CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A singular cochain need not have finite support on singular simplices

Statement refuted

Refuted claim: every singular cochain on a singular chain group is supported on only finitely many singular simplices.

On the interval $I = [0, 1]$ , define a homomorphism $φ : C_{0} (I; Z) \to Z$ by $φ (σ) = 1$ for every singular $0$ -simplex $σ : Δ^{0} \to I$ . This is a perfectly valid singular $0$ -cochain, but it is nonzero on every singular $0$ -simplex.

Facts & Assumptions

Given: The interval $I = [0, 1]$ .

[L1]

Singular $0$ -chains are finite integer linear combinations of singular $0$ -simplices (Singular simplices and singular chain groups with coefficients).

[L2]

For each point $x \in I$ , the constant map $c_{x} : Δ^{0} \to I$ is a singular $0$ -simplex (Singular simplices and singular chain groups with coefficients).

Counterexample

technique · direct

1.1

The assignment $φ (σ) = 1$ on each singular $0$ -simplex extends uniquely to a homomorphism from the free abelian group $C_{0} (I; Z)$ to $Z$ , so it is a singular $0$ -cochain in the usual dual-group sense.

L1given

2.1

By [L2], each point $x \in I$ determines a singular $0$ -simplex $c_{x}$ , and distinct points give distinct maps, so $I$ has infinitely many singular $0$ -simplices. The cochain $φ$ takes the value $1$ on every one of them, so its support is infinite.

L2step 1.1

3.1

Thus $φ$ is a singular cochain whose support is not finite, refuting the claim.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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 (standard reference, not scraped)