Poincaré duality

If $M$ is a compact, oriented, $n$-dimensional manifold, then there is a canonical (though non-natural (http://planetmath.org/NaturalTransformation)) isomorphism

$$D:H^q(M,\mathbb{Z})\to H_{n-q}(M,\mathbb{Z})$$

(where $H^k(M,\mathbb{Z})$ is the $k$th homology group of $M$ with integer coefficients and $H_k(M,\mathbb{Z})$ the $k$th cohomology (http://planetmath.org/DeRhamCohomology) group) for all $q$, which is given by cap product with a generator of $H_n(M,\mathbb{Z})$(a choice of a generator here corresponds to an orientation). This isomorphism exists withcoefficients in $\mathbb{Z}/2\mathbb{Z}$ regardless of orientation.

This isomorphism gives a nice interpretation to cup product. If $X,Y$ are transverse submanifolds of $M$, then $X\cap Y$ is also a submanifold. All of these submanifolds represent homology classes of $M$ in the appropriate dimensions, and

$$D^{-1}([X])\cup D^{-1}([Y])=D^{-1}([X\cap Y]),$$

where $\cup$ is cup product, and $\cap$ in intersection, not cap product.