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I was wondering if there is an easy way to compute the topological index (in the sense of Atiyah-Singer) of a (Fredholm) elliptic operator on a Riemann surface. I'm aware the Todd class is very simple in dimension 1, but I'm not sure how to compute the Chern class of such an operator.

Is there a reasonable way to do this in general? If not, I am particularly interested in working it out for the Laplacian operator, is there a reference where that is done?

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    A reference that might be relevant: Michael Taylor's Partial Differential Equations, Volume II, Chapter 10 (Dirac Operators and Index Theory), especially Section 10 (Direct Attack in 2-D). Taylor's PDE series, published by Springer, is available electronically for free through some universities' libraries, so you might check your institution's library. – 2017-02-04
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    Thank you! I'll check that book! – 2017-02-09

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