A solution to the Arf-Kervaire invariant problem

Douglas Ravenel
University of Rochester

Abstract: In the past three decades, homotopy theorists have found increasingly deep connections between stable homotopy theory and algebraic geometry. This series of talks will outline the necessary background and describe some recent progress that is joint work with Mike Hill and Mike Hopkins. Here is a brief description. The best approach to the stable homotopy groups of spheres is the Adams-Novikov spectral sequence. This is the Adams spectral sequence based on complex cobordism theory (MU) or equivalently (after localizing at a prime p) Brown-Peterson theory (BP). The relevant algebra is controlled by the theory of 1-dimensional formal group laws. This connection was first discovered by Quillen in 1969, then explored more deeply by Morava in the early 1970s. This was followed by the discovery of the chromatic filtration of the stable homotopy category in the 1980s and the work of Hopkins and Miller in the 1990s.
A formal group law over an algebraically closed field in characteristic p is determined up to isomorphism by an invariant called the height, which is a positive integer n. A height n formal group law has an automorphism group S_n known as the Morava stabilizer group. It is a pro-p-group with interesting arithmetic properties. We now know that it has a canonical action on a certain E_\infty ring spectrum E_n which is difficult to describe explicitly, but very valuable to know. Knowing the cohomology of this action would tell us a lot about stable homotopy, but the problem is prohibitively difficult for n>2.
A more tractable problem involves finite subgroups G of S_n, which have been classified by Hewett. The most interesting cases are ones with order divisible by p, which occur when n is divisible by p-1. In these cases we can look at the homotopy fixed point set of G acting on E_n. Studying this in depth requires some techniques from equivariant stable homotopy theory, which we will introduce as needed.


An article in Nature magazine announcing the solution of the Kervaire invariant problem.

See this webpage for much more information about the Kervaire invariant problem including slides of the lectures below.

Here is a picture from the lectures (with thanks to Luciano Lomonaco).













The lectures will be held in Room P3.10, Department of Mathematics, Instituto Superior Técnico.

Recommended background reading: The Adams spectral sequence is treated in [1] and [3] and there is also a summary in Appendix A of [4]. The first half of [2] gives enough background on equivariant stable homotopy. The chromatic filtration and the Morava stabilizer group are described in the first 40 pages of [4].
  1. J. F. Adams, "Stable homotopy and generalised homology", Chicago Lectures in Mathematics. University of Chicago Press, Chicago, Ill.-London, 1974. x+373 pp.
  2. J. Greenlees and J. P. May, "Equivariant stable homotopy theory", in Handbook of algebraic topology, 277--323, North-Holland, Amsterdam, 1995. Online version .
  3. D. Ravenel, "Complex cobordism and stable homotopy groups of spheres." Pure and Applied Mathematics, 121. Academic Press, Inc., Orlando, FL, 1986. xx+413 pp. Online version .
  4. D. Ravenel, "Nilpotence and periodicity in stable homotopy theory.", Annals of Mathematics Studies, 128. Princeton University Press, Princeton, NJ, 1992. xiv+209 pp.

Preparatory lectures: All in room 4.35 (Mathematics building)

Organizers: Sharon Hollander and Gustavo Granja. Please feel free to contact us for further details.
Sponsors: Center for Mathematical Analysis, Geometry and Dynamical Systems and Fundação para a Ciência e a Tecnologia