Sum of topological charges is the Euler characteristic

Question

I have seen many places claiming that the given a collection of topological defects on a 2-dimensional surface, the sum of the topological charges is $2pichi$ (where $chi$ is the Euler characteristic).
What is the proof of this statement? Can you give me any references for the general case (not in a special case, like liquid crystals)? If not, what is the proof for ordinary 2 dimensional crystals?
Thanks.

mike stone · Accepted Answer

You don't explain what you mean by "topological charges" but I expect you  mean the sum of the Hopf indices of the zeros  of a tangent-vector field on a manifold. The resulting
Poincare-Hopf theorem  says that the sum of these numbers is indeed $chi$.
A discussion and sketch of a proof can be found  starting on on page 547 of my book with Paul Goldbart  Mathematics for Physics,  an online  draft version of which can be found  here.

Sum of topological charges is the Euler characteristic

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