We calculate the homotopy type of L1LK.2/S0 and LK.1/LK.2/S0 at the prime 2, where LK.n/ is localization with respect to Morava K–theory and L1 localization with respect to 2–local K–theory. In L1LK.2/S0 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*.G2; E0/, where G2 is the Morava stabilizer group and E0 D W ŒŒu1 is the ring of functions on the height 2 Lubin–Tate space. We show that the inclusion of the constants W ! E0 induces an isomorphism on group cohomology, a radical simplification.
ASJC Scopus subject areas
- Geometry and Topology