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Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E * E is flat over E * . We wish to address the following question: given a commutative E * -algebra A in E * E-comodules, is there an E∞-ring spectrum X with E * X ∼ = A as comodule algebras? We will formulate this as a moduli problem, and give a way -suggested by work of Dwyer, Kan, and Stover -of dissecting the resulting moduli space as a tower with layers governed by appropriate André-Quillen cohomology groups. A special case is A = E * E itself. The final section applies this to discuss the Lubin-Tate or Morava spectra En.Some years ago, Alan Robinson developed an obstruction theory based on Hochschild cohomology to decide whether or not a homotopy associative ring spectrum actually has the homotopy type of an A ∞ -ring spectrum. In his original paper on the subject [35] he used this technique to show that the Morava K-theory spectra K(n) can be realized as an A ∞ -ring spectrum; subsequently, in [3], Andrew Baker used these techniques to show that a completed version of the Johnson-Wilson spectrum E(n) can also be given such a structure. Then, in the mid-90s, the second author and Haynes Miller showed that the entire theory of universal deformations of finite height formal group laws over fields of non-zero characteristic can be lifted to A ∞ -ring spectra in an essentially unique way. This implied, in particular, that the Morava E-theory (or Lubin-Tate) spectra E n were A ∞ (which could have been deduced from Baker's work), but it also showed much more. Indeed, the theory of Lubin and Tate [25] gives a functor from a category of finite height formal group laws to the category of complete local rings, and one way to state the results of [34] is that this functor factors in an essentially unique way through A ∞ -ring spectra. It was the solution of the diagram lifting problem that gave this result its additional heft; for example, it implied that the Morava stabilizer group acted on E nsimply because Lubin-Tate theory implied that this group acted on (E n ) * .In this paper, we would like to carry this program several steps further. One step forward would be to address E ∞ -ring spectra rather than A ∞ -ring spectra. There is an existing literature on this topic developed by Robinson and others, some based on Γ-homology. See [36], [37], and [4]. This can be used, to prove, among other things, that the spectra E n are E ∞ , and we guess that the obstruction theory we uncover here reduces to that theory. Another * The authors were partially supported by the National Science Foundation. 1 step forward would be to write down and try to solve the realization problem as a moduli problem: what is the space of all possible realizations of a spectrum as an A ∞ or E ∞ -ring spectrum, and how can one calculate the homotopy type of this space? Robinson's original work on Morava K-theory implied that this space would often have many components or, put another way, that there could be many A ∞ -realizations of a fixed homotopy ass...

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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

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