The 𝐿₂-localization of 𝑊(𝑛)

Thompson, Robert

doi:10.1090/S0002-9947-98-02149-7

The $L_2$-localization of $W(n)$
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Trans. Amer. Math. Soc. 350 (1998), 1931-1944 Request permission

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

A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281. MR 551009, DOI 10.1016/0040-9383(79)90018-1
A. K. Bousfield, $K$-localizations and $K$-equivalences of infinite loop spaces, Proc. London Math. Soc. (3) 44 (1982), no. 2, 291–311. MR 647434, DOI 10.1112/plms/s3-44.2.291
A. K. Bousfield, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994), no. 4, 831–873. MR 1257059, DOI 10.1090/S0894-0347-1994-1257059-7
J. Caruso, F. R. Cohen, J. P. May, and L. R. Taylor, James maps, Segal maps, and the Kahn-Priddy theorem, Trans. Amer. Math. Soc. 281 (1984), no. 1, 243–283. MR 719669, DOI 10.1090/S0002-9947-1984-0719669-2
Frederick R. Cohen, Thomas J. Lada, and J. Peter May, The homology of iterated loop spaces, Lecture Notes in Mathematics, Vol. 533, Springer-Verlag, Berlin-New York, 1976. MR 0436146
F. R. Cohen, J. P. May, and L. R. Taylor, Splitting of certain spaces $CX$, Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 3, 465–496. MR 503007, DOI 10.1017/S0305004100055298
F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Torsion in homotopy groups, Ann. of Math. (2) 109 (1979), no. 1, 121–168. MR 519355, DOI 10.2307/1971269
Frederick R. Cohen, The unstable decomposition of $\Omega ^{2}\Sigma ^{2}X$ and its applications, Math. Z. 182 (1983), no. 4, 553–568. MR 701370, DOI 10.1007/BF01215483
Donald M. Davis and Mark Mahowald, $v_1$-periodicity in the unstable Adams spectral sequence, Math. Z. 204 (1990), no. 3, 319–339. MR 1107466, DOI 10.1007/BF02570877
E. Dror Farjoun and J. H. Smith, Homotopy localization nearly preserves fibrations, preprint.
Brayton Gray, Desuspension at an odd prime, Algebraic topology, Aarhus 1982 (Aarhus, 1982) Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 360–370. MR 764589, DOI 10.1007/BFb0075577
Brayton Gray, On the iterated suspension, Topology 27 (1988), no. 3, 301–310. MR 963632, DOI 10.1016/0040-9383(88)90011-0
Brayton Gray, $EHP$ spectra and periodicity. I. Geometric constructions, Trans. Amer. Math. Soc. 340 (1993), no. 2, 595–616. MR 1152323, DOI 10.1090/S0002-9947-1993-1152323-X
Brayton Gray, $EHP$ spectra and periodicity. II. $\Lambda$-algebra models, Trans. Amer. Math. Soc. 340 (1993), no. 2, 617–640. MR 1152324, DOI 10.1090/S0002-9947-1993-1152324-1
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 664045, DOI 10.1090/S0002-9947-1982-0664045-2
M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory, II, to appear in Ann. of Math.
Daniel S. Kahn, On the stable decomposition of $\Omega ^{\infty }S^{\infty }A$, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 206–214. MR 513576
Nicholas J. Kuhn, The geometry of the James-Hopf maps, Pacific J. Math. 102 (1982), no. 2, 397–412. MR 686560
Nicholas J. Kuhn, The transfer and James-Hopf invariants, Math. Z. 196 (1987), no. 3, 391–405. MR 913664, DOI 10.1007/BF01200360
Lisa Langsetmo and Robert D. Thompson, Some applications of $K(1)_*W(n)$, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 339–354. MR 1320999, DOI 10.1090/conm/181/02041
Mark Mahowald, On the double suspension homomorphism, Trans. Amer. Math. Soc. 214 (1975), 169–178. MR 438333, DOI 10.1090/S0002-9947-1975-0438333-5
Mark Mahowald, The image of $J$ in the $EHP$ sequence, Ann. of Math. (2) 116 (1982), no. 1, 65–112. MR 662118, DOI 10.2307/2007048
Mark Mahowald and Robert Thompson, A commentary on: “Image of $J$ in the $EHP$ sequence” [Ann. of Math. (2) 116 (1982), no. 1, 65–112; MR0662118 (83i:55019)] by Mahowald, Algebraic topology (Seattle, Wash., 1985) Lecture Notes in Math., vol. 1286, Springer, Berlin, 1987, pp. 298–304. MR 922933, DOI 10.1007/BFb0078748
Mark Mahowald and Robert D. Thompson, The $K$-theory localization of an unstable sphere, Topology 31 (1992), no. 1, 133–141. MR 1153241, DOI 10.1016/0040-9383(92)90066-Q
Mark Mahowald and Robert D. Thompson, The fiber of the secondary suspension map, Amer. J. Math. 116 (1994), no. 1, 179–205. MR 1262430, DOI 10.2307/2374985
Binhua Mao, On the realizations of certain ${A}^*$-modules and their properties, Ph.D. thesis, University of Rochester, 1993.
R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger, $H_\infty$ ring spectra and their applications, Lecture Notes in Mathematics, vol. 1176, Springer-Verlag, Berlin, 1986. MR 836132, DOI 10.1007/BFb0075405
Douglas C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), no. 2, 351–414. MR 737778, DOI 10.2307/2374308
V. P. Snaith, A stable decomposition of $\Omega ^{n}S^{n}X$, J. London Math. Soc. (2) 7 (1974), 577–583. MR 339155, DOI 10.1112/jlms/s2-7.4.577
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 1010890, DOI 10.1090/S0002-9947-1990-1010890-8