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The wiki entry on the Poincaré-Hopf theorem claims that it "relies heavily on integral, and, in particular, Stokes' theorem". However, in the sketch of proof given there which is more or less the one in Milnor's Topology from the Differentiable Viewpoint there is no integration.

Does the proof become easier by using Stokes' theorem? Is there a good reference?

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Do you mean the WikiPEDIA article? – Henning Makholm Apr 24 '12 at 14:28
@t.b. Thanks for including the link – Jochen Wengenroth Apr 24 '12 at 16:03
That wasn't me, I only fixed some typos :) You can see what was changed by whom by clicking on edited x time ago above the last editor's name. – t.b. Apr 24 '12 at 16:06

1 Answer 1

In step 3 of the scketch you read "the degree of the Gauss map". Now one has to look for the definition of the degree of a map $ f:M\to N$. But that map gives a linear map in de Rham cohomology $f^*:H^m(N)\to H^m(M), n=\dim(M)=\dim(N)$. Stokes theorem says that the integral of $m$-forms induces an isomorphism $\int:H^m\to R$ an this transforms $f^*$ (by change of variables in the integral) into a linear map $R\to R$. Such a linear map is the product times an scalar $d$: this is the degree of $f$.

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