{
  "id": "poincare-duality-on-manifolds",
  "title": "Poincaré duality on manifolds",
  "description": "Poincaré duality for topological manifolds with coefficients in any commutative ring: local homology and orientations, the canonical orientation of complex manifolds, fundamental classes, cap products and compactly supported cohomology, duality over fields and over self-injective rings such as Z/m, vanishing above the dimension, and the identification of the sheaf cohomology of a manifold with singular cohomology, with supports and with compact supports.",
  "units": [
    {
      "id": "orientations-and-fundamental-classes",
      "title": "Orientations and fundamental classes",
      "reader": "orientations-and-fundamental-classes.html",
      "source": "src/orientations-and-fundamental-classes.md",
      "source_sha256": "67dd3599269a0c780cc7f5329f9895a83e573a784965635c2e108fb5d123c3a2",
      "status": "author_self_checked"
    },
    {
      "id": "cap-products-and-cohomology-with-compact-supports",
      "title": "Cap products and cohomology with compact supports",
      "reader": "cap-products-and-cohomology-with-compact-supports.html",
      "source": "src/cap-products-and-cohomology-with-compact-supports.md",
      "source_sha256": "f5af7c50e50c5556a2dc1f83fb732251503b07fcab2566eab8d1c47794d15efc",
      "status": "author_self_checked"
    },
    {
      "id": "poincare-duality",
      "title": "Poincaré duality",
      "reader": "poincare-duality.html",
      "source": "src/poincare-duality.md",
      "source_sha256": "2f3fd894e8bafbd7963cb6bd33f4ab089c9cf5a1bd0ea4130201e017006cd39b",
      "status": "author_self_checked"
    },
    {
      "id": "sheaf-cohomology-and-singular-cohomology-on-manifolds",
      "title": "Sheaf cohomology and singular cohomology on manifolds",
      "reader": "sheaf-cohomology-and-singular-cohomology-on-manifolds.html",
      "source": "src/sheaf-cohomology-and-singular-cohomology-on-manifolds.md",
      "source_sha256": "a40b64ead046bb786b1d0bebc3719aa3ce14aebd1430cabc223e1fd23954bca2",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "Core course D60 Algebraic Topology (Fomberg: Theorem 1.13, Corollaries 1.14 and 1.17, Theorems 1.21, 1.23, 1.26, Propositions 1.25, 1.29 and 1.35; Roberts: Theorems 13-15, Propositions 24, 26 and 27)",
    "Characteristic classes, Thom classes and Euler classes (DG-CHAR-06): Lemma 5.1 and Theorem 5.2",
    "Derived categories of sheaves: Injective modules and bounded-below derived functors (Lemmas 1.1 and 2.1, Proposition 1.3, Theorem 4.1); Sections with support and the localization triangle (Section 1, Theorem 2.1, Lemma 4.1)",
    "AI Integrated Stacks Project, Cohomology of Sheaves: cohomology of a quasi-compact subset as a colimit over neighbourhoods"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}