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I am looking for a good introductory level textbook (or set of lecture notes) on classical topological K-Theory that would be suitable for a one-semester graduate course. Ideally, it would require minimal background: standard introductory courses in algebraic topology and differential geometry, would cover core topics (Bott periodicity, Chern character, representation rings, etc) mostly in a self-contained way, and would give interesting examples and exercises.

As I learned the subject from multiple books and papers, I don't know a "canonical" reference that gives a coherent picture of the subject. Any suggestions ?

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Allen Hatcher's book: math.cornell.edu/~hatcher/VBKT/VB.pdf is awesome, but unfinished :( –  Tom Boardman Jul 16 '10 at 16:44
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Do you want something that goes beyond Atiyah's book, "K-Theory"? I suppose it's not the most up-to-date reference, but as an introductory text it is magnificent. I learned most of what I know about the topological side of things from that book. The only problem is that there are no exercises. –  Paul Siegel Jul 16 '10 at 17:10
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Community wiki? –  Dan Ramras Jul 16 '10 at 17:50
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I've made this community wiki. Please remember that all questions asking for a sorted list of resources, e.g. textbooks, should be community wiki. –  Scott Morrison Jul 16 '10 at 17:57
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5 Answers

I wrote a book that may be what you are looking for. It's called "Complex Topological K-Theory," and it is published by Cambridge University Press. As the title suggests, I do not discuss real (KO) theory in the book, and I also do not talk about representation rings. But the other topics you mentioned are covered, and the only background required for the book are introductory courses in point-set topology and abstract algebra.

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@Efton I've seen your book,Efton. It looks very nice,but it's quite a bit harder then either Atiyah's or Karoubi's. –  Andrew L Jul 16 '10 at 17:13
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To me, this was an excellent book for learning topological K-Theory from scratch. I do no think that it is harder to read than Karoubi's. –  Rasmus Bentmann Jul 16 '10 at 18:05
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You have written a good, self-contained, user-friendly book, Efton. I especially like the numerous explicit calculations in it (idempotent matrices, etc.): +1 for this reference . –  Georges Elencwajg Jul 16 '10 at 18:12
    
When I was a grad student, another student and I read through this book one summer. I liked it quite a bit. –  jd.r Jul 16 '10 at 20:13
    
Thanks. I did not know that book and, indeed, it covers a lot of what I need. –  Martin Pinsonnault Jul 18 '10 at 13:52
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The standard texts on the subject are by Michael Atiyah and Max Karoubi, both called K-Theory,I believe. The Atiyah book is more readable and has fewer prerequisites,but the Karoubi book covers a great deal more.

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I have lecture notes on my website that you might find helpful. They're from a one-semester graduate course (the second such course I've taught). Sadly, they're not yet typed...

They're a mix of material from Milnor and Stasheff, Hatcher's notes, and Husemoller's book Fibre Bundles. They cover vector bundles and principle bundles, characteristic classes and the Chern Character, and complex Bott periodicity. They don't cover representation rings or real K-theory. (I assume that in mentioning representation rings, you're talking about the Atiyah-Segal Theorem, or at least Atiyah's version for finite groups? I don't know any textbook reference for that.)

The proof of Bott periodicity that I give in the notes is a mixture of Hatcher's proof with some observations from Husemoller's book, and it uses the Chern Character to prove that the Bott map is injective. This is nice, because the proof of injectivity in Hatcher's notes (or Atiyah's book) is a bit more complicated that the proof of surjectivity. So if you're covering the Chern character anyway, this is a nice route to take.

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If you do look at the notes, comments and corrections would be welcomed! –  Dan Ramras Jul 16 '10 at 17:41
    
Well,this isn't exactly what's being asked for-but it's certainly overlapping subject matter. And let's face it,we don't see lecture notes on characteristic classes that are this nice just anywhere. Good job,Dan-let's see if we can get these TeX-ed sometime,either by you or someone else. –  Andrew L Jul 16 '10 at 18:01
    
Thanks. I will let you know if I find any typos. –  Martin Pinsonnault Jul 18 '10 at 14:46
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I don't know of a single book that does what you want. Perhaps that's because it's hard to top Atiyah & Segal's writings. Pity that Atiyah's book is so expensive. On the other hand, Segal's paper on equivariant K-theory is freely available.

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@Efton I should say I like your book. It is a well written one. @Andrew Atiyah's book really needs some work. I don't think it is easier.

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