{
  "id": "a-counterexample-to-walls-d2-problem",
  "title": "A counterexample to Wall's D(2) problem",
  "description": "Wall's finite D(2) problem: is a finite complex of dimension three that looks two-dimensional to homology and cohomology with all coefficients homotopy equivalent to a finite complex of dimension two? Chain complexes twisted by a character of the fundamental group; Fox derivatives; an obstruction for complexes of dimension two proved by a tower of infinite cyclic covers; the finite plus construction; an explicit group that gives a counterexample. The new arguments are proved in full; of six standard theorems of algebraic topology, five are proved in the programme and the homotopy invariance of twisted homology is a named hypothesis. The course is an exposition of OpenAI's construction (2026).",
  "units": [
    {
      "id": "walls-d2-problem-and-twisted-homology",
      "title": "Wall's D(2) problem and twisted homology",
      "reader": "walls-d2-problem-and-twisted-homology.html",
      "source": "src/walls-d2-problem-and-twisted-homology.md",
      "source_sha256": "8e8830373894e83baa6d6224bf06f70699ec99d892d79dd51df0ef3b9060b84d",
      "status": "author_self_checked"
    },
    {
      "id": "the-tower-obstruction",
      "title": "The tower obstruction",
      "reader": "the-tower-obstruction.html",
      "source": "src/the-tower-obstruction.md",
      "source_sha256": "d252653af89df74cbe9b54f28f3ce6873c33dc3942290609a039277dd51f9504",
      "status": "author_self_checked"
    },
    {
      "id": "the-finite-plus-construction",
      "title": "The finite plus construction",
      "reader": "the-finite-plus-construction.html",
      "source": "src/the-finite-plus-construction.md",
      "source_sha256": "78ce50d6c90d76f8e67678a16ca6adbb7c56093082f71fb405f95f37c372a46a",
      "status": "author_self_checked"
    },
    {
      "id": "the-counterexample",
      "title": "The counterexample",
      "reader": "the-counterexample.html",
      "source": "src/the-counterexample.md",
      "source_sha256": "349f2c7d7f665fe68a3a31fcaf781662eaa86d1f8b557b18b0dbcfc70a0f8cf0",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Algebraic Topology (D60): CW complexes, fundamental group, covering spaces, homology",
    "groups, group rings and modules; tensor products over a ring",
    "the standard theorems TOP1 to TOP6 of the first lesson: all proved in the programme (course cell-complexes-and-covering-spaces, lesson Frame fields and primary obstructions of the course on characteristic classes, core course Algebraic Topology) except the homotopy invariance in TOP5, a named hypothesis; everything else is proved in the four lessons"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}