# Euler characteristic of a manifold and self-intersection

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

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The normal bundle to M in MxM is isomorphic to the tangent bundle of M, so a tubular neighborhood N of M in MxM 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 MxM 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.

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Thanks, that's very pretty! –  Sam Derbyshire Oct 16 '09 at 2:34