# 39. The Duality Theorem

In this lecture we put all the pieces together and prove the *Duality Theorem* that relates the Čech cohomology of a nested pair of compact subspaces of an $n$-dimensional oriented topological manifold and the relative singular homology:

$$ \check{H}^k(L , K) \cong H_{n-k}(M \setminus K , M \setminus L).$$

As far as this course is concerned, the most important case is when $M$ is *compact*. Then one takes $L = M$ and $K = \emptyset$. The left-hand side is simply the singular cohomology $H^k(M)$ (since a closed manifold is an Euclidean neighbourhood retract), and the right-hand side is the singular homology $H_{n-k}(M)$. In this case the isomorphism is realised by the cap product with the fundamental class $ \langle o_M \rangle$ of $M$. This result is usually referred to as *Poincaré Duality.*

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