Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The $L_2$-localization of $W(\lowercase{n})$


Author: Robert D. Thompson
Journal: Trans. Amer. Math. Soc. 350 (1998), 1931-1944
MSC (1991): Primary 55P60; Secondary 55Q52, 55T15
DOI: https://doi.org/10.1090/S0002-9947-98-02149-7
MathSciNet review: 1467475
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we analyze the localization of $W(n)$, the fiber of the double suspension map $S^{2n-1}\to \Omega^{2}S^{2n+1}$, with respect to $E(2)$. If four cells at the bottom of $D_pM^{2np-1}$, the $p$th extended power spectrum of the Moore spectrum, are collapsed to a point, then one obtains a spectrum $C$. Let $QM^{2np-1}\to QC$ be the James-Hopf map followed by the collapse map. Then we show that the secondary suspension map $BW(n)\to QM^{2np-1}$ has a lifting to the fiber of $QM^{2np-1}\to QC$ and this lifting is shown to be a $v_2$-periodic equivalence, hence an $E(2)$-equivalence.


References [Enhancements On Off] (What's this?)

  • 1. A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), 257-281. MR 80m:55006
  • 2. -, $K$-localizations and $K$-equivalences of infinite loop spaces, Proc. London Math. Soc. (3) 44 (1982), 291-311. MR 83g:55008
  • 3. -, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994), no. 4, 831-873. MR 95c:55010
  • 4. J. Caruso, F. Cohen, J. P. May, and L. Taylor, James maps, Segal maps, and the Kahn-Priddy theorem, Trans. Amer. Math. Soc. 218 (1984), 243-283. MR 86g:55007
  • 5. F. Cohen, T. Lada, and J. P. May, The homology of iterated loop spaces, Lecture Notes in Math., vol. 533, Springer-Verlag, 1976. MR 55:9096
  • 6. F. Cohen, J. P. May, and L. R. Taylor, Splitting of certain spaces ${CX}$, Math. Proc. Cambridge Philos. Soc. 84 (1978), 465-496. MR 80a:55010
  • 7. F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Torsion in homotopy groups, Ann. of Math. 109 (1979), 121-168. MR 80e:55024
  • 8. Frederick R. Cohen, The unstable decomposition of ${\Omega^2}{\Sigma^2}{X}$ and its applications, Math. Z. (1983), 553-568. MR 85d:55008
  • 9. Donald M. Davis and Mark Mahowald, $v_1$-periodicity in the unstable Adams spectral sequence, Math. Z. 204 (1990), 319-339. MR 92e:55018
  • 10. E. Dror Farjoun and J. H. Smith, Homotopy localization nearly preserves fibrations, preprint.
  • 11. Brayton Gray, Desuspension at an odd prime, Algebraic Topology, Proceedings, Aarhus 1982, Lecture Notes in Math., vol. 1051, Springer-Verlag, 1984, pp. 360-370. MR 86d:55017
  • 12. -, On the iterated suspension, Topology 27 (1988), no. 3, 301-310. MR 89h:55016
  • 13. -, EHP spectra and periodicity. I: Geometric constructions, Trans. Amer. Math. Soc. 340 (1993), no. 2, 595-616. MR 94c:55015
  • 14. -, EHP spectra and periodicity. II: $\Lambda$-algebra models, Trans. Amer. Math. Soc. 340 (1993), no. 2, 617-640. MR 94c:55016
  • 15. J. R. Harper and H. R. Miller, On the double suspension homomorphism at odd primes, Trans. Amer. Math. Soc. 273 (1982), no. 1, 319-331. MR 83h:55037
  • 16. M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory, II, to appear in Ann. of Math.
  • 17. D. Kahn, On the stable decomposition of ${\Omega^{\infty}}{\Sigma^{\infty}} A$, Lecture Notes in Mathematics, vol. 658, Springer-Verlag, Berlin and New York, 1978, pp. 206-213. MR 80d:55010
  • 18. Nicholas J. Kuhn, The geometry of James-Hopf maps, Pacific J. Math. 102 (1982), no. 2, 397-412. MR 85c:55006
  • 19. -, The transfer and James-Hopf invariants, Math. Z. 196 (1987), 391-405. MR 89d:55033
  • 20. Lisa Langsetmo and Robert D. Thompson, Some applications of $K(1)_*W(n)$, The \v{C}ech Centennial, Cont. Math., vol. 181, Amer. Math. Soc., 1995, pp. 339-354. MR 95m:55019
  • 21. Mark Mahowald, On the double suspension homomorphism, Trans. Amer. Math. Soc. 214 (1975), 169-178. MR 55:11248
  • 22. -, The image of $J$ in the EHP sequence, Ann. of Math. (2) 116 (1982), 65-112. MR 83i:55019
  • 23. Mark Mahowald and Robert Thompson, A commentary on the `image of J in the EHP sequence', Algebraic Topology, Proceedings, Seattle 1985, Lecture Notes in Math., vol. 1286, Springer-Verlag, 1987, pp. 298-304. MR 88j:55012
  • 24. -, The $K$-theory localization of an unstable sphere, Topology 31 (1992), no. 1, 133-141. MR 92k:55017
  • 25. -, The fiber of the secondary suspension map, Amer. J. Math. 116 (1994), 179-206. MR 95f:55012
  • 26. Binhua Mao, On the realizations of certain ${A}^*$-modules and their properties, Ph.D. thesis, University of Rochester, 1993.
  • 27. J. E. McClure, The mod $p$ ${K}$-theory of ${QX}$, $H_{\infty}$-ring spectra and their applications, Lecture Notes in Math., vol. 1176, Springer-Verlag, 1986, pp. 291-377. MR 88e:55001
  • 28. Douglas C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), 351-414. MR 85k:55009
  • 29. V. Snaith, A stable decomposition for ${\Omega^n}{\Sigma^n} {X}$, J. London Math. Soc. 7 (1974), 577-583. MR 49:3918
  • 30. Robert D. Thompson, The $v_1$-periodic homotopy groups of an unstable sphere at odd primes, Trans. Amer. Math. Soc. 319 (1990), no. 2, 535-559. MR 90j:55021

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 55P60, 55Q52, 55T15

Retrieve articles in all journals with MSC (1991): 55P60, 55Q52, 55T15


Additional Information

Robert D. Thompson
Affiliation: Hunter College and the Graduate Center, CUNY, 695 Park Avenue, New York, New York 10021
Email: thompson@math.hunter.cuny.edu

DOI: https://doi.org/10.1090/S0002-9947-98-02149-7
Keywords: \protect{$L_2$}-localization, double suspension, \protect{$v_2$}-periodicity
Received by editor(s): July 23, 1996
Additional Notes: The author was partially supported by PSC-CUNY Grant 667399
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society