TAILIEUCHUNG - Đề tài " 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 0 as the inverse limit of a tower of hF fibrations with four layers. The successive fibers are of the form E2 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. | Annals of Mathematics A resolution of the K 2 -local sphere at the prime 3 By P. Goerss . Henn M. Mahowald and C. Rezk Annals of Mathematics 162 2005 777-822 A resolution of the K 2 -local sphere at the prime 3 By P. Goerss . Henn M. Mahowald and C. Rezk 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 LK 2 S0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E F 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 .

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