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

Abstract

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 L K(2) S 0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E hF 2 where F is a finite subgroup of the Morava stabilizer group and E 2 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

The f-invariant and index theory

“…The algebraic approximation E s, * 2 to π s * (S 0 ) is known completely only for s ≤ 2 and represents the current edge of computational knowledge about π s * (S 0 ), c.f. for example [GHMR05]. We have E 0, * 2 = E 0,0 2 = Z, the groups E 1,t 2 are finite cyclic with order given by denominators of Bernoulli numbers, and E 2, * 2 is very complicated but known explicitly by [MRW77].…”

Morava Modules and the -Local Picard group

“…We use this to give an explicit identification of the E 2 -term of the K(n)-local E n -based Adams spectral sequence (see [1,Appendix A]. This turns out to be related to work of Goerss et al [2], as constructed as part of their resolution of the K(2)-local sphere at the prime 3.…”

“…Then, specialising to the case n = p − 1, we discuss the decomposition of the group of exotic elements, by studying the map from the Picard group of the K(n)-local category to the Picard group of E hG n -modules. We finish by explaining the connection to Gross-Hopkins duality, and [2] outline an approach to constructing elements of κ n when n > 2. In fact, this method already allows us to (independently) construct elements of κ 2 that are nonzero in I 2 , the Gross-Hopkins dual of the sphere.…”

“…The first is to closely study the category of Morava modules, as defined in [2]. It turns out that this category is equivalent to the category of complete E ∨ * E-comodules.…”

On Hopkins’ Picard group Pic2at the prime 3

“…Recall that E n is acted on by the (big) Morava stabilizer group G n D S n Ì Gal.F p n = F p / by E 1 -maps; see Goerss and Hopkins [4]. Let EG n be the category of profinite E n OEOEG n -modules, ie E nmodules with a continuous G n -action compatible with the action of G n on E n (see Goerss, Henn, Mahowald and Rezk [3] or Hopkins, Mahowald and Sadofsky [6] for details). The tensor product (over .E n / ) gives a monoidal structure on EG n .…”