Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)
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A prime-power torsion linking form need not have a metabolizer

Statement refuted

Assume AC. The assertion that every nonsingular linking form on a finite group of odd prime-power order has a metabolizer is false: the double branched cover of the positive trefoil has cyclic first homology of order three and no metabolizer.

Facts & Assumptions

Given: The positive trefoil, its Seifert matrix V, and the double branched-cover orientation convention.

[A1]

Full AC is assumed wherever the cover and orientation conventions use the choice axiom. The Axiom of Choice

[F1]

V=(−10−1−1). A positive Hopf plumbing gives the trefoil Seifert matrix

[F2]

The double-cover surgery matrix is A=V+VT. Surgery presentation from a Seifert surface

[F3]

The oriented linking form on the surgery cokernel is −A−1 mod Z. Inverse-matrix formula for a surgery linking form

[F4]

A finite nonsingular linking form has a metabolizer only if some subgroup has square-root order. Finite nonsingular linking forms and metabolizers

[F5]

The surgery linking matrix computes the same oriented torsion linking pairing as the signed bounding-chain formula. Bounding-chain expression for a torsion linking pairing Inverse-matrix formula for a surgery linking form

[F6]

A closed oriented 3-manifold with finite first homology has nonsingular torsion linking form. The torsion linking pairing of a rational homology sphere is nonsingular

Counterexample

1.1F1F2F3F5givenalgebraA1

Computing gives A=(−2−1−1−2) with determinant 3, and unimodular row and column operations reduce it to diag⁡(1,3). Therefore H1(Σ2(T);Z)≅Z/3. The matrix formula [F3] gives generator self-linking 2/3≡−1/3 mod Z under [F5], so the full group is not isotropic.

2.1F4F6step 1.1algebra∎

The only subgroups of Z/3 have order 1 or 3, while a metabolizer would have order 3 by [F4]. The form is nonsingular by [F6] and step 1.1, so it satisfies the assertion's nonsingularity hypothesis. No subgroup has that order, so no metabolizer exists. This finite nonsingular form of odd prime-power order refutes the assertion.

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