Van Kampen diagrams

Van Kampen diagrams without topology: plane maps and their operations, free groups, presentations and Cayley graphs, and van Kampen's lemma in three parts; the foundation for the diagram arguments of the courses on irrational traces, the Kadison-Kaplansky conjecture and reduced Baum-Connes injectivity.

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. Plane maps
2. Words, free groups and Cayley graphs
3. Van Kampen's lemma

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Sphere maps (Barnette's conjecture)
• Finite graphs and permutations; groups, normal subgroups and quotients.

Used in

• Graphical presentations and coned diagrams (A projection of irrational trace and the Baum–Connes conjecture)
• The Kadison–Kaplansky conjecture and scalar compression (A torsion-free counterexample to the Kadison–Kaplansky conjecture)
• The reduced assembly map and the torus class (A torsion-free counterexample to reduced Baum–Connes injectivity)