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The evaluated Fox Jacobian is the twisted cellular boundary

Statement

Let $X$ be the based presentation $2$ -complex with one vertex, oriented $1$ -cells $x_{1}, \dots, x_{n}$ , and oriented $2$ -cells attached along words $r_{1}, \dots, r_{m}$ in the free group $F_{n}$ . Let $G = ⟨ x_{1}, \dots, x_{n} ∣ r_{1}, \dots, r_{m} ⟩$ be its presented fundamental group, let $ρ : G \to {G L}_{d} (F)$ be a representation, and let $ψ : G \to H$ be the declared free-abelian deck map. For $Φ : F [F_{n}] \to M_{d} (Q)$ induced by $x_{j} \mapsto ρ (x_{j}) ψ (x_{j})$ , the twisted differential $\partial_{1}$ has blocks $ρ (x_{j}) ψ (x_{j}) - I_{d}$ , and the $i$ th block row of $\partial_{2}$ is

$(Φ (\frac{\partial r_{i}}{\partial x_{1}}), \dots, Φ (\frac{\partial r_{i}}{\partial x_{n}})),$

where coefficient vectors are written as rows and differentials act on the right: the matrix of $\partial_{1}$ is the stacked column of blocks $ρ (x_{j}) ψ (x_{j}) - I_{d}$ , while the matrix of $\partial_{2}$ is the $m d$ -by- $n d$ block matrix whose $i$ th row is the displayed Fox row. The Fox derivatives use the product and inverse recursions in the Wada definition. Thus the evaluated Fox-Jacobian blocks are exactly the cellular boundary blocks of the based twisted complex of this presentation $2$ -complex.

Statement references

Twisted chain convention

Fox derivative and Wada matrix conventions

Conway, §§2.4 and 3.1

Kirk and Livingston, §4

Ranicki, pp. 3–5

Facts & Assumptions

Given: The finite based presentation $2$ -complex, its presentation generators and relators, $ρ$ , $ψ$ , and the twisted coefficient map $Φ$ .

[A1]

AC is inherited from the twisted-chain convention; the finite word calculation uses no choice principle. The Axiom of Choice

[F1]

A presentation $2$ -complex has one vertex, one oriented $1$ -cell for each generator, and one oriented $2$ -cell attached by each relator word. Group presentation by generators and relations; Free group on a set of generators

[F2]

Fox derivatives satisfy $\partial x_{k} / \partial x_{j} = δ_{k j}$ , $\partial (u v) / \partial x_{j} = \partial u / \partial x_{j} + u \partial v / \partial x_{j}$ , and $\partial x_{k}^{- 1} / \partial x_{j} = - x_{k}^{- 1} δ_{k j}$ . The admissible Wada quotient of a deficiency-one exterior spine

[F3]

The group ring has basis indexed by group elements and multiplication $[g] [h] = [g h]$ ; the coefficient map $Φ$ is the ring homomorphism prescribed in the Wada definition. The group ring $R [G]$ of finitely supported formal $R$ -linear combinations of group elements; The group ring $R [G]$ is a unital $R$ -algebra with basis $G$ , and each $g \in G$ is a unit of $R [G]$ ; Twisted chain complex and based Reidemeister torsion for link exteriors

Proof

1.1F2F3giveninduction

For every word $w \in F_{n}$ and each generator index $j$ , the recursive Fox derivative satisfies $\sum_{j = 1}^{n} \frac{\partial w}{\partial x_{j}} (x_{j} - 1) = w - 1$ . For $w = x_{k}$ this is $x_{k} - 1$ , and for $w = x_{k}^{- 1}$ it is $- x_{k}^{- 1} (x_{k} - 1) = x_{k}^{- 1} - 1$ . If it holds for $u$ and $v$ , then the product recursion in [F2] makes the sum for $u v$ equal $(u - 1) + u (v - 1) = u v - 1$ . Induction on the length of a reduced word proves the identity.

1.2A1F1F3givenconstruct

Under the coefficient convention [A1], lift the vertex of $X$ to $\tilde{v}$ . The lift of the oriented edge $x_{j}$ from $\tilde{v}$ ends at $x_{j} \tilde{v}$ , so its cellular boundary is $(x_{j} - 1) \tilde{v}$ . Tensoring with the right coefficient module of the twisted-chain definition sends this block to $ρ (x_{j}) ψ (x_{j}) - I_{d}$ . Thus the $j$ th block of $\partial_{1}$ is the stated matrix in the stacked-block convention.

2.1F1F2F3step 1.1construct

Write a relator $r_{i}$ as a finite word in letters $x_{j}^{\pm 1}$ . Along its lifted attaching path, a positive occurrence of $x_{j}$ after prefix $u$ traverses the translate $u {\tilde{e}}_{j}$ and contributes $u$ to the $x_{j}$ coefficient; a negative occurrence $x_{j}^{- 1}$ after prefix $u$ traverses the reverse of $u x_{j}^{- 1} {\tilde{e}}_{j}$ and contributes $- u x_{j}^{- 1}$ . The product/inverse recursions [F2] give exactly these same contributions to $\partial r_{i} / \partial x_{j}$ term by term. Summing the finitely many occurrences shows $\partial_{2} ({\tilde{e}}_{r_{i}}) = \sum_{j} (\partial r_{i} / \partial x_{j}) {\tilde{e}}_{x_{j}}$ .

3.1F1F3step 1.2step 2.1algebra∎

Tensor the displayed cellular boundary with the coefficient module and extend scalars to $Q$ . Since tensoring sends each group-ring coefficient to its image under the ring homomorphism $Φ$ , the resulting $i$ th block row is exactly $(Φ (\partial r_{i} / \partial x_{1}), \dots, Φ (\partial r_{i} / \partial x_{n}))$ . This is the asserted twisted cellular boundary matrix.

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Sources

Anthony Conway, Burau Maps and Twisted Alexander Polynomials, §§2.4 and 3.1 (standard reference, not scraped)
Paul Kirk and Charles Livingston, Twisted Alexander Invariants, Reidemeister Torsion, and Casson–Gordon Invariants, §4 (standard reference, not scraped)
Andrew Ranicki, Notes on Reidemeister Torsion, pp. 3–5 (standard reference, not scraped)