induced map into stage $i + 1$ , with opposite signs. Multiplying each odd-indexed overlap summand by $- 1$ , and reordering the target even/odd direct sums by stage, identifies it with $1 - s$ on $⨁_{i} h_{n} (X^{i}, A^{i})$ . These changes of signs leave the outgoing sum map equal to the canonical stage-to-telescope map.

F1step 1.1

3.1

F3 says $1 - s$ is injective also in degree $n - 1$ . Exactness therefore makes the map from the stage direct sum onto telescope homology surjective, with kernel $im (1 - s)$ . Its cokernel is the sequential colimit by F3. F2 identifies telescope homology with $h_{n} (X, A)$ by the actual projection. Its composite on every stage is the canonical inclusion, so the isomorphism obtained is the claimed canonical map.

F2F3step 2.1

4.1

Each skeletal pair is finite-dimensional, so F4 identifies its group with the colimit of the finite subcomplex pairs it contains. Every finite CW subcomplex of X lies in some skeleton, as its finitely many cells have bounded dimensions. Thus the iterated colimit is precisely the colimit over all finite subcomplex pairs, proving that assertion.

F4step 3.1

5.1

A continuous map takes a finite CW subcomplex into a finite subcomplex by compact-cell support, the same fact used in F4. Restrict the map to those finite pairs and use ordinary functoriality; their maps to the full pair commute. Since every class has finite support, this proves naturality for arbitrary maps, without assuming such maps preserve skeleta. Empty X and all zero groups give zero colimits, and all degrees, including negative ones, are covered by the same exact sequences.

F4step 4.1∎

Depends on

Used by

Finite additivity alone does not prove infinite cw uniqueness Counterexample
Eilenberg steenrod uniqueness on all cw pairs Theorem

Dependency tree · two levels

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Sources

May, A Concise Course in Algebraic Topology, 14§6, theorem and telescope proof pp.114–116 (standard reference, not scraped)
Hatcher, Algebraic Topology, Theorem 3F.8 pp.314–315 (standard reference, not scraped)