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The Kervaire theorem for groups

OpenAI · September 24, 2026

Lean

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Abstract

We prove that no free product of a nontrivial group with an infinite cyclic group is normally generated by one element. This resolves the Kervaire conjecture positively.

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Notes

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Automatically collected notes (unreviewed)

work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Follow-up works

  • OpenAI, Nonsingular systems of equations over arbitrary groups (OpenAI Math Release preprint, 5 Oct 2026; copy hosted by the Kourovka Notebook). Same-family companion that names this paper's spectral-phase and planar argument as its direct antecedent and extends it to finite nonsingular systems. [AI-generated per source]

Comments and discussion

Formalization

  • OpenAI math release, Lean scope notes for family 256 (lean/docs/256.md) (release Lean documentation, 6 Oct 2026). States that the linked formalization proves this paper's coefficient-injectivity theorem for any group $A$ and any relator in $A*\mathbb Z$ with exponent sum $\pm1$, with a comparator statement in ComparatorChallenges/Kervaire.lean. [family-level]
  • OpenAI, lean/OAI/GroupTheory/Kervaire/Main.lean (github.com/openai/math, 6 Oct 2026). Contains Kervaire.coefficient_injective for unimodular relators (Theorem 1.1) and Kervaire.quotient_nontrivial for arbitrary relators (Corollary 1.2).

Digests and expositions

  • Open Mathematics Courses, The Kervaire conjecture (undated). Six-lesson exposition of the manuscript developing spectral phase, the fixed-space lemma and disk-and-band surfaces; it identifies its author and self-checker as Claude Opus 5.5, with independent review incomplete. [AI-generated per source]

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:The-Kervaire-Theorem-for-Groups-September-24-2026,
  author = {{OpenAI}},
  title = {{The Kervaire theorem for groups}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/The-Kervaire-Theorem-for-Groups-September-24-2026.pdf}{OAI:The-Kervaire-Theorem-for-Groups-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/256.md. The Lean statements have not been checked by people against the paper's claims.

Family 256: Nonsingular systems of equations over arbitrary groups

Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group embeds in the presented quotient. A companion proves Kervaire's conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group.

Other papers in this family: