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Register a new userChromatic splitting for the $K(2)$-local sphere at $p=2$
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We calculate the homotopy type of $L_1L_{K(2)}S^0$ 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 $L_1L_{K(2)}S^0$ 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 $H^ast_c(mathbb{G}_2,E_0)$ where $mathbb{G}_2$ is the Morava stabilizer group and $E_0 = mathbb{W}[[u_1]]$ is the ring of functions on the height $2$ Lubin-Tate space. We show that the inclusion of the constants $mathbb{W} to E_0$ induces an isomorphism on group cohomology, a radical simplification.
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We calculate the rational homotopy and the K(1)-local homotopy of the K(2)-local sphere at the prime 3 and level 2. We use this to verify the chromatic splitting conjecture in this case.
In this note, we compute the image of the $alpha$-family in the homotopy of the $K(2)$-local sphere at the prime $p=2$ by locating its image in the algebraic duality spectral sequence. This is a stepping stone for the computation of the homotopy groups of the $K(2)$-local sphere at the prime $2$ using the duality spectral sequences.
We calculate the homotopy type of the Brown-Comenetz dual $I_2$ of the K(2)-local sphere at the prime 3 and show that there is a twisting by a non-trivial element $P$ in the exotic part of the Picard group. We give a complete characterization of $P$ as well. The main technique is to give a sequence of calculations of the homotopy groups of elements of the Picard group after smashing with the Smith-Toda complex V(1).
In this paper we use the approach introduced in an earlier paper by Goerss, Henn, Mahowald and Rezk in order to analyze the homotopy groups of L_{K(2)}V(0), the mod-3 Moore spectrum V(0) localized with respect to Morava K-theory K(2). These homotopy groups have already been calculated by Shimomura. The results are very complicated so that an independent verification via an alternative approach is of interest. In fact, we end up with a result which is more precise and also differs in some of its details from that of Shimomura. An additional bonus of our approach is that it breaks up the result into smaller and more digestible chunks which are related to the K(2)-localization of the spectrum TMF of topological modular forms and related spectra. Even more, the Adams-Novikov differentials for L_{K(2)}V(0) can be read off from those for TMF.
Recently, Lawson has shown that the 2-primary Brown-Peterson spectrum does not admit the structure of an $E_{12}$ ring spectrum, thus answering a question of May in the negative. We extend Lawsons result to odd primes by proving that the p-primary Brown-Peterson spectrum does not admit the structure of an $E_{2(p^2+2)}$ ring spectrum. We also show that there can be no map $MU to BP$ of $E_{2p+3}$ ring spectra at any prime.
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