Poincaré Duality

Algebraic Topology · Axiom Academy

One of the most beautiful theorems in algebraic topology, relating cohomology and homology of manifolds For a closed (compact without boundary), orientable n-manifold M, we have a canonical isomorphism: This isomorphism is given by the cap product with the fundamental class [M], a distinguished element in H n (M): The cap product is a bilinear pairing that takes a cohomology class and a homology chain and produces a homology chain of lower dimension: The Poincaré duality isomorphism is explicitly given by: This map takes a cohomology class α and caps it with the fundamental class [M], producing a homology class in the complementary dimension. Let's see how Poincaré duality works for familiar manifolds: 3-Sphere S³: For the 3-sphere, we have: Notice the beautiful symmetry: H⁰(S³) ≅ H₃(S³) and H¹(S³) ≅ H₂(S³) (both zero!) Orientability is crucial for Poincaré duality. For non-orientable manifolds, the theorem fails with integer coefficients! However, Poincaré duality holds for any closed manifold with ℤ/2 coefficients : For non-orientable manifolds with ℤ coefficients, we get a twisted form: where w is the orientation bundle (local coefficient system). 5. Applications to Intersection Theory Poincaré duality provides a beautiful interpretation of intersection numbers. In an orientable n-manifold M, the intersection of a k-cycle and an (n-k)-cycle is a number. Through Poincaré duality, this becomes the cup product in cohomology:

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