The Kervaire conjecture

Adding one generator and one relation to a nontrivial group never gives the trivial group: the Kervaire conjecture, proved by OpenAI in 2026. For a word whose exponent sum in the new generator is +-1, a relation killing a coefficient is drawn as a planar surface of disks and bands; the boundary is measured by the trace of the spectral phase in the group von Neumann algebra, which exists for every group, and a degree computation modulo two supplies a unitary with a large fixed space. Comparing an upper bound from subadditivity with a computation along the boundary circles shows that one boundary word cannot be the only nontrivial one. The course is an exposition of OpenAI's proof (2026).

Six lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Equations over groups and the Kervaire conjecture
2. The regular trace and spectral phase
3. A unitary with a large fixed space
4. Disks, bands and boundary words
5. The planar boundary theorem
6. From a relation to a planar surface

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Idempotents and integral traces (The Bass trace conjecture for group rings)
• Regular values and degree modulo two (Borsuk's conjecture fails in dimension nine)
• Fundamental groups of graphs and two-complexes (Cell complexes, fundamental groups and covering spaces)
• Word maps on unit quaternions and degree (A finitely generated counterexample to the Eilenberg-Ganea conjecture)
• Order, local units and quotients of C*-algebras · Foundations of von Neumann algebras
• The spectral theorem for bounded self-adjoint operators · Foundations of von Neumann algebras
• Directly finite group algebras (Kaplansky's direct-finiteness conjecture)
• Simultaneous fixed spaces (Nonsingular systems of equations over groups)
• Systems of equations over groups (Nonsingular systems of equations over groups)
• The planar boundary obstruction (Nonsingular systems of equations over groups)
• Cone pictures and asphericity (A torsion-free group algebra with zero divisors)
• Spherical arrangements and planar intervals (A torsion-free group algebra with zero divisors)
• Group presentations and free products (recalled in the first lesson).
• Hilbert spaces, the spectral theorem and the Borel functional calculus (earlier programme lessons).
• Smooth manifolds, degree modulo two and the fundamental group (earlier programme lessons and the core course Algebraic Topology).