Poincaré duality on manifolds

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.

Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Orientations and fundamental classes
2. Cap products and cohomology with compact supports
3. Poincaré duality
4. Sheaf cohomology and singular cohomology on manifolds

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Resolutions, Tor and Ext
• Thom classes and Euler classes · Characteristic classes
• Manifold duality, the diagonal and Wu classes · Characteristic classes
• Poincaré duality for curves
• Injective modules, flasque sheaves and bounded-below derived functors
• Sections with support and the localization triangle
• Singular homology and cohomology: the core course Algebraic Topology.
• For smooth manifolds, Poincaré duality with compact supports is also proved in the course on characteristic classes (Manifold duality, the diagonal and Wu classes, Theorem 2.2).