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I'm mainly interested (graduate student) in surgery theory and geometric topology.

If I have a chance to suggest "must read" papers in geometric topology for beginner, I'm very glad to suggest "Topological Library" books volume 1,2,3 (including monumental papers of Smale,Milnor,Kervaire-Milnor,Thom,Serre,Novikov...) available in the following cite.(volume 3 is not available in English edition up to now)

Question: What are "must read" papers in algebraic K-theory? (I hope that most of them can be readable with basic understanding about classical K-theory such as Rosenberg's text or Milnor's ann. math. studies book)

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Only tangentially related, but Lurie is giving a course on surgery theory and L-theory, which might have to do with what you're looking for. He has course notes on his website. – Harry Gindi Feb 5 '11 at 12:58
Warning, it does use a fair bit of black magic, but somehow, as far as I've seen, nothing makes essential use of results from HTT or DAG I. – Harry Gindi Feb 5 '11 at 13:08
Thanks Harry! I did not see that, and I am taking a course on surgery this semester, thes will be fun to compare with my notes. – Sean Tilson Feb 6 '11 at 4:58
@Harry: Lurie's course does not even mention K-theory. It is on L-theory; a completely different story. – Johannes Ebert Jul 23 '12 at 9:44

I'd say, of course Quillen's "Higher algebraic K-theory I", the "K-theory Handbook".

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The first few sections of the Thomason/Trobaugh paper constitute an exceptionally readable overview of the Waldhausen approach to K-theory, with very few prerequisites.

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Steven, could you give a (more) complete reference? – Ryan Budney Jul 23 '12 at 0:33
I see there's two, one you reviewed on MathSciNet and one reviewed by Weibel. – Ryan Budney Jul 23 '12 at 0:37
Ryan Budey: I meant the paper "Higher Algebraic K-Theory of Schemes and of Derived Categories", published in the Grothendieck Festschrift (and apparently unavailable online). I forget whether this is the one I reviewed. I've just spent a couple of hours refreshing my memory of this paper, and I've been reminded anew that there is major enlightenment in every section. I would not hesitate to recommend this to a novice (with a good first-year graduate education); it covers a fantastic amount of ground and is amazingly easy to read. – Steven Landsburg Jul 23 '12 at 1:57

Quillen's "Higher Algebraic K-Theory I" is probably the best source for understanding the basics and the original intuition.

Thomason/Trobaugh is also an excellent paper, but it is a fairly large paper and very fundamental (so the first half of the paper is dedicated to construction of the basic objects).

And if you want a deeper understanding, you could have a look at some of Thomason's older papers, as well as some of Waldhausen's papers.

When I was learning Algebraic K-Theory, I kind of found it easier to understand by going backwards (i.e. I would think of something to get a kind of big picture, ask myself questions about why something might be true, and use that approach to go backwards through Thomason/Trobaugh and if necessary back to older papers). Not everyone will agree with this approach, but I felt that it helped me to build the intuition needed to progress in the subject.

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