Volume 10, issue 1 (2010)

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On Hopkins' Picard group $\mathrm{Pic}_2$ at the prime $3$

Nasko Karamanov

Algebraic & Geometric Topology 10 (2010) 275–292
Abstract

In this paper we calculate the algebraic Hopkins Picard group ${Pic}_{2}^{alg}$ at the prime $p=3$, which is a subgroup of the group of isomorphism classes of invertible $K\left(2\right)$–local spectra, ie of Hopkins’ Picard group ${Pic}_{2}$. We use the resolution of the $K\left(2\right)$–local sphere introduced by Goerss, Henn, Mahowald and Rezk in [Ann. of Math (2) 162 (2005) 777-822] and the methods from Henn, Karamanov and Mahowald [to appear in Math. Zeit. arXiv:0811.0235] and Karamanov [PhD thesis, Universite Louis Pasteur (2006)].

Keywords
Hopkins Picard group, Morava $K$–theory, $K(2)$–local sphere
Mathematical Subject Classification 2000
Primary: 55N22, 55P42, 55Q10, 55Q51, 55Q52
Publication
Received: 30 March 2009
Revised: 20 November 2009
Accepted: 24 November 2009
Published: 12 February 2010
Authors
 Nasko Karamanov Augsburger Strasse 36E 93051 Regensburg Germany