{
  "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": "f5bc7a81cccccc41c23ed997de144a25cb22fa211fb20cb71bb82c7c1f86c174",
      "status": "available"
    },
    {
      "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": "75f22b5ebef22664d515fd38e9a9c98dd8022685fa5be6474e9d07fa99217750",
      "status": "available"
    },
    {
      "id": "poincare-duality",
      "title": "Poincaré duality",
      "reader": "poincare-duality.html",
      "source": "src/poincare-duality.md",
      "source_sha256": "2ab79198c17a6172ed9fd73f29dbc7549a84f4c6dca1386f42b2b17777cb4a7e",
      "status": "available"
    },
    {
      "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": "319887bf61c21414fa75fc9c2094c79e8959d86347d8be066f8af596448f9865",
      "status": "available"
    }
  ],
  "prerequisites": [
    "Core course D60 Algebraic Topology (Fomberg, Spring 2025: 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 and 27)",
    "Characteristic classes, Thom classes and Euler classes (DG-CHAR-06): Lemma 1.1, Lemma 2.1, Corollary 2.2 and Section 3 (complete chain-level proofs, including restriction to small cochains); 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": "available",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0",
  "contributors": [
    {
      "name": "GPT-6 Astra (OpenAI)",
      "contribution": "Small-cochain restriction, chain-homotopy explanation and source references."
    },
    {
      "name": "GPT-6.1 Sol (OpenAI)",
      "contribution": "Degree-zero cap products, zero-dimensional orientations and local degree, and reader preparation."
    }
  ]
}