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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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    $\begingroup$ Allen Hatcher's book: math.cornell.edu/~hatcher/VBKT/VB.pdf is awesome, but unfinished :( $\endgroup$
    – Tom Boardman
    Commented Jul 16, 2010 at 16:44
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    $\begingroup$ 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. $\endgroup$
    – Paul Siegel
    Commented Jul 16, 2010 at 17:10

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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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  • $\begingroup$ @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. $\endgroup$
    – The Mathemagician
    Commented Jul 16, 2010 at 17:13
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    $\begingroup$ 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. $\endgroup$
    – Rasmus
    Commented Jul 16, 2010 at 18:05
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    $\begingroup$ 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 . $\endgroup$
    – Georges Elencwajg
    Commented Jul 16, 2010 at 18:12
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    $\begingroup$ When I was a grad student, another student and I read through this book one summer. I liked it quite a bit. $\endgroup$
    – Josh
    Commented Jul 16, 2010 at 20:13
  • $\begingroup$ Thanks. I did not know that book and, indeed, it covers a lot of what I need. $\endgroup$
    – Martin Pinsonnault
    Commented Jul 18, 2010 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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  • $\begingroup$ If you do look at the notes, comments and corrections would be welcomed! $\endgroup$
    – Dan Ramras
    Commented Jul 16, 2010 at 17:41
  • $\begingroup$ 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. $\endgroup$
    – The Mathemagician
    Commented Jul 16, 2010 at 18:01
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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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  • $\begingroup$ The link to Segal's paper at springerlink.com is broken. I'm also unable to find any snapshot saved on the Wayback Machine. $\endgroup$
    – The Amplitwist
    Commented Jul 29, 2022 at 20:24
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There are two(or three maybe) way to go to the topological K-theory, one is from the algebraic topology(or vector bundles), the other is from(download) the operator K-theory(the K-theory of C*-algebras).

Form the algebraic topology: there are many second course book mention it, for example:

May J P. A concise course in algebraic topology[M]. University of Chicago Press, 1999.

Switzer R M. Algebraic topology--homotopy and homology[M]. Springer, 1975.

Aguilar M, Gitler S, Prieto C. Algebraic topology from a homotopical viewpoint[M]. Springer Science & Business Media, 2008.

From the vector bundle:

Hatcher A. Vector bundles and K-theory[J]. Im Internet unter http://www.math.cornell.edu/~hatcher, 2003.

D. Husemoller, Fibre bundles. Graduate Texts in Mathematics, 20. Springer-Verlag, New York, 1994.

There is also an online course note:Algebraic Topology II: Topological K-Theory (Spring 2015) http://www.math.ru.nl/~gutierrez/k-theory2015.html

From the operator K-theory(K-theory of C*-algebras): maybe the only one is:

Park E. Complex topological K-theory[M]. Cambridge University Press, 2008.

Some K-theory of C*-algebras books also mention a little topological K-theory as a background, you can see this book:

Blackadar B. K-theory for operator algebras[M]. Cambridge University Press, 1998.

I am making some videos of K-theory(from topological to operator) in my language Chinese, if you can read Chinese or have some friend help to translate, you can see them in my blog: http://blog.sina.com.cn/s/articlelist_1215048895_12_1.html

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