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I am looking for some elementary books (may be introduction) about Spin structure in general, and Spin structure on Riemannian manifolds. Someone can help me? Thanks a lot!

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Here is a short paper of lecture notes. The first couple of pages introduce $Spin$ and $Spin^c$ structures on 4-manifolds. – Neal Mar 13 '13 at 17:55
You might find these notes helpful: – Henry T. Horton Mar 13 '13 at 17:59
Milnor has a classic paper in l'enseignement mathematique on this topic. – Ryan Budney Mar 13 '13 at 18:39
Thanks every one! – mapping Mar 13 '13 at 20:09

Spin Geometry by Lawson and Michelson is highly regarded. It seems to cover a lot of very important material.

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