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Lately my studies have been focusing on learning the machinery of K-Theory, and I thought that learning the Atiyah-Singer Index Theorem would be a good way to see K-Theory in action a bit and to learn a deep result on the way. From what I have read, there are a few methods of proof of Atiyah-Singer, one of which uses K-Theory. Also from what I have read, it seems that I have most of the background knowledge to approach the proof of Atiyah-Singer.

However, it doesn't seem that there is a standard reference or sequence of references to go to in order to learn the proof of this theorem. In particular, I am not sure which book(s) would be best to look at to see a proof of Atiyah-Singer which utilizes K-Theory. I have found a few that seem to take the K-Theory approach to the theorem, but I have no way of telling how good or useful they are.

So what I am asking for is a roadmap or a reference to a proof of the Atiyah-Singer Index Theorem that uses K-Theory, and of course other advice concerning learning this theorem is welcome as well.

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3 Answers 3

up vote 15 down vote accepted

The original paper by Atiyah and SInger at http://www.jstor.org/stable/1970715 is as good as anything

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Thanks! It seems I was confused and/or misinformed. I did not know their paper used K-Theory. –  Eric A. Bunch Jul 26 '10 at 22:01

Although perhaps different enough to be not a duplicate, the answers in this question should be linked to.

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Not to nitpick, but shouldn't this be a comment? I agree though- except that I think this question is extremely close to being a duplicate, especially in light of the analytic approach to K-theory as in Higson and Roe's book- in particular, I doubt any answer here will be any more useful to anyone, including the poster, than the answers in the question you linked to. –  Daniel Moskovich Jul 27 '10 at 14:09
Possibly, I debated adding it to the question itself, actually. –  Loop Space Jul 27 '10 at 14:23

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