LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Torsion first homology forces twisted acyclicity for a deficiency-one exterior spine

Statement

Assume the Axiom of Choice convention of the twisted-chain definition. Let $X$ be a connected finite based two-dimensional CW complex with one $0$ -cell, $n \geq 2$ $1$ -cells and $n - 1$ $2$ -cells, so $χ (X) = 0$ . Let $G = π_{1} (X)$ , let $ψ : G ↠ H ≅ Z^{m}$ be an epimorphism with $m \geq 1$ , let $ρ : G \to {G L}_{d} (F)$ be a representation over a field $F$ , set $Λ = F [H]$ and $Q = Frac (Λ)$ , and give $Λ^{d}$ the right $F [G]$ -action $v \cdot g = v ρ (g) ψ (g)$ , with $ψ (g)$ acting by its central monomial. Form

$C_{*}^{ρ, ψ} (X; Λ^{d}) = Λ^{d} \otimes_{F [G]} C_{*} (\tilde{X}; F),$

where $\tilde{X}$ is the universal cover, and set $C_{*}^{ρ, ψ} (X; Q^{d}) = Q \otimes_{Λ} C_{*}^{ρ, ψ} (X; Λ^{d})$ . If every element of $H_{1} (C_{*}^{ρ, ψ} (X; Λ^{d}))$ is annihilated by some nonzero element of $Λ$ , then $C_{*}^{ρ, ψ} (X; Q^{d})$ is acyclic. No assumption that its degree-two homology vanishes before localization is required.

Statement references

Twisted coefficient complex and deck action

Conway, Lemma 2.6

Kirk and Livingston, §§3–4

Ranicki, Example 17

Facts & Assumptions

Given: AC; the finite based $2$ -complex and its cell counts; a positive-rank free abelian quotient $ψ : G ↠ H$ ; a field $F$ ; and the torsion hypothesis on $H_{1}$ over $Λ = F [H]$ .

[A1]

AC is inherited from the maximal-abelian-cover convention used to define the coefficient complex. The finite-dimensional linear-algebra argument below itself uses no choice principle. The Axiom of Choice

[F1]

The twisted complex is formed from the chosen based universal-cover cellular chains using the right action $v \cdot g = v ρ (g) ψ (g)$ ; the boundary of a lifted oriented $1$ -cell $x$ is multiplication by $ρ (x) ψ (x) - I_{d}$ . Twisted chain complex and based Reidemeister torsion for link exteriors

[F2]

The field of fractions of an integral domain is a field, and the canonical map embeds the domain. The field of fractions $Frac (D) = (D ∖ {0})^{- 1} D$ of an integral domain; $Frac (D)$ is a field and $d \mapsto d / 1$ embeds the integral domain $D$

Proof

1.1A1givenF2algebra

By [A1], use the prescribed AC-based twisted coefficient convention. Choose an ordered basis $t_{1}, \dots, t_{m}$ of $H$ . Then $Λ$ is the Laurent-polynomial ring $F [t_{1}^{\pm 1}, \dots, t_{m}^{\pm 1}]$ . Order exponent vectors lexicographically. For nonzero Laurent polynomials $f, g$ , their finite supports have largest exponents $a, b$ with nonzero coefficients $f_{a}, g_{b}$ ; the largest exponent of $f g$ is $a + b$ and its coefficient is $f_{a} g_{b} \neq 0$ . Thus $Λ$ is an integral domain, and [F2] makes $Q$ a field containing $Λ$ .

2.1givenF1step 1.1algebra

Let $z / s \in C_{1} \otimes_{Λ} Q$ be a fraction-field cycle, with $z \in C_{1}$ and $0 \neq s \in Λ$ . Since $d_{1} (z) / s = 0$ and each coordinate of $C_{0}$ lies in the domain $Λ$ , $d_{1} (z) = 0$ . The torsion hypothesis gives $0 \neq f \in Λ$ and $w \in C_{2}$ with $f z = d_{2} (w)$ . Hence $z / s = d_{2} (w / (f s))$ , so every fraction-field $1$ -cycle is a boundary and $H_{1} (C_{*} \otimes Q) = 0$ .

2.2givenF1step 1.1algebra

Because $ψ$ is onto a positive-rank free abelian group, some $1$ -cell generator $x$ has $ψ (x) = h \neq 1$ . Lift that oriented edge from the unique $0$ -cell to the universal-cover vertex; its cellular boundary is $(x - 1)$ times that vertex. Tensoring with the stated right action gives the corresponding $d_{1}$ block $A = ρ (x) h - I_{d}$ . Its determinant is a polynomial in the distinct monomials $1, h, \dots, h^{d}$ whose top coefficient is $\det ρ (x) \neq 0$ , so $\det A \neq 0$ in $Λ$ . Thus $A$ is invertible over $Q$ , this block makes $d_{1} : C_{1} \otimes Q \to C_{0} \otimes Q$ surjective, and $H_{0} (C_{*} \otimes Q) = 0$ .

3.1F1step 2.1step 2.2algebra∎

The fraction-field chain dimensions are $d (n - 1), d n, d$ , whose alternating sum is $d ((n - 1) - n + 1) = 0$ . Euler–Poincaré for this finite-dimensional complex gives $\dim_{Q} H_{0} - \dim_{Q} H_{1} + \dim_{Q} H_{2} = 0$ . Steps 2.1 and 2.2 make the first two terms zero, hence $H_{2} = 0$ ; there are no chain groups above degree $2$ . The localized complex is acyclic.

Depends on

Used by

Dependency tree · two levels

16 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

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, §§3–4 (standard reference, not scraped)
Andrew Ranicki, Notes on Reidemeister Torsion, Example 17 (standard reference, not scraped)