×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv
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 <span class=”textbf”>1</span>1 (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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.