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Question

Suppose you grasped and enjoyed reading Quillen's "Higher Algebraic K-theory I". Now, if you could go back in time to when you started studying algebraic topology and create a reading list / roadmap with the above paper as a goal, what would this plan look like?

Here's another version of the question: suppose you're the PhD advisor of a student that's recently finished their undergrad degree and in terms of books background has only read and completed most of the exercises in:

  • algebraic geometry: all of Q. Liu's book
  • algebraic topology: first two chapters of W. Massey's intro book
  • differential geometry: first half of J. Lee's smooth manifolds book

Which books and papers (and in what order) should this student master in order to understand (or at least appreciate) most of Quillen 1?

Outlook

The above questions are likely quite vague wrt to "how do I learn modern algebraic k theory?", but hopefully they're somewhat concrete by stating i) the goal [Quillen 1] and ii) the starting maths background.

If it helps, assume a secondary goal is to eventually focus on studying/appreciating arithmetic problems like Parshin's conjecture.

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    $\begingroup$ What do you mean with "has never studied algebraic topology proper"? Has he seen the fundamental group? Homology groups (even deRham)? Some differential topology? $\endgroup$ – Denis Nardin May 23 at 9:37
  • $\begingroup$ @DenisNardin: great question -- I've attempted to clarify some of the assumed book-level background for the hypothetical new grad student $\endgroup$ – Quetzalcoatl May 23 at 18:16
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    $\begingroup$ It's not quite what you're asking but Weibel's K-book is quite nice. The first several chapters are from a more historical perspective so you go through elementary matrices, the whitehead group, K_0, K_1, K_2, Milnor K-theory, and then you start going through various constructions of the algebraic K-theory spectrum. $\endgroup$ – Joe Berner May 23 at 18:41
  • $\begingroup$ On a minimal level, I'd say that reading Kan's On c.s.s. complexes is enough. Of course the real answer is that to learn K-theory it would be a very good idea to learn (the basis of) modern homotopy theory first. $\endgroup$ – Denis Nardin May 24 at 10:35

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