Abstract
In the 1980’s, remarkable advances were made by Ravenel, Hopkins, Devinatz, and Smith toward a global understanding of stable homotopy theory, showing that some major features arise “chromatically” from an interplay of periodic phenomena arranged in a hierarchy (see [20], [21], [28]). We would like very much to achieve a similar understanding in unstable homotopy theory and shall describe some progress in that direction. In particular, we shall explain and extend some results of our papers [4], [11], and some closely related results of Dror Farjoun and Smith [17], [18], [19].
The author was partially supported by the National Science Foundation.
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References
J.F. Adams, Stable homotopy and generalized homology, University of Chicago Press, 1974.
D.W. Anderson, Localizing CW-complexes, Illinois J. Math. 16 (1972), 519–525.
A.K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133–150.
A.K. Bousfield, Constructions of factorization systems in categories, J. Pure Appl. Algebra 9 (1977), 207–220.
A.K. Bousfield, The Boolean algebra of spectra, Comment. Math. Helv. 54 (1979), 368–377.
A.K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), 257–281.
A.K. Bousfield, Cohomological localizations of spaces and spectra, unpublished preprint (1979).
A.K. Bousfield, K-localizations and K-equivalences of infinite loop spaces, Proc. London Math. Soc. 44 (1982), 291–311.
A.K. Bousfield, On homology equivalences and homological localizations of spaces, Amer. J. Math. 104 (1982), 1025–1042.
A.K. Bousfield, Uniqueness of infinite deloopings for K-theoretic spaces, Pacific J. Math. 129 (1987), 1–31.
A.K. Bousfield, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994), 831–873.
A.K. Bousfield and E.M. Friedlander, Homotopy theory of T-spaces, spectra, and bisimplicial sets, Lecture Notes in Math, vol. 658, Springer-Verlag, 1978, pp. 80–130.
A.K. Bousfield and D.M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math., vol. 304, Springer-Verlag, 1972.
C. Casacuberta, Anderson localization from a modern point of view, Contemp. Math. (to appear).
F. Cohen and J. Neisendorfer, A note on desuspending the Adams map, Math. Proc. Camb. Philos. Soc. 99 (1986), 59–64.
D.M. Davis and M. Mahowald, v 1 -localizations of finite torsion spectra and spherically resolved spaces, Topology 32 (1993), 543–550.
E. Dror Farjoun, Homotopy localization and v 1 -periodic spaces, Lecture Notes in Math., vol. 1509, Springer-Verlag, 1992, pp. 104–113.
E. Dror Farjoun, Localizations, fibrations and conic structures (to appear).
E. Dror Farjoun and J.H. Smith, Homotopy localization nearly preserves fibrations, Topology (to appear).
M.J. Hopkins, Global methods in homotopy theory, London Math. Soc. Lecture Note Ser., vol. 117, Cambridge Univ. Press, 1987, pp. 73–96.
M.J. Hopkins and J.H. Smith, Nilpotence and stable homotopy II, Ann. of Math. (to appear).
N.J. Kuhn, Morava K-theories and infinite loop spaces, Lecture Notes in Math., vol. 1370, Springer-Verlag, 1989, pp. 243–257.
L. Langsetmo, The K-theory localization of an odd sphere and applications, Topology 32 (1993), 577–585.
M. Mahowald and R. Thompson, The K-theory localization of an unstable sphere, Topology 31 (1992), 133–141.
H. Miller and V. Snaith, On the K-theory of the Kahn-Priddy map, J. London Math. Soc. 20 (1979), 339–342.
S.A. Mitchell, Finite complexes with A(n)-free cohomology, Topology 24 (1985), 227–246.
D.C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), 351–414.
D.C. Ravenel, Nilpotence and periodicity in stable homotopy theory., Ann. of Math. Stud., no. 128, Princeton Univ. Press, 1992.
D.C. Ravenel, Life after the telescope conjecture, in: P.G. Goerss and J.F. Jardine, eds., Algebraic K-Theory and Algebraic Topology, Kluwer Academic Publishers, 1993, 205–222.
D.C. Ravenel and W.S. Wilson, The Morava K-theories of Eilenberg-MacLane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980), 691–748.
R. Switzer, Algebraic topology-homotopy and homology, Springer-Verlag, 1975.
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Bousfield, A.K. (1996). Unstable localization and periodicity. In: Broto, C., Casacuberta, C., Mislin, G. (eds) Algebraic Topology: New Trends in Localization and Periodicity. Progress in Mathematics, vol 136. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9018-2_3
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