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

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

2007-06-14

arxiv.org/abs/0706.2175v1

Ann. of Math. (2) 162 (2005), no. 2, 777--822

Mathematics

Algebraic Topology

46 pages, published version

Scientific paper

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_2^hF 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.

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