A torsion-free counterexample to reduced Baum–Connes injectivity

The Baum-Connes conjecture asserts that the assembly map from the K-homology of a classifying space to the K-theory of the reduced group C*-algebra is an isomorphism. OpenAI constructed in 2026 a finitely generated torsion-free group in which the K-homology class of a torus has infinite order but reduced assembly image zero, so reduced assembly is not even rationally injective. The group contains the Heisenberg group and is defined by the closed walks of a voltage cover of an infinite labelled tower with Heisenberg voltages; an integral cocycle of the Heisenberg group extends to the whole group and detects the torus, while bounded families of spread Bott symbols, realized by finite Cayley formulas, kill the Bott class by a swindle in a norm quotient. The course is an exposition of OpenAI's proof (2026).

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

1. The reduced assembly map and the torus class
2. The Heisenberg group and its laws
3. Streams, coefficient pools and universal corrections
4. Auxiliary incidences, deletions and recovery
5. Expansion and the sequential law
6. Coherent returns, reconstruction and accumulated rank
7. Terminal compatibility, key separation and base girth
8. Thetas and the pruned old graph
9. Compact Bott projections and spreading
10. The Fourier envelope and the product sampler
11. Polynomial avoidance, norm tests and moments
12. The private completion and the graphical group
13. Polygons through sheets and finite peeling
14. Cycle coordinates and disjoint domains
15. Cone chains, torsion-freeness and the integral detector
16. Carriers, trimming and constant extraction
17. Finite Cayley formulas for prescribed profiles
18. Parity cancellation and the finite witness

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Homotopy, associativity, the index pairing and KK-equivalence · Kasparov’s KK-theory
• Dirac classes and the cotangent Dolbeault element · Kasparov’s KK-theory
• The Baum–Connes assembly map and the conjecture · Kasparov’s KK-theory
• Commutative and noncommutative tori · K-theory of C*-algebras
• Idempotents, projections and their equivalences · K-theory of C*-algebras
• Nonunital algebras: unitization, relative classes and half-exactness · K-theory of C*-algebras
• Graphical presentations and coned diagrams (A projection of irrational trace and the Baum–Connes conjecture)
• Traces of projections and the Baum–Connes conjecture (A projection of irrational trace and the Baum–Connes conjecture)
• From the cover band to the Cayley projection (A torsion-free counterexample to the Kadison–Kaplansky conjecture)
• Girth and centered moments (A torsion-free counterexample to the Kadison–Kaplansky conjecture)
• Separated bands and equivariant traces (A torsion-free counterexample to the Kadison–Kaplansky conjecture)
• Streams, slots and slot evaluations (A torsion-free counterexample to the Kadison–Kaplansky conjecture)
• The Kadison–Kaplansky conjecture and scalar compression (A torsion-free counterexample to the Kadison–Kaplansky conjecture)
• The mean incidence and its gap (A torsion-free counterexample to the Kadison–Kaplansky conjecture)
• Torsion-freeness (A torsion-free counterexample to the Kadison–Kaplansky conjecture)
• Plane maps (Van Kampen diagrams)
• Van Kampen's lemma (Van Kampen diagrams)
• Words, free groups and Cayley graphs (Van Kampen diagrams)
• Kasparov products, assembly and the K-theory of the torus (courses Kasparov’s KK-theory and K-theory of C*-algebras).
• Towers, voltage covers and graphical presentations (courses A torsion-free counterexample to the Kadison–Kaplansky conjecture and A projection of irrational trace and the Baum–Connes conjecture).
• Graphs, walks and Cayley graphs; basic probability with independent random variables; classifying spaces and cellular cohomology.