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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.
##### MSC:
 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
##### Keywords:
stable homotopy theory; chromatic homotopy theory
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##### References:
 [1] Almkvist, Gert; Fossum, Robert, Decomposition of exterior and symmetric powers of indecomposable $$\textbf{Z}/p\textbf{Z}$$-modules in characteristic $$p$$ and relations to invariants. S\'eminaire d’Alg\ebre Paul Dubreil, 30\eme ann\'ee (Paris, 1976–1977), Lecture Notes in Math. 641, 1-111, (1978), Springer, Berlin [2] [And69]Anderson D. W. Anderson, \emph Universal coefficient theorems for $$K$$-theory, mimeographed notes, Univ. of California, Berkeley (1969). [3] [BBGS18]det T. Barthel, A. Beaudry, P. G. Goerss, and V. Stojanoska, \emph Constructing the determinant sphere using a Tate twist, arXiv e-prints, 2018. [4] Behrens, Mark, A modular description of the $$K(2)$$-local sphere at the prime 3, Topology, 45, 2, 343-402, (2006) · Zbl 1099.55002 [5] Behrens, Mark, The Goodwillie tower and the EHP sequence, Mem. Amer. Math. Soc., 218, 1026, xii+90 pp., (2012) · Zbl 1330.55012 [6] [BG16]goerss_bobkova I. Bobkova and P. G. Goerss, \emph Topological resolutions in K(2)-local homotopy theory at the prime 2, Journal of Topology 11 (2018), no. 4, 918–957. · Zbl 1419.55014 [7] Goerss, Paul G.; Henn, Hans-Werner, The Brown-Comenetz dual of the $$K(2)$$-local sphere at the prime 3, Adv. Math., 288, 648-678, (2016) · Zbl 1339.55009 [8] Goerss, P.; Henn, H.-W.; Mahowald, M.; Rezk, C., A resolution of the $$K(2)$$-local sphere at the prime 3, Ann. of Math. (2), 162, 2, 777-822, (2005) · Zbl 1108.55009 [9] Greenlees, J. P. C.; May, J. P., Generalized Tate cohomology, Mem. Amer. Math. Soc., 113, 543, viii+178 pp., (1995) · Zbl 0876.55003 [10] Greenlees, J. P. C.; Sadofsky, Hal, The Tate spectrum of $$v_n$$-periodic complex oriented theories, Math. Z., 222, 3, 391-405, (1996) · Zbl 0849.55005 [11] [Hea15]heard_eop D. Heard, \emph The Tate spectrum of the higher real $$K$$-theories at height $$n=p-1$$, arXiv e-prints, 2015. [12] Henn, Hans-Werner, On finite resolutions of $$K(n)$$-local spheres. Elliptic cohomology, London Math. Soc. Lecture Note Ser. 342, 122-169, (2007), Cambridge Univ. Press, Cambridge · Zbl 1236.55015 [13] Hopkins, M. J.; Gross, B. H., Equivariant vector bundles on the Lubin-Tate moduli space. Topology and representation theory, Evanston, IL, 1992, Contemp. Math. 158, 23-88, (1994), Amer. Math. Soc., Providence, RI · Zbl 0807.14037 [14] Hopkins, M. J.; Gross, B. H., The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory, Bull. Amer. Math. Soc. (N.S.), 30, 1, 76-86, (1994) · Zbl 0857.55003 [15] Hopkins, Michael J.; Mahowald, Mark; Sadofsky, Hal, Constructions of elements in Picard groups. Topology and representation theory, Evanston, IL, 1992, Contemp. Math. 158, 89-126, (1994), Amer. Math. Soc., Providence, RI · Zbl 0799.55005 [16] Heard, Drew; Mathew, Akhil; Stojanoska, Vesna, Picard groups of higher real $$K$$-theory spectra at height $$p-1$$, Compos. Math., 153, 9, 1820-1854, (2017) · Zbl 1374.14006 [17] Hovey, Mark; Strickland, Neil P., Morava $$K$$-theories and localisation, Mem. Amer. Math. Soc., 139, 666, viii+100 pp., (1999) · Zbl 0929.55010 [18] Heard, Drew; Stojanoska, Vesna, $$K$$-theory, reality, and duality, J. K-Theory, 14, 3, 526-555, (2014) · Zbl 1325.55003 [19] Mahowald, Mark; Rezk, Charles, Brown-Comenetz duality and the Adams spectral sequence, Amer. J. Math., 121, 6, 1153-1177, (1999) · Zbl 0942.55012 [20] Nave, Lee S., The Smith-Toda complex $$V((p+1)/2)$$ does not exist, Ann. of Math. (2), 171, 1, 491-509, (2010) · Zbl 1194.55017 [21] Ravenel, Douglas C., Localization with respect to certain periodic homology theories, Amer. J. Math., 106, 2, 351-414, (1984) · Zbl 0586.55003 [22] Ravenel, Douglas C., Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics 121, xx+413 pp., (1986), Academic Press, Inc., Orlando, FL · Zbl 0608.55001 [23] Stojanoska, Vesna, Duality for topological modular forms, Doc. Math., 17, 271-311, (2012) · Zbl 1366.55005 [24] Strickland, N. P., Gross-Hopkins duality, Topology, 39, 5, 1021-1033, (2000) · Zbl 0957.55003 [25] Symonds, Peter, The Tate-Farrell cohomology of the Morava stabilizer group $$S_{p-1}$$ with coefficients in $$E_{p-1}$$. Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $$K$$-theory, Contemp. Math. 346, 485-492, (2004), Amer. Math. Soc., Providence, RI · Zbl 1069.55010
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