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I have been trying to learn about spin geometry, Dirac operators, and index theory by reading Lawson/Michelson's "Spin Geometry" and Friedrich's "Dirac Operators in Riemannian Geometry." Both are abstract, and basically no explicit examples are worked all the way through.

For example, I have been trying to find the spinor bundles, Dirac operators, and various indices for relatively simple manifolds: spheres and tori. However often these computations are detailed and even when I get to the end, it's not clear that I've done it correctly.

Is there another book, or perhaps online notes, which have a bunch of examples worked through in detail so that I can make sure what I'm doing is correct and also have a bank of examples to look at as I progress?

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Take a look at "Twistors and Killing spinors on Riemannian manifolds", by Baum, Friedrich, Grunewald and Kath. It has many examples. –  José Figueroa-O'Farrill Dec 12 '12 at 3:28
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I recommend Bott's paper "The Index Theorem for Homogeneous Differential Operators" and papers on index theory over homogeneous spaces in general. The computations can be done very systematically and explicitly using representation theory, and it includes most basic examples of manifolds. –  Paul Siegel Dec 12 '12 at 4:16
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In terms of additional resources, pretty much all introductory accounts of spectral triples will at least sketch basic theory for and examples of Dirac operators. Joseph Varilly's lecture notes on spectral triples (toknotes.mimuw.edu.pl/sem3/index.html) include very detailed, step-by-step exercises working through the spin geometry of the circle, 2-torus, and 2-sphere, though without covering any index theory at all. –  Branimir Ćaćić Dec 12 '12 at 5:33
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Try Chapter 11 of these notes nd.edu/~lnicolae/Lectures.pdf –  Liviu Nicolaescu Dec 12 '12 at 14:57
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Appendix A to Chapter 9 of the book Elements of Noncommutative Geometry by Gracia-Bondia, Varilly, and Figueroa is titled "Spin geometry of the Riemann sphere". It is 15 pages long and goes into quite some detail. (Some might call that level of detail excruciating, but YMMV.)

As Paul Siegel notes, computations on homogeneous spaces can be done quite effectively using representation theory. Some years ago, in the course of learning about that approach, I wrote up an account of the construction of the spinor bundle, Dirac operator, etc on $S^2$, viewed as the homogeneous space $SU(2)/U(1)$. If you're interested, email me (you can find my email address at my website, linked in my profile) and I can send it to you.

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