Duality on manifolds: statements and assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Poincaré duality — Theorem 1.1. R-oriented topological n-manifolds, any commutative ring R: the cap product with the orientation classes is an isomorphism from compactly supported cohomology in degree k to homology in degree n-k.
• Self-injective coefficients — Corollary 2.3. A commutative ring injective over itself, for example Z/m, and a manifold oriented over it: ordinary cohomology in degree k is the dual of compactly supported cohomology in degree n-k; over Z/m finite groups have equal orders.
• Dimension — Corollary 2.4. Every n-manifold: integral homology vanishes above degree n and is free in degree n; cohomology with any coefficient group vanishes above degree n.
• Complex orientation — Corollary 3.5. Complex manifolds of complex dimension d: a canonical orientation as real 2d-manifolds, given by the standard orientation in every holomorphic chart.
• Classes on compact subsets — Theorem 4.1. n-manifolds, compact subsets K, any commutative ring: homology relative to the complement of K vanishes above degree n, and a locally consistent family of local classes comes from a unique class.
• Duality and Mayer–Vietoris — Proposition 3.2. R-oriented manifolds and open covers by two sets: the duality maps commute with the Mayer–Vietoris sequences, the connecting square up to the sign (-1)^(i+1).
• Sheaf and singular cohomology — Theorem 1.3. Topological manifolds and their open subsets, any abelian group: the sheaf of singular cochains is a flasque resolution of the constant sheaf, and sheaf cohomology is singular cohomology.
• Supports — Theorem 2.2. Closed subsets Z of a manifold: sheaf cohomology with supports in Z is singular cohomology relative to the complement of Z.
• Extension by zero into a compact space — Theorem 4.1. An open embedding j of a manifold into a compact Hausdorff space and any sheaf F: the cohomology of j_!F is the compactly supported cohomology of F.
• Duality for sheaf cohomology — Corollary 5.2. Manifolds oriented over a self-injective ring such as Z/m, for example complex manifolds: sheaf cohomology in degree q is the dual of compactly supported sheaf cohomology in degree n-q.

Reading order

The first lesson sets up local homology and orientations and proves the structure theorem on compact subsets. The second defines cap products and compactly supported cohomology and constructs the duality map. The third proves Poincaré duality and its consequences, and the fourth identifies the sheaf cohomology of a manifold with singular cohomology, which is the form used in étale cohomology.