Counterexamples to Baum–Connes and Kadison–Kaplansky. Disproves the coefficient-free reduced Baum–Connes conjecture through a failure of rational injectivity and a separate failure of surjectivity of assembly. A finitely generated torsion-free example witnesses the injectivity failure. Separately, a torsion-free group has a nontrivial projection in its reduced group C∗-algebra, disproving the Kadison–Kaplansky conjecture.
released 2026-09-23 | 1 theorem · 49 lemmas · 58 proofs · 41,401 words |
PLAY LEVEL 1 »(pdf)
We disprove rational injectivity of the coefficient-free reduced Baum–Connes assembly map for torsion-free groups. Specifically, we construct a finitely generated torsion-free discrete group whose degree-zero assembly map has a kernel class of infinite order.
released 2026-09-23 | 5 theorems · 34 lemmas · 52 proofs · 36,258 words |
PLAY LEVEL 2 »(pdf)
We construct a finitely generated torsion-free discrete group whose reduced group C∗-algebra contains a projection other than zero and the identity. This disproves the Kadison–Kaplansky projection conjecture.
released 2026-09-23 | 2 theorems · 10 lemmas · 16 proofs · 10,170 words |
PLAY LEVEL 3 »(pdf)
We construct a finitely generated discrete group whose reduced group C∗-algebra contains a projection of irrational canonical trace. Its K0-class lies outside the image of the coefficient-free reduced Baum–Connes assembly map. This disproves the coefficient-free reduced Baum–Connes conjecture for countable discrete groups.