{
  "id": "the-kervaire-conjecture",
  "title": "The Kervaire conjecture",
  "description": "Adding one generator and one relation to a nontrivial group never gives the trivial group: the Kervaire conjecture, proved by OpenAI in 2026. For a word whose exponent sum in the new generator is +-1, a relation killing a coefficient is drawn as a planar surface of disks and bands; the boundary is measured by the trace of the spectral phase in the group von Neumann algebra, which exists for every group, and a degree computation modulo two supplies a unitary with a large fixed space. Comparing an upper bound from subadditivity with a computation along the boundary circles shows that one boundary word cannot be the only nontrivial one. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "equations-over-groups-and-the-kervaire-conjecture",
      "title": "Equations over groups and the Kervaire conjecture",
      "reader": "equations-over-groups-and-the-kervaire-conjecture.html",
      "source": "src/equations-over-groups-and-the-kervaire-conjecture.md",
      "source_sha256": "01d448c2a79ba648a38058fa70da31ad209566bd4a3a20b4385b0f2d0e96dc35",
      "status": "author_self_checked"
    },
    {
      "id": "the-regular-trace-and-spectral-phase",
      "title": "The regular trace and spectral phase",
      "reader": "the-regular-trace-and-spectral-phase.html",
      "source": "src/the-regular-trace-and-spectral-phase.md",
      "source_sha256": "a00a6d62a068a82e170d0f3e3bc4acffd3902662e6e44c97df541ae7981eb6bb",
      "status": "author_self_checked"
    },
    {
      "id": "a-unitary-with-a-large-fixed-space",
      "title": "A unitary with a large fixed space",
      "reader": "a-unitary-with-a-large-fixed-space.html",
      "source": "src/a-unitary-with-a-large-fixed-space.md",
      "source_sha256": "8e588e819cf7f1c64a80ab8aa926b4ad66948d916e77803df38f1847a98d542d",
      "status": "author_self_checked"
    },
    {
      "id": "disks-bands-and-boundary-words",
      "title": "Disks, bands and boundary words",
      "reader": "disks-bands-and-boundary-words.html",
      "source": "src/disks-bands-and-boundary-words.md",
      "source_sha256": "e4a959c7c79548bd07995a71f76bbbe4a0912a68393979ec2e25153722392032",
      "status": "author_self_checked"
    },
    {
      "id": "the-planar-boundary-theorem",
      "title": "The planar boundary theorem",
      "reader": "the-planar-boundary-theorem.html",
      "source": "src/the-planar-boundary-theorem.md",
      "source_sha256": "2d342aac6baae537d7aa968f804e131e0718ca5acab36796c3fb042032439e37",
      "status": "author_self_checked"
    },
    {
      "id": "from-a-relation-to-a-planar-surface",
      "title": "From a relation to a planar surface",
      "reader": "from-a-relation-to-a-planar-surface.html",
      "source": "src/from-a-relation-to-a-planar-surface.md",
      "source_sha256": "34d98aaac4dd22a7311d10754765f7f2764ba40c0eeb37372d23ab3d294ebb27",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "programme lesson The spectral theorem for bounded self-adjoint operators (course foundations-of-von-neumann-algebras), Theorems 3.1 and 8.1, Remark 4.5: the Borel functional calculus of normal operators",
    "programme lesson C*-algebras, the continuous functional calculus, automatic continuity and positive cones (course foundations-of-von-neumann-algebras), Theorem 5.1",
    "programme lesson Regular values and degree modulo two (course borsuks-conjecture-in-dimension-nine), Definition 4.1, Lemma 4.2, Corollary 4.3: degree modulo two",
    "core course Real Analysis II (Lebl, Basic Analysis II): Theorem 8.5.1, the inverse function theorem",
    "programme lessons Spherical arrangements and planar intervals, Section 3, and Cone pictures and asphericity, Section 2 (course torsion-free-group-algebra-with-zero-divisors): the named statement (SURF) and the statement (PI1)",
    "programme course Cell complexes, fundamental groups and covering spaces (cell-complexes-and-covering-spaces): lesson Fundamental groups of graphs and two-complexes, Corollary 2.4 and Theorem 3.3, which prove the statement (PI1)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}