Abstract
Let V(1) be the Toda-Smith complex for the prime 3. We give a complete calculation of the homotopy groups of the L2-localization of V(1) by making use of the higher real K-theory EO 2 of Hopkins and Miller and related homotopy fixed point spectra. In particular we resolve an ambiguity which was left in an earlier approach of Shimomura whose computation was almost complete but left an unspecified parameter still to be determined.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P. Deligne, Courbes elliptiques: Formulaire (d’après J. Tate), in: Modular Functions of One Variable IV, Lecture Notes in Math. 476 (1975), Springer Verlag.
E. Devinatz and M. Hopkins, The action of the Morava stabilizer group on the Lubin-Tate moduli space of lifts, Amer. J. Math. 117 (1995), 669–710.
E. Devinatz and M. Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, to appear in Topology.
V. Gorbounov and P. Symonds, Towards the homotopy groups of the higher real K theory EO2, Homotopy theory via algebraic geometry and group representations. Contemp. Math. 220 (1998), 103–115.
V. Gorbounov, S. Siegel and P. Symonds, Cohomology of the Morava stabilizer group S2 at the prime 3, Proc. Amer. Math. Soc. 126 (1998), 933–941.
H.-W. Henn, Centralizers of elementary abelian p-subgroups and mod-p cohomology of profinite groups, Duke Math. J. 91 (1998), 561–585.
[LT] J. Lubin and J. Tate, Formal moduli for one-parameter formel Lie groups, Bull. Soc. Math. France 94 (1966), 49–60.
L. Nave, On the nonexistence of Smith-Toda complexes, Preprint (available at http://hopf.math.purdue.edu)
D. Ravenel, The Cohomology of the Morava Stabilizer Algebras, Math. Zeit. 152 (1977), 287–297.
D. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press, 1986.
D. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Ann. of Math. Studies, Princeton University Press, 1992.
C. Rezk, Notes on the Hopkins-Miller theorem, Homotopy theory via algebraic geometry and group representations. Contemp. Math. 220 (1998), 313–366.
K. Shimomura, The homotopy groups of the L2-localized Toda-Smith complex V(1) at the prime 3, Trans. Amer. Math. Soc. 349 (1997), 1821–1850.
K. Shimomura, The homotopy groups of the L2-localized mod-3 Moore spectrum at the prime 3, J. Math. Soc. Japan 52 (2000), 65–90.
K Shimomura and A. Yabe, The homotopy groups x.(L2S0), Topology 34 (1995), 261–289.
K. Shimomura and X. Wang, The homotopy groups 7,42S0) at the prime 3, Topology 41 (2002), 1183–1198.
N. Strickland, Gross-Hopkins duality, Topology 39 (2000), 1021–1033.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Basel AG
About this chapter
Cite this chapter
Goerss, P., Henn, HW., Mahowald, M. (2003). The Homotopy of L 2 V(1) for the Prime 3. In: Arone, G., Hubbuck, J., Levi, R., Weiss, M. (eds) Categorical Decomposition Techniques in Algebraic Topology. Progress in Mathematics, vol 215. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7863-0_8
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7863-0_8
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9601-6
Online ISBN: 978-3-0348-7863-0
eBook Packages: Springer Book Archive