{
  "id": "reduced-baum-connes-injectivity",
  "title": "A torsion-free counterexample to reduced Baum–Connes injectivity",
  "description": "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).",
  "units": [
    {
      "id": "the-reduced-assembly-map-and-the-torus-class",
      "title": "The reduced assembly map and the torus class",
      "reader": "the-reduced-assembly-map-and-the-torus-class.html",
      "source": "src/the-reduced-assembly-map-and-the-torus-class.md",
      "source_sha256": "4cc41cdcfc0325f77d848553e92f2dc09dfc27456d390b35f3059ef19ea1d3b8",
      "status": "author_self_checked"
    },
    {
      "id": "the-heisenberg-group-and-its-laws",
      "title": "The Heisenberg group and its laws",
      "reader": "the-heisenberg-group-and-its-laws.html",
      "source": "src/the-heisenberg-group-and-its-laws.md",
      "source_sha256": "d53ccaa941bf4a87929c9dd1693a007053dd1c65823c310c5a26a76634688e0d",
      "status": "author_self_checked"
    },
    {
      "id": "streams-coefficient-pools-and-universal-corrections",
      "title": "Streams, coefficient pools and universal corrections",
      "reader": "streams-coefficient-pools-and-universal-corrections.html",
      "source": "src/streams-coefficient-pools-and-universal-corrections.md",
      "source_sha256": "62c6cd0b41f0a2c49b69f2611f1cd09bb0289091b742352b58575aa760aedb6a",
      "status": "author_self_checked"
    },
    {
      "id": "auxiliary-incidences-deletions-and-recovery",
      "title": "Auxiliary incidences, deletions and recovery",
      "reader": "auxiliary-incidences-deletions-and-recovery.html",
      "source": "src/auxiliary-incidences-deletions-and-recovery.md",
      "source_sha256": "57a3cef229f46cabc51234978c5cd0938072ea09bafd04dc75492cb30248eb3d",
      "status": "author_self_checked"
    },
    {
      "id": "expansion-and-the-sequential-law",
      "title": "Expansion and the sequential law",
      "reader": "expansion-and-the-sequential-law.html",
      "source": "src/expansion-and-the-sequential-law.md",
      "source_sha256": "2daf419b1f5bb18373d88cf098fc2f8e1985d3899d89a80e66bdaba1ebbc67bc",
      "status": "author_self_checked"
    },
    {
      "id": "coherent-returns-reconstruction-and-accumulated-rank",
      "title": "Coherent returns, reconstruction and accumulated rank",
      "reader": "coherent-returns-reconstruction-and-accumulated-rank.html",
      "source": "src/coherent-returns-reconstruction-and-accumulated-rank.md",
      "source_sha256": "a46fa373308be6518e1d534367109e4eb908ac80c41a48cfb64fe8efcd824961",
      "status": "author_self_checked"
    },
    {
      "id": "terminal-compatibility-key-separation-and-base-girth",
      "title": "Terminal compatibility, key separation and base girth",
      "reader": "terminal-compatibility-key-separation-and-base-girth.html",
      "source": "src/terminal-compatibility-key-separation-and-base-girth.md",
      "source_sha256": "d602636dfb26b9e17466d5af7c7ecc6e0dd2498b8b30d7e5a46cd3fea0218180",
      "status": "author_self_checked"
    },
    {
      "id": "thetas-and-the-pruned-old-graph",
      "title": "Thetas and the pruned old graph",
      "reader": "thetas-and-the-pruned-old-graph.html",
      "source": "src/thetas-and-the-pruned-old-graph.md",
      "source_sha256": "04f8a24649f9dc9d08fba43ca89625653a5aa2f8140d15cf4b3af3a0165e95e5",
      "status": "author_self_checked"
    },
    {
      "id": "compact-bott-projections-and-spreading",
      "title": "Compact Bott projections and spreading",
      "reader": "compact-bott-projections-and-spreading.html",
      "source": "src/compact-bott-projections-and-spreading.md",
      "source_sha256": "a3df25c6dcfacacef3a787ec442df9de0a215119c13d6ca3572e047094d9f6a5",
      "status": "author_self_checked"
    },
    {
      "id": "the-fourier-envelope-and-the-product-sampler",
      "title": "The Fourier envelope and the product sampler",
      "reader": "the-fourier-envelope-and-the-product-sampler.html",
      "source": "src/the-fourier-envelope-and-the-product-sampler.md",
      "source_sha256": "1799f731cc71979738d3bf8294ae05e636bef36eb665187ef6249cb2a3437e25",
      "status": "author_self_checked"
    },
    {
      "id": "polynomial-avoidance-norm-tests-and-moments",
      "title": "Polynomial avoidance, norm tests and moments",
      "reader": "polynomial-avoidance-norm-tests-and-moments.html",
      "source": "src/polynomial-avoidance-norm-tests-and-moments.md",
      "source_sha256": "683d096c465509a5ea0c701c5f4cdb0133b7ea00418f159d2eba2e0828a69edc",
      "status": "author_self_checked"
    },
    {
      "id": "the-private-completion-and-the-graphical-group",
      "title": "The private completion and the graphical group",
      "reader": "the-private-completion-and-the-graphical-group.html",
      "source": "src/the-private-completion-and-the-graphical-group.md",
      "source_sha256": "1b7ed128d5d6e27f78cd02d72e99ed6d292855c6d3d44a30dd6a0db6473eaed8",
      "status": "author_self_checked"
    },
    {
      "id": "polygons-through-sheets-and-finite-peeling",
      "title": "Polygons through sheets and finite peeling",
      "reader": "polygons-through-sheets-and-finite-peeling.html",
      "source": "src/polygons-through-sheets-and-finite-peeling.md",
      "source_sha256": "e9feaaa86438dba3587d81a3ee27006fe77af3d8da19a2d080afeebce6899c4d",
      "status": "author_self_checked"
    },
    {
      "id": "cycle-coordinates-and-disjoint-domains",
      "title": "Cycle coordinates and disjoint domains",
      "reader": "cycle-coordinates-and-disjoint-domains.html",
      "source": "src/cycle-coordinates-and-disjoint-domains.md",
      "source_sha256": "01518037086418dc11c7a44e6b1aaa6754395b5fcf74bb8e2b6846b2a49b879e",
      "status": "author_self_checked"
    },
    {
      "id": "cone-chains-torsion-freeness-and-the-integral-detector",
      "title": "Cone chains, torsion-freeness and the integral detector",
      "reader": "cone-chains-torsion-freeness-and-the-integral-detector.html",
      "source": "src/cone-chains-torsion-freeness-and-the-integral-detector.md",
      "source_sha256": "a8728da2a64567cfb3806ec2fa80e4c7932b2625c3e6a18c50d7091bc31ae6f2",
      "status": "author_self_checked"
    },
    {
      "id": "carriers-trimming-and-constant-extraction",
      "title": "Carriers, trimming and constant extraction",
      "reader": "carriers-trimming-and-constant-extraction.html",
      "source": "src/carriers-trimming-and-constant-extraction.md",
      "source_sha256": "bfa2de2b59fba26623bf026d8a92d859b5fd411c0beb586e1616ed389cdf7dc5",
      "status": "author_self_checked"
    },
    {
      "id": "finite-cayley-formulas-for-prescribed-profiles",
      "title": "Finite Cayley formulas for prescribed profiles",
      "reader": "finite-cayley-formulas-for-prescribed-profiles.html",
      "source": "src/finite-cayley-formulas-for-prescribed-profiles.md",
      "source_sha256": "d50601510d83f83fcbbcfa9d97aa8ca0416435b6c31c60d6f53c22f7fc88e5dc",
      "status": "author_self_checked"
    },
    {
      "id": "parity-cancellation-and-the-finite-witness",
      "title": "Parity cancellation and the finite witness",
      "reader": "parity-cancellation-and-the-finite-witness.html",
      "source": "src/parity-cancellation-and-the-finite-witness.md",
      "source_sha256": "515988157dfef8e63f644471b654a14db55aba97415d825837465bdd0cd94945",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "programme course Kasparov's KK-theory (KT-KK): lessons The Baum-Connes assembly map and the conjecture (Theorems A.4, A.7, A.8, A.9 and A.10, Lemmas A.7b, A.9a and A.10a, formulas (Q5) and (U8)), Homotopy, associativity, the index pairing and KK-equivalence (Proposition 5.1) and Dirac classes and the cotangent Dolbeault element (the flat torus)",
    "programme course K-theory of C*-algebras (KT-OPK): lessons Idempotents, projections and their equivalences (Corollary 3.3), Nonunital algebras: unitization, relative classes and half-exactness (Theorem 1.1) and Commutative and noncommutative tori (Corollary 3.7)",
    "programme course A torsion-free counterexample to the Kadison-Kaplansky conjecture (kadison-kaplansky-projections): lessons Streams, slots and slot evaluations (Lemma 5.1), The mean incidence and its gap (Lemma 2.1), Separated bands and equivariant traces (Lemma 2.1), Torsion-freeness (Lemmas 3.1 and 3.2), Girth and centered moments (Lemmas 1.1 and 3.1) and From the cover band to the Cayley projection (Lemma 4.2)",
    "programme course A projection of irrational trace and the Baum-Connes conjecture (irrational-trace-and-baum-connes): lesson Graphical presentations and coned diagrams (Section 3, the statements (VK) and (MAP))",
    "programme course Van Kampen diagrams (van-kampen-diagrams): lessons Plane maps (Propositions 1.2, 2.1, 2.2 and 2.3) and Van Kampen's lemma (Definition 1.1, Theorems 3.1, 4.1 and 5.1), which prove the statements (VK) and (MAP)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}