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Gross-Hopkins duals of higher real K-theory spectra. (English) Zbl 1426.55001
To understand the \(K(n)\)-local sphere \(\mathbb{S}_{K(n)}\) and its homotopy groups, it is useful to consider \(\mathbb{S}_{K(n)}\) as homotopy fixed point spectrum of the \(n^{th}\) Morava \(E\)-theory \(E_n\) under the action of the Morava stabilizer group \(\mathbb{G}_n\). Using finite subgroups of \(\mathbb{G}_n\), one can hope to construct a resolution of \(\mathbb{S}_{K(n)}\) as in [P. G. Goerss et al., Ann. Math. (2) 162, No. 2, 777–822 (2005; Zbl 1108.55009)]. To understand \(E_n^{hH}\) for a finite subgroup \(H\), it is helpful to then know that \(I_nE_n^{hH} = \Sigma^{k_I}E_n{hH}\) for some number \(k_I\), where \(I_n\) denotes the Gross-Hopkins dual.
This paper determines \(k_I\) for the groups \(C_p\), \(F\) (the maximal finite subgroup of the small Morava stabilizer group \(\mathbb{S}_n\)) and \(G\) (the maximal finite subgroup of \(\mathbb{G}_n\)) in the case \(p \ge 3, n = p-1\), using the homotopy fixed point spectral sequence, Tate spectral sequence and homotopy orbit spectral sequence.
55M05 Duality in algebraic topology
55P42 Stable homotopy theory, spectra
20J06 Cohomology of groups
55Q91 Equivariant homotopy groups
55Q51 \(v_n\)-periodicity
55P60 Localization and completion in homotopy theory
Full Text: DOI arXiv
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