A resolution of the K ( 2 ) - local sphere at the prime 3

We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum LK(2)S as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form EhF 2 where F is a finite subgroup of the Morava stabilizer group and E2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n = 2 at p = 3 represents the edge of our current knowledge: n = 1 is classical and at n = 2, the prime 3 is the largest prime where the Morava stabilizer group has a p-torsion subgroup, so that the homotopy theory is not entirely algebraic. The problem of understanding the homotopy groups of spheres has been central to algebraic topology ever since the field emerged as a distinct area of mathematics. A period of calculation beginning with Serre’s computation of the cohomology of Eilenberg-MacLane spaces and the advent of the Adams spectral sequence culminated, in the late 1970s, with the work of Miller, Ravenel, and Wilson on periodic phenomena in the homotopy groups of spheres and Ravenel’s nilpotence conjectures. The solutions to most of these conjectures by Devinatz, Hopkins, and Smith in the middle 1980s established the primacy of the “chromatic” point of view and there followed a period in which the community absorbed these results and extended the qualitative picture of stable homotopy theory. Computations passed from center stage, to some extent, although there has been steady work in the wings – most notably by Shimomura and his coworkers, and Ravenel, and more lately by Hopkins and *The first author and fourth authors were partially supported by the National Science Foundation (USA). The authors would like to thank (in alphabetical order) MPI at Bonn, Northwestern University, the Research in Pairs Program at Oberwolfach, the University of Heidelberg and Université Louis Pasteur at Strasbourg, for providing them with the opportunity to work together. 778 P. GOERSS, H.-W. HENN, M. MAHOWALD, AND C. REZK his coauthors in their work on topological modular forms. The amount of interest generated by this last work suggests that we may be entering a period of renewed focus on computations. In a nutshell, the chromatic point of view is based on the observation that much of the structure of stable homotopy theory is controlled by the algebraic geometry of formal groups. The underlying geometric object is the moduli stack of formal groups. Much of what can be proved and conjectured about stable homotopy theory arises from the study of this stack, its stratifications, and the theory of its quasi-coherent sheaves. See for example, the table in Section 2 of [11]. The output we need from this geometry consists of two distinct pieces of data. First, the chromatic convergence theorem of [21, §8.6] says the following. Fix a prime p and let E(n)∗, n ≥ 0 be the Johnson-Wilson homology theories and let Ln be localization with respect to E(n)∗. Then there are natural maps LnX → Ln−1X for all spectra X, and if X is a p-local finite spectrum, then the natural map X−→ holimLnX is a weak equivalence. Second, the maps LnX → Ln−1X fit into a good fiber square. Let K(n)∗ denote the n-th Morava K-theory. Then there is a natural commutative diagram