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, S1 in the torus) do not seem to help much.


The [Wikipedia article about the Euler class](http://en.wikipedia.org/wiki/Euler_class) mentions very briefly something about the self-intersection and that does seem relevant, but there are too few details.



Euler characteristic of a manifold and self-intersection
The normal bundle to $M$ in $M\times M$ is isomorphic to the tangent bundle of $M$, so a tubular neighborhood $N$ of $M$ in $M\times M$ is isomorphic to the tangent bundle of $M$. A section $s$ of the tangent bundle with isolated zeros thus gives a submanifold $M'$ of $N \subset M\times M$ with the following properties:


1) $M'$ is isotopic to $M$.


2) The intersections of $M'$ with $M$ are in bijection with the zeros of $s$ (and their signs are given by the indices of the zeros).


The desired result then follows from the Hopf index formula.


