Coboundary expanders of bounded degree

Coboundary expansion of simplicial complexes over the field with two elements, and the complexes of OpenAI's theorem (2026) that bounded-degree coboundary expanders exist in every dimension. Definitions, weights, the relation to cohomology, examples, the passage to skeletons; coset complexes of groups; the groups of polynomial matrices and their coset complexes, with bounded degree and growth. The theorem is derived from two named statements, an expansion inequality relative to the cocycles and the vanishing of cohomology, which this course does not prove.

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

1. Coboundary expansion
2. Coset complexes
3. The groups and the complexes

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Finite sets, counting, and vector spaces over the field with two elements (core course Linear Algebra), and groups, cosets and quotient groups (core course Abstract Algebra I); the lessons prove the other facts they use.