CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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A face cocycle need not be the meridian abelianization

Statement refuted

For a finite triangulation of the compact exterior of any nonempty oriented link, requiring only that the edge labels sum to zero around every oriented 2-simplex is sufficient to ensure that the labels represent the full meridian abelianization and are admissible input to the chain-model procedure.

Statement references

Build the maximal-abelian-cover chain complex from a marked triangulated exterior

Facts & Assumptions

Given: A nonempty oriented $r$ -component link with $r \geq 1$ , a finite triangulation of its compact exterior, its certified positive meridian edge loops, and the label checks in the chain-model procedure. Exhibit labels satisfying every face-sum check but not the prescribed meridian homomorphism.

[A1]

AC is the stated hypothesis of the full maximal-abelian-cover definition (The Axiom of Choice).

[F1]

The procedure requires edge reversal, zero sum around each oriented 2-simplex, and value $e_{i}$ around the $i$ th certified positive meridian loop; failure of any check returns invalid-labeling (Build the maximal-abelian-cover chain complex from a marked triangulated exterior).

[F2]

The full meridian abelianization sends the $r$ positive meridian basis classes to the standard basis of $Z^{r}$ (The maximal-abelian-cover Alexander module of an oriented link).

Counterexample

Assign the zero vector in $Z^{r}$ to every oriented edge. Edge reversal then holds, and the sum around every 2-simplex is zero. However every positive meridian loop also has label sum zero, which differs from $e_{i} \neq 0$ for every component. The induced homomorphism is trivial, not the full meridian abelianization, and the chain-model procedure must reject this labeling.

1.1givenF1algebra

For each oriented edge $e$ , the zero labeling gives $ℓ (\overset{ˉ}{e}) = 0 = - 0 = - ℓ (e)$ . Summing the three edge labels of any oriented 2-simplex also gives $0 + 0 + 0 = 0$ , so every face cocycle check passes.

2.1A1F1F2step 1.1algebra∎

Every edge path, including each certified meridian loop, has total label zero. Since $r \geq 1$ , each required standard basis vector $e_{i}$ is nonzero, so the meridian-image check in [F1] fails and the induced map is not the full meridian abelianization in [F2]. Therefore the face-sum condition alone is insufficient, and the procedure returns invalid-labeling on this witness.

Sources

Build the maximal-abelian-cover chain complex from a marked triangulated exterior

The maximal-abelian-cover Alexander module of an oriented link

The Axiom of Choice

Depends on

Used by

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Sources

Daniel S. Silver and Susan G. Williams, Mahler Measure, Links and Homology Growth, §4 (standard reference, not scraped)
Allen Hatcher, Algebraic Topology, §2.2 (standard reference, not scraped)