Poincarédualiteetti

Poincarédualiteetti, also known as Poincaré duality, is a fundamental theorem in topology that relates the homology and cohomology of a manifold. It states that for a compact, orientable manifold of dimension $n$, there is a natural isomorphism between its $k$-th homology group and its $(n-k)$-th cohomology group for all integers $k$. This isomorphism is established by pairing homology classes with cohomology classes using the cap product.

More formally, if $M$ is a compact, orientable $n$-dimensional manifold, then the cap product with the fundamental class $[M] \in H_n(M, \mathbb{Z})$ induces an isomorphism $D: H^k(M, \mathbb{Z}) \to H_{n-k}(M, \mathbb{Z})$ given by $D(\alpha) = \alpha \cap [M]$. This duality holds more generally when coefficients are taken in a field, or for non-orientable manifolds with coefficients in $\mathbb{Z}_2$.

Poincaré duality provides a powerful tool for understanding the topological structure of manifolds. It implies that if a manifold has trivial homology in certain degrees, then its cohomology in corresponding degrees must also be trivial. This relationship is crucial in various areas of mathematics, including differential geometry, algebraic topology, and theoretical physics, where it appears in contexts like the study of vector bundles, characteristic classes, and quantum field theory. The theorem also has extensions to manifolds with boundary and to more general topological spaces like topological manifolds.