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)



On the telescopic homotopy theory of spaces

Author: A. K. Bousfield
Journal: Trans. Amer. Math. Soc. 353 (2001), 2391-2426
MSC (2000): Primary 55P60; Secondary 55N20, 55P42, 55P65, 55U35
Published electronically: July 18, 2000
MathSciNet review: 1814075
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In telescopic homotopy theory, a space or spectrum $X$ is approximated by a tower of localizations $L^{f}_{n}X$, $n\ge 0$, taking account of $v_{n}$-periodic homotopy groups for progressively higher $n$. For each $n\ge 1$, we construct a telescopic Kuhn functor $\Phi _{n}$ carrying a space to a spectrum with the same $v_{n}$-periodic homotopy groups, and we construct a new functor $\Theta _{n}$ left adjoint to $\Phi _{n}$. Using these functors, we show that the $n$th stable monocular homotopy category (comprising the $n$th fibers of stable telescopic towers) embeds as a retract of the $n$th unstable monocular homotopy category in two ways: one giving infinite loop spaces and the other giving ``infinite $L^{f}_{n}$-suspension spaces.'' We deduce that Ravenel's stable telescope conjectures are equivalent to unstable telescope conjectures. In particular, we show that the failure of Ravenel's $n$th stable telescope conjecture implies the existence of highly connected infinite loop spaces with trivial Johnson-Wilson $E(n)_{*}$-homology but nontrivial $v_{n}$-periodic homotopy groups, showing a fundamental difference between the unstable chromatic and telescopic theories. As a stable chromatic application, we show that each spectrum is $K(n)$-equivalent to a suspension spectrum. As an unstable chromatic application, we determine the $E(n)_{*}$-localizations and $K(n)_{*}$-localizations of infinite loop spaces in terms of $E(n)_{*}$-localizations of spectra under suitable conditions. We also determine the $E(n)_{*}$-localizations and $K(n)_{*}$-localizations of arbitrary Postnikov $H$-spaces.

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

  • [Ada] J. F. Adams, Stable Homotopy and Generalized Homology, University of Chicago Press, 1974. MR 53:6534
  • [Bor] F. Borceux, Handbook of Categorical Algebra 2, Categories and Structures, Encyclopedia of Mathematics and its Applications, vol. 51, Cambridge University Press, 1994. MR 96g:18001b
  • [Bou1] A.K. Bousfield, Boolean algebra of spectra, Comment. Math. Helv 54 (1979), 368-377; Correction, Comment Math. Helv. 58 (1983), 599-600. MR 81a:55015; MR 85h:55013
  • [Bou2] -, The localization of spectra with respect to homology, Topology 18 (1979), 257-281. MR 80m:55006
  • [Bou3] -, Cohomological localizations of spaces and spectra, preprint, 1979.
  • [Bou4] -, $K$-localizations and $K$-equivalences of infinite loop spaces, Proc. London Math. Soc. 44 (1982), 291-311. MR 83g:55008
  • [Bou5] -, On homology equivalences and homological localizations of spaces, Amer. J. Math. 104 (1982), 1025-1042. MR 84g:55014
  • [Bou6] -, Uniqueness of infinite deloopings for $K$-theoretic spaces, Pacific J. Math. 129 (1987), 1-31. MR 89g:55017
  • [Bou7] -, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994), 831-873. MR 95c:55010
  • [Bou8] -, Unstable localization and periodicity, Algebraic Topology: New Trends in Localization and Periodicity, Progress in Mathematics, vol. 136, Birkhauser-Verlag, 1996, pp. 33-50. MR 98c:55014
  • [Bou9] -, Homotopical localizations of spaces, Amer. J. Math. 119 (1997), 1321-1354. MR 98m:55009
  • [Bou10] -, On $K(n)$-equivalences of spaces, Homotopy invariant algebraic structures: a conference in honor of J. Michael Boardman, Contemp. Math., vol. 239. CMP 2000:03
  • [BF] A.K. Bousfield and E.M. Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Lecture Notes in Math., vol. 658, Springer-Verlag, 1978, pp. 80-130. MR 80e:55021
  • [BK] A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions and Localizations, Lecture Notes in Math., vol. 304, Springer-Verlag, 1972. MR 51:1825
  • [Dro] E. Dror Farjoun, Cellular spaces, null spaces and homotopy localizations, Lecture Notes in Math., vol. 1622, Springer-Verlag, 1996. MR 98f:55010
  • [DDK] E. Dror Farjoun, W.G. Dwyer, and D.M. Kan, An arithmetic square for virtually nilpotent spaces, Illinois J. Math. 21 (1977), 242-254. MR 55:11246
  • [DS] W.G. Dwyer and J. Spalinski, Homotopy theories and model categories, Handbook of Algebraic Topology, North-Holland, 1995, pp. 73-126. MR 96h:55014
  • [GJ] P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics, vol. 174, Birkhauser-Verlag, 1999. CMP 2000:02
  • [Hir] P.S. Hirschhorn, Localization in Model Categories, in preparation.
  • [Hop] M.J. Hopkins, Global methods in homotopy theory, Homotopy Theory - Proceedings of the Durham Symposium 1985, London Math. Soc. Lecture Note Series, vol. 117, Cambridge Univ. Press, 1987, pp. 73-96. MR 89g:55022
  • [HRW] M.J. Hopkins, D.C. Ravenel, and W.S. Wilson, Morava Hopf algebras and spaces $K(n)$ equivalent to finite Postnikov systems, Stable and Unstable Homotopy, Fields Institute Communications, vol. 19, 1998, pp. 137-163. MR 99e:55005
  • [HSm] M.J. Hopkins and J.H. Smith, Nilpotence and stable homotopy theory II, Ann. Math. 148 (1998), 1-49. MR 99h:55009
  • [Hov] M. Hovey, Model Categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, 1998. MR 99h:55031
  • [HP] M. Hovey and J.H. Palmieri, The structure of the Bousfield lattice, Homotopy invariant algebraic structures: a conference in honor of J. Michael Boardman, Contemp. Math., vol. 239. CMP 2000:03
  • [HPS] M. Hovey, J.H. Palmieri, and N.P. Strickland, Axiomatic Stable Homotopy Theory, Mem. Amer. Math. Soc. (1997). MR 98a:55017
  • [HSS] M. Hovey, B. Shipley, and J.H. Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149-208. CMP 2000:02
  • [HSt] M. Hovey and N.P. Strickland, Morava $K$-theories and localization, Mem. Amer. Math. Soc. 139 (1999). MR 99b:55017
  • [Kas] T. Kashiwabara, On Brown-Peterson cohomology of $QX$ (to appear).
  • [Kuh1] N.J. Kuhn, Suspension spectra and homology equivalences, Trans. Amer. Math. Soc. 283 (1984), 303-313. MR 85g:55014
  • [Kuh2] -, Morava $K$-theories and infinite loop spaces, Lecture Notes in Math., vol. 1370, Springer-Verlag, 1989, pp. 243-257. MR 90d:55014
  • [Mah] M.E. Mahowald, The image of $J$ in the EHP sequences, Ann. Math. 116 (1982), 65-112. MR 86d:55018
  • [MS] M. Mahowald and H. Sadofsky, $v_{n}$-telescopes and the Adams spectral sequence, Duke Math. J. 78 (1995), 101-129. MR 96h:55006
  • [Mil1] H.R. Miller, On relations between Adams spectral sequences with an application to stable homotopy theory, J. Pure Appl. Algebra 20 (1981), 287-312. MR 82f:55029
  • [Mil2] -, Finite localizations, Bol. Soc. Mat. Mexicana (Homenaje a Jose Adem) 37 (1992), 383-390. MR 96h:55009
  • [Qui] D.G. Quillen, Homotopical Algebra, Lecture Notes in Math., vol. 43, Springer-Verlag, 1967. MR 36:6480
  • [Rav1] D.C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), 351-414. MR 85k:55009
  • [Rav2] -, Progress report on the telescope conjecture, London Math. Soc. Lecture Note Series 176, Adams Memorial Symposium on Algebraic Topology, Cambridge Univ. Press, 1992, pp. 1-21. MR 94h:55023
  • [Rav3] -, Nilpotence and Periodicity in Stable Homotopy Theory, Ann. of Math. Studies 128, Princeton Univ. Press, 1992. MR 94b:55015
  • [Rav4] -, Life after the telescope conjecture, Algebraic $K$-Theory and Algebraic Topology (P.G. Goerss and J.F. Jardine, eds.), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 407, Kluwer, 1993, pp. 205-222. MR 96i:55016
  • [RW] D.C. Ravenel and W.S. Wilson, The Morava $K$-theories of Eilenberg-Mac Lane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980), 691-748. MR 81i:55005
  • [War] R.B. Warfield, Jr., Nilpotent Groups, Lecture Notes in Math., vol. 513, Springer-Verlag, 1976. MR 53:13413

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 55P60, 55N20, 55P42, 55P65, 55U35

Retrieve articles in all journals with MSC (2000): 55P60, 55N20, 55P42, 55P65, 55U35

Additional Information

A. K. Bousfield
Affiliation: Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, Chicago, Illinois 60607

Received by editor(s): March 29, 1999
Published electronically: July 18, 2000
Additional Notes: Research partially supported by the National Science Foundation.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society