Which of Quillen's Papers Should I read?
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.
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.
Quillen's book on Homotopical Algebra is a great pleasure to read, and likely to appeal to a geometer.