## Abstract

We calculate the homotopy type of L_{1}L_{K.2/}S^{0} and L_{K.1/}L_{K.2/}S^{0} at the prime 2, where L_{K.n/} is localization with respect to Morava K–theory and L_{1} localization with respect to 2–local K–theory. In L_{1}L_{K.2/}S^{0} we find all the summands predicted by the Chromatic Splitting Conjecture, but we find some extra summands as well. An essential ingredient in our approach is the analysis of the continuous group cohomology H^{*}.G_{2}; E_{0}/, where G_{2} is the Morava stabilizer group and E_{0} D W ŒŒu_{1} is the ring of functions on the height 2 Lubin–Tate space. We show that the inclusion of the constants W ! E_{0} induces an isomorphism on group cohomology, a radical simplification.

Original language | English (US) |
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Pages (from-to) | 377-476 |

Number of pages | 100 |

Journal | Geometry and Topology |

Volume | 26 |

Issue number | 1 |

DOIs | |

State | Published - 2022 |

## ASJC Scopus subject areas

- Geometry and Topology