Nonsingular systems of equations over groups

A system of equations over a group G has coefficients in G and unknowns x_1, ..., x_n; it is nonsingular if the matrix of exponent sums of the unknowns has full row rank over Q. Howie conjectured that every nonsingular system has a solution in some group containing G; OpenAI proved this in 2026 for every group, which includes the Kervaire-Laudenbach conjecture. The proof draws a relation that kills an element of G as a planar surface of disks and bands and measures its boundary words with the spectral phase of unitary operators over the group von Neumann algebra. The new ingredient is a theorem on simultaneous fixed spaces of block unitary matrices in several unitary variables, proved with the rational cohomology of unitary groups. The course is an exposition of OpenAI's proof (2026) and builds on the course on the Kervaire conjecture.

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

1. Systems of equations over groups
2. Rational cohomology of unitary groups
3. Simultaneous fixed spaces
4. The planar boundary obstruction
5. From a kernel relation to a solution

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Fundamental groups of graphs and two-complexes (Cell complexes, fundamental groups and covering spaces)
• Poincaré duality (Poincaré duality on manifolds)
• A unitary with a large fixed space (The Kervaire conjecture)
• Disks, bands and boundary words (The Kervaire conjecture)
• Equations over groups and the Kervaire conjecture (The Kervaire conjecture)
• From a relation to a planar surface (The Kervaire conjecture)
• The planar boundary theorem (The Kervaire conjecture)
• The regular trace and spectral phase (The Kervaire conjecture)
• 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).
• Singular cohomology, cup products and Poincaré duality (programme course Poincaré duality on manifolds).
• The course on the Kervaire conjecture: the trace and spectral phase, unitaries with fixed spaces, disk-and-band surfaces.