This is probably quite easy, but how do you show that the Euler characteristic of a manifold *M* (defined for example as the alternating sum of the dimensions of integral cohomology groups) is equal to the self intersection of *M* in the diagonal (of *M* × *M*)?

The few cases which are easy to visualise (ℝ in the plane, S^{1} in the torus) do not seem to help much.

The Wikipedia article about the Euler class mentions very briefly something about the self-intersection and that does seem relevant, but there are too few details.