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In Connes's book on noncommutative geometry, he outlines a rather short "algebraic" proof of the Gauss-Bonnet theorem that uses multilinear forms. (Start reading on page 19 of the book) This is given as motivation for cyclic cohomology.

Where can I find a fleshed-out version of this proof?

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    Where exactly in the book is this outline?2012-01-11
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    @Jonas: I've added the location to the question.2012-01-16

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