LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Torsion first homology makes the localized exterior spine complex acyclic

Statement

Assume the Axiom of Choice convention of the twisted-chain definition. Let $L$ be a nonempty oriented link and let $X$ be a certified finite based two-dimensional spine of its compact exterior with one $0$ -cell, $n \geq 2$ $1$ -cells and $n - 1$ $2$ -cells. Let $ψ : π_{1} (X) ↠ H ≅ Z^{m}$ be an epimorphism with $m \geq 1$ , let $ρ : π_{1} (X) \to {G L}_{d} (F)$ be a representation over a field, and form the twisted chain complex over $Λ = F [H]$ and its localization over $Q = Frac (Λ)$ . If its degree-one homology over $Λ$ is a torsion module, then its fraction-field complex is acyclic.

Statement references

Twisted exterior chain complex

General localized acyclicity lemma

Conway, Lemma 2.6

Kirk and Livingston, §3

Facts & Assumptions

Given: AC; a based two-dimensional simple spine of a nonempty link exterior with one $0$ -cell, $n$ $1$ -cells and $n - 1$ $2$ -cells; a positive-rank free abelian deck quotient; a field-valued representation; and the torsion hypothesis on degree-one homology.

[A1]

AC is inherited from the full abelianization convention. The Axiom of Choice

[F1]

The based twisted complex is built from the exterior spine and its prescribed coefficient action; a certified spine has the same based simple-homotopy type as the selected exterior cell structure. Twisted chain complex and based Reidemeister torsion for link exteriors; The admissible Wada quotient of a deficiency-one exterior spine

[F2]

For a finite based two-dimensional complex with one $0$ -cell, $n$ $1$ -cells and $n - 1$ $2$ -cells, positive-rank free-abelian deck map and torsion $H_{1}$ over the Laurent coefficient ring, localization yields an acyclic complex. Torsion first homology forces twisted acyclicity for a deficiency-one exterior spine

Proof

1.1A1F1F2givenalgebra

The stated cell counts give $χ (X) = 1 - n + (n - 1) = 0$ , and the two-dimensional spine has no cellular chain groups above degree $2$ . The declared deck group has positive rank, the coefficient field and representation are as in [F1], and the degree-one torsion hypothesis is exactly the remaining module assumption of [F2].

2.1F1F2step 1.1∎

Apply [F2] to this $X$ , $ρ$ , and $ψ$ . It gives $H_{i} (C_{*}^{ρ, ψ} (X; Q^{d})) = 0$ in every degree; hence the fraction-field complex is acyclic.

Depends on

Used by

A certified knot-spine Wada quotient equals twisted Reidemeister torsion Theorem

Dependency tree · two levels

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Sources

Anthony Conway, Burau Maps and Twisted Alexander Polynomials, Lemma 2.6 (standard reference, not scraped)
Paul Kirk and Charles Livingston, Twisted Alexander Invariants, Reidemeister Torsion, and Casson–Gordon Invariants, §3 (standard reference, not scraped)