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

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PTFA Ore localization alone does not imply Blanchfield nonsingularity

Statement refuted

For every closed oriented three-manifold $M$ , PTFA coefficient homomorphism $ϕ : π_{1} (M) \to Γ$ , group ring $Λ = Z Γ$ and fraction skew field $K$ , the condition $H_{1} (M; K) = 0$ implies that the Blanchfield pairing on $T H_{1} (M; Λ)$ is nonsingular over $Λ$ .

Statement references

COT, Theorem 2.13, printed page 451

Facts & Assumptions

Given: $M = T^{3} = (R / Z)^{3}$ with its product orientation, $Γ = Z^{2} = ⟨ x, y ⟩$ , and $ϕ (m, n, p) = (m, n)$ . Put $Λ = Z [x^{\pm 1}, y^{\pm 1}]$ , $K = Q (x, y)$ , $a = x - 1$ , $b = y - 1$ , and $I = (a, b)$ .

[F1]

Supplied lifted oriented cells compute local homology by their group-ring incidence boundaries. Cellular chains compute local homology

Counterexample

1.1F1givenconstructalgebra

The group $Γ$ is torsion-free abelian, hence PTFA with its one-step series. The ring $Λ$ is a commutative domain: after multiplying nonzero Laurent polynomials by monomials, lexicographically leading nonzero terms of ordinary polynomials have nonzero product over $Z$ . Its fraction field is $K$ . Give each circle its one-vertex, one-edge cell structure, use their eight product cells on $T^{3}$ , and lift them to the oriented coordinate cells of $R^{3}$ . Opposite faces differ by $x, y, 1$ , respectively. Thus [F1] identifies the coefficient complex with the exterior algebra on $e_{1}, e_{2}, e_{3}$ , with differential the graded derivation $d (e_{1}) = a$ , $d (e_{2}) = b$ , $d (e_{3}) = 0$ . In particular $d_{1} (u, v, w) = a u + b v$ and $d_{2} (p, q, r) = (- b p, a p, a q + b r)$ in the ordered bases $(e_{1}, e_{2}, e_{3})$ and $(e_{1} \land e_{2}, e_{1} \land e_{3}, e_{2} \land e_{3})$ .

2.1step 1.1algebra

Evaluation at $x = 1$ identifies $Λ / (a)$ with the domain $Z [y^{\pm 1}]$ ; its kernel is exactly $(a)$ by multiplying by a power of $x$ and applying the ordinary identity $x^{j} - 1 = (x - 1) (1 + \dots + x^{j - 1})$ . If $a u + b v = 0$ , reduction modulo $a$ gives $b \overset{ˉ}{v} = 0$ , whence $v = a t$ and $u = - b t$ . Therefore every degree-one cycle has a unique expression $(- b t, a t, w)$ . The image of $d_{2}$ consists precisely of those expressions with arbitrary $t = p$ and $w \in I$ . Consequently $H_{1} (M; Λ) ≅ Λ / I ≅ Z$ , generated by the third circle. It is nonzero, and all of it is $Λ$ -torsion because $a \neq 0$ annihilates it.

2.2step 1.1constructalgebra

Over $K$ , multiplication by $a$ is invertible. The operator $h (ω) = a^{- 1} e_{1} \land ω$ satisfies $d h + h d = 1$ by the displayed graded-derivation differential. Hence the entire coefficient chain complex over $K$ is contractible and in particular $H_{1} (M; K) = 0$ . This verifies the vanishing assumption of the refuted statement without any finiteness or rank inference.

3.1step 2.1algebra

Any $Λ$ -linear map $Λ / I \to K / Λ$ is determined by the image $q + Λ$ of $1 + I$ , where $a q, b q \in Λ$ . Set $r = a q$ and $s = b q$ . Then $b r = a s$ . Reducing this identity modulo $a$ and using the domain $Λ / (a)$ from step 2.1 gives $r = a t$ for some $t \in Λ$ . Since $a \neq 0$ in the field $K$ , the identity $a q = a t$ forces $q = t \in Λ$ . Thus the image of $1 + I$ is zero, and ${Hom}_{Λ} (Λ / I, K / Λ) = 0$ . The conjugate dual is also zero; the involution $x \mapsto x^{- 1}, y \mapsto y^{- 1}$ preserves $I$ because $x^{- 1} - 1 = - x^{- 1} a$ and $y^{- 1} - 1 = - y^{- 1} b$ .

4.1step 2.1step 2.2step 3.1algebra∎

By steps 2.1 and 3.1, the adjoint of any sesquilinear pairing on $T H_{1} (M; Λ)$ with target $K / Λ$ has nonzero source and zero target, and therefore is not an isomorphism. This includes the coefficient Bockstein–duality pairing whenever expressed in those conventions. Together with step 2.2 this refutes the asserted implication. Every object and calculation is explicit and finite; no choice assumption is needed.

Remarks

COT Theorem 2.13 does not assert the refuted generality. Its printed statement assumes $β_{1} (M) = 1$ and a nontrivial coefficient system, and obtains nonsingularity with the extra condition that the intermediate coefficient ring is a principal ideal domain or an augmentation-ideal element becomes invertible. Those conditions are absent from the refuted statement. The example above has $β_{1} (T^{3}) = 3$ and no such coefficient localization.

Depends on

Cellular chains compute local homology

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Cochran, Orr and Teichner, Knot concordance, Whitney towers and L2-signatures, Theorem 2.13 (additional localization hypotheses) (standard reference, not scraped)