Which of Quillen's Papers Should I read?

Question

I just heard that Daniel 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 geometer 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.

Here 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.

He won the fields medal largely for his article on algebraic K-theory (LNM 341). I recommend starting there. — Dan Ramras, May 06 '11 at 04:44
Sad to hear that. IMO one should put his article free online. Memories from Landsburg: http://www.thebigquestions.com/2011/05/03/the-architect/ — Thomas Riepe, May 06 '11 at 04:50
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-Álvarez, May 06 '11 at 05:16
@Mariano Suárez-Alvarez: Which paper contain his computation of the K-theory of a finite field? — jlk, May 06 '11 at 05:45
@jlk: the paper is [Quillen, Daniel. On the cohomology and $K$-theory of the general linear groups over a finite field. Ann. of Math. (2) 96 (1972), 552--586. MR0315016 (47 #3565)] — Mariano Suárez-Álvarez, May 06 '11 at 05:59
@jlk: ''On the cohomology and K-theory of the general linear groups over finite fields'', Annals Math. 96. — Johannes Ebert, May 06 '11 at 06:33
@Thomas Riepe: "IMO one should put his article free online.". Of course the company who owns the copyright might have other opinions :-) especially if we're talking about LNM 341 and hence that company is Sptinger... — Kevin Buzzard, May 06 '11 at 06:56
From meta, regarding his death: http://tea.mathoverflow.net/discussion/1036/dan-quillen/ (a message from his wife). — danseetea, May 06 '11 at 07:23
@Kevin Buzzard: Can companies have opinions? A still working pre-Internet download site, i.e. a "books for all and for free"-shop with unfortunately underfed math department: http://www.fensterzumhof.eu/bilder/v/Fassaden-der-Hauptstadt/Buecher-fuer-alle-sozialer-Buchladen.html — Thomas Riepe, Jun 02 '11 at 19:54

score 22 · Accepted Answer · edited May 06 '11 at 07:29

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.

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.
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.

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 09 '11 at 21:02

score 18 · Answer 2 · answered May 06 '11 at 05:51

18

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.

answered May 06 '11 at 05:51

Tim Porter

9,187

Is it somewhere free online? – Thomas Riepe May 06 '11 at 06:07
It would seem to be on Google books (without out any pages being omitted). – Tim Porter May 06 '11 at 06:39
4

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 06 '11 at 14:56
@Charles: I'd love to see that. – Mariano Suárez-Álvarez May 06 '11 at 16:40
15

Here's a digital copy of the unpublished version, hopefully no one minds the linking: http://chromotopy.org/paste/quillen.djvu . – Eric Peterson May 06 '11 at 17:34
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Anyone knows what's the state of conjecture 10.16 in those notes? – Mariano Suárez-Álvarez May 06 '11 at 18:47
@Peterson: Many thanks! I had been looking for these notes for a long time. – SGP May 07 '11 at 02:49
1

These notes are from MIT 1968; see item 23 in the bibliography of Iyengar's notes on Andre-Quillen homology http://www.math.uic.edu/~bshipley/iyengar.pdf – SGP May 07 '11 at 02:52
1

@Mariano: it seems tha conjecture has been proved by Avramov, see Iyengar's paper.. – SGP May 07 '11 at 02:54
Ah. He was my candidate for having settled it :) – Mariano Suárez-Álvarez May 09 '11 at 15:25
As of right now, this article is available from AMS books online here: http://www.ams.org/books/pspum/017. (Also as of now, Google doesn't seem to know this.) – Edgardo Jun 24 '14 at 05:26

score 13 · Answer 3 · answered May 06 '11 at 12:54

13

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

answered May 06 '11 at 12:54

Steven Landsburg

22,477

1

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 06 '11 at 12:58

score 11 · Answer 4 · answered May 06 '11 at 12:43

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.)

score 8 · Answer 5 · answered May 06 '11 at 06:39

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").

score 4 · Answer 6 · answered May 06 '11 at 07:46

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...

Which of Quillen's Papers Should I read?

6 Answers6