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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 $2\pi\chi$ (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.

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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.

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