### 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 _{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.

Original language | English (US) |
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Pages (from-to) | 777-822 |

Number of pages | 46 |

Journal | Annals of Mathematics |

Volume | 162 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2005 |

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Annals of Mathematics*,

*162*(2), 777-822. https://doi.org/10.4007/annals.2005.162.777