# Which of Quillen's Papers Should I read?

I just heard that Dan Quillen passed on. I am not familiar with his work and want to celebrate his life by reading some of his papers. Which one(s?) should I read?

I am an algebraic geometry who is comfortable with cohomological methods in his field, but knows almost nothing about homotopy theory. My goal is to gain a better appreciation of Quillen's work, not to advance my own research.

I tagged this question as "at.algebraic-topology, algebraic-k-theory" because I think these are the main fields in which Quillen worked. Please add or change this if other tags are appropriate.

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He won the fields medal largely for his article on algebraic K-theory (LNM 341). I recommend starting there. – Dan Ramras May 6 '11 at 4:44
Sad to hear that. IMO one should put his article free online. Memories from Landsburg: thebigquestions.com/2011/05/03/the-architect – Thomas Riepe May 6 '11 at 4:50
His "Higher K-theory" paper, mentioned by Dan, is a real pleasure to read. His computation of K-theory of a finite field, on the other hand, is a nice example of how one can compute difficult things while writing clearly :) – Mariano Suárez-Alvarez May 6 '11 at 5:16
This is a very touching request, jlk. – Georges Elencwajg May 6 '11 at 6:50
From meta, regarding his death: tea.mathoverflow.net/discussion/1036/dan-quillen (a message from his wife). – danseetea May 6 '11 at 7:23

Can I be the first to recommend Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 1971 29–56 (1971).

From the MR review: "In this important and elegant paper the author gives new elementary proofs of the structure theorems for the unoriented cobordism ring $N^\ast$ and the complex cobordism ring $U^\ast$, together with new results and methods. Everyone working in cobordism theory should read this paper."

The paper was revolutionary in (at least) two ways.

1. The proofs are almost entirely geometric, with cobordism classes represented by proper oriented maps of manifolds. Quillen cites Grothendieck as inspiration for this, and such methods should appeal to algebraic geometers familiar with the Chow ring.
2. Formal group methods are used to prove results in stable homotopy theory. It's hard to overestimate the impact this has had. Indeed almost all of the modern connections between homotopy theory and algebraic geometry seem to go through formal groups, drawing influence from Quillen's idea.
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I quite agree with your line "it's hard to overestimate the impact this has had." The connection between formal groups and complex cobordism has always struck me as one of the most unexpected connections in mathematics. Professor Quillen seemed adept at finding such unexpected connections. – David White May 9 '11 at 21:02

Perhaps as an algebraic geometer, you might read his paper:On the (co-)homology of commutative rings , in Proc. Symp. on Categorical Algebra , (1970), 65–87, American Math. Soc.

This is fairly short and yet is one of the key papers in the area. It introduces ideas of homotopical algebra, which have been crucial in algebraic topology and algebraic geometry, yet the paper does not need a whole load of prior knowledge. The lecture notes on homotopical algebra and the paper on rational homotopy theory are beautiful in their use of techniques from adjacent areas to solve hard general problems and follow on from that initial work.

The algebraic K-theory papers are another thread, but these have been mentioned above.

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This is an important paper, but it's a little bit terse. There's an unpublished mimeographed typescript called "Cohomology of Commutative Rings" by Quillen, which goes over the same ground as this paper, but more gently. I don't know from where or when it comes from, and I don't know anywhere online you can get a hold of it. But if you can find it, it's good; I much prefer it to the published paper. – Charles Rezk May 6 '11 at 14:56
Here's a digital copy of the unpublished version, hopefully no one minds the linking: chromotopy.org/paste/quillen.djvu . – Eric Peterson May 6 '11 at 17:34
Anyone knows what's the state of conjecture 10.16 in those notes? – Mariano Suárez-Alvarez May 6 '11 at 18:47
These notes are from MIT 1968; see item 23 in the bibliography of Iyengar's notes on Andre-Quillen homology math.uic.edu/~bshipley/iyengar.pdf – SGP May 7 '11 at 2:52
@Mariano: it seems tha conjecture has been proved by Avramov, see Iyengar's paper.. – SGP May 7 '11 at 2:54

Quillen's book on Homotopical Algebra is a great pleasure to read, and likely to appeal to a geometer.

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It is really good, I am spending time with it now. I was going to suggest it if no one else did. – Sean Tilson May 6 '11 at 12:58

There are a number of papers that I have affection for. Those which I don't see listed:

"On the group completion of a simplicial monoid." Before I read this paper I never really understood the appearance of the plus-construction. This essentially proves the +=Q theorem. (A little harder to find: Appendix Q in Friedlander-Mazur's "Filtrations on the homology of algebraic varieties.")

"On the formal group laws of unoriented and complex cobordism theory." A very influential paper, along the likes that Mark Grant mentioned, and one which we've been trying to unravel the consequences of ever since. (Bull. Amer. Math. Soc. 75 1969, 1293–1298.)

"The Adams conjecture." Quillen's proof of the Adams conjecture by making use of a Brauer lift is short but wonderful. (Topology 10 1971 67–80.)

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I would definitely suggest his two-part paper, The Spectrum of an Equivariant Cohomology Ring.

He develops a ton of information concerning $H_G^*(X)$, and forms the basis for much of its use in future papers.

In terms of studying group cohomology, you should also check out his short paper Cohomology of Finite Groups and Elementary Abelian Subgroups (by Quillen and Venkov) which establishes the celebrated result: If $u\in H^*(G,\mathbb{Z}_p)$ restricts to zero on every elementary abelian $p$-subgroup of $G$ (a finite group), then $u$ is nilpotent. This paper is not even two pages long, although his original proof was contained in a different paper ("A Cohomological Criterion for p-Nilpotence").

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Being a mathematical physicist, I have a particular fondness for his papers in Topology in the 1980s, alone or with Mathai, on the subject of superconnections:

• Superconnections and the Chern character, Topology 24 (1985) 89-95

• Superconnections, Thom classes, and equivariant differential forms, Topology 25 (1986) 85-100

• Superconnection character forms and the Cayley transform, Topology 27 (1988) 211-238

All are sadly hidden behind an Elsevier paywall...

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