{
  "id": "eilenberg-ganea-counterexample",
  "title": "A finitely generated counterexample to the Eilenberg-Ganea conjecture",
  "description": "A finitely generated residually finite group with cohomological dimension 2 and geometric dimension 3: the Eilenberg-Ganea conjecture fails. The group is the kernel of the height homomorphism of the right-angled Artin group of an acyclic flag complex whose fundamental group maps nontrivially into the unit quaternions; a degree argument on finite quotients and compactness show that a two-dimensional classifying space would force small quaternion labellings that are trivial around triangles to be trivial around every closed path, and a slowly varying map of the level complex into the flag complex produces a labelling that is not. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "cohomological-and-geometric-dimension",
      "title": "Cohomological and geometric dimension",
      "reader": "cohomological-and-geometric-dimension.html",
      "source": "src/cohomological-and-geometric-dimension.md",
      "source_sha256": "eb43ef364027c4b9457a5b519f215d29c693a8e25bf3763e633bea5a408d6533",
      "status": "author_self_checked"
    },
    {
      "id": "word-maps-on-unit-quaternions-and-degree",
      "title": "Word maps on unit quaternions and degree",
      "reader": "word-maps-on-unit-quaternions-and-degree.html",
      "source": "src/word-maps-on-unit-quaternions-and-degree.md",
      "source_sha256": "b75db1559a5a527dd9d8e34cc331b44bf8cf1b5f3175fa294f096388a49211c3",
      "status": "author_self_checked"
    },
    {
      "id": "aligned-presentations-of-the-cycle-module",
      "title": "Aligned presentations of the cycle module",
      "reader": "aligned-presentations-of-the-cycle-module.html",
      "source": "src/aligned-presentations-of-the-cycle-module.md",
      "source_sha256": "23e589c903a65aeb50715f07c94198f0b916939cc557d604fabae631002324fe",
      "status": "author_self_checked"
    },
    {
      "id": "exact-counts-and-a-quadratic-estimate",
      "title": "Exact counts and a quadratic estimate",
      "reader": "exact-counts-and-a-quadratic-estimate.html",
      "source": "src/exact-counts-and-a-quadratic-estimate.md",
      "source_sha256": "208be7a6ec93457ddb5da0e4694bcff1befd1a0038b5930d5a23c1a22bb3a8ec",
      "status": "author_self_checked"
    },
    {
      "id": "finite-quotients-and-the-comparison-theorem",
      "title": "Finite quotients and the comparison theorem",
      "reader": "finite-quotients-and-the-comparison-theorem.html",
      "source": "src/finite-quotients-and-the-comparison-theorem.md",
      "source_sha256": "894d22283cf2efabe63b39db8ac97b36637c371ef0328b94155061fac780c49d",
      "status": "author_self_checked"
    },
    {
      "id": "right-angled-artin-groups-and-residual-finiteness",
      "title": "Right-angled Artin groups and residual finiteness",
      "reader": "right-angled-artin-groups-and-residual-finiteness.html",
      "source": "src/right-angled-artin-groups-and-residual-finiteness.md",
      "source_sha256": "f5800c94829f1275d39e7175692806af7016d4d4a16d5fd54d73aa0f264b7013",
      "status": "author_self_checked"
    },
    {
      "id": "the-cube-complex-and-its-universal-cover",
      "title": "The cube complex and its universal cover",
      "reader": "the-cube-complex-and-its-universal-cover.html",
      "source": "src/the-cube-complex-and-its-universal-cover.md",
      "source_sha256": "4a8e3fdb5879a067c150e6b36e337d8a44de7733c15f369a05f403cf6aa089ca",
      "status": "author_self_checked"
    },
    {
      "id": "the-level-complex-and-its-acyclicity",
      "title": "The level complex and its acyclicity",
      "reader": "the-level-complex-and-its-acyclicity.html",
      "source": "src/the-level-complex-and-its-acyclicity.md",
      "source_sha256": "648eff57c251a6a1c1db4efa06fd7facb50dc18fc386fd2046a8e17af28080b4",
      "status": "author_self_checked"
    },
    {
      "id": "an-acyclic-seed-with-quaternion-monodromy",
      "title": "An acyclic seed with quaternion monodromy",
      "reader": "an-acyclic-seed-with-quaternion-monodromy.html",
      "source": "src/an-acyclic-seed-with-quaternion-monodromy.md",
      "source_sha256": "f9f2aacd22ec2475824816d3a60ed699fd8ebb384e43ec64d4ff55e5359d6121",
      "status": "author_self_checked"
    },
    {
      "id": "traces-and-direction-vectors",
      "title": "Traces and direction vectors",
      "reader": "traces-and-direction-vectors.html",
      "source": "src/traces-and-direction-vectors.md",
      "source_sha256": "7cfab68ac3dc566c6a2c9b59a4566f2a05c2a2bc4e46de224e7586c1b9673920",
      "status": "author_self_checked"
    },
    {
      "id": "small-transport-and-the-counterexample",
      "title": "Small transport and the counterexample",
      "reader": "small-transport-and-the-counterexample.html",
      "source": "src/small-transport-and-the-counterexample.md",
      "source_sha256": "2a1574c587299916976c7e225c0c807f0ae773512f68b48cffeaa3f521dac2ba",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Algebraic Topology (Fomberg's notes): homotopy invariance of singular homology (Theorem 1.13, Corollary 1.14); the three-sphere is simply connected (Roberts's lectures, Lecture 12, Example 12.2) and unique path lifting (Lecture 5, Theorem 5.1)",
    "core course Point-Set Topology: the Lebesgue number lemma for compact metric spaces",
    "programme lesson Explicit CW models and the first Hurewicz map (course CG-S6), Theorem 4.1: the Hurewicz theorem",
    "programme lesson Homotopy fibres and the Serre spectral sequence (course DG-CHAR), Section B.4: Whitehead's theorem for CW complexes",
    "programme lesson Regular values and degree modulo two (course borsuks-conjecture-in-dimension-nine), Theorem 2.1, Corollary 2.2, Proposition 3.1: Sard's theorem and regular level sets",
    "programme lesson Weak topologies, Tychonoff, Banach-Alaoglu (course foundations-of-von-neumann-algebras), Theorem 2.3: Tychonoff's theorem",
    "programme lesson Cone pictures and asphericity (course torsion-free-group-algebra-with-zero-divisors), Section 2: the facts (PI1), (COV), (CELL 1)",
    "programme course Cell complexes, fundamental groups and covering spaces (cell-complexes-and-covering-spaces): Cell complexes and homotopy equivalences, Lemma 4.3, Propositions 2.1, 3.1 and 4.2; Fundamental groups of graphs and two-complexes, Theorem 3.3 and Corollaries 2.5 and 4.2; Universal covers and cellular chains, Theorem 1.1, Propositions 3.1, 4.1 and 5.3 and Section 5"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}