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Fibrewise localization and completion


Author: J. P. May
Journal: Trans. Amer. Math. Soc. 258 (1980), 127-146
MSC: Primary 55P60
DOI: https://doi.org/10.1090/S0002-9947-1980-0554323-8
MathSciNet review: 554323
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Abstract: The behavior of fibrewise localization and completion on the classifying space level is analyzed. The relationship of these constructions to fibrewise joins and smash products and to orientations of spherical fibrations is also analyzed. This theory is essential to validate Sullivan's proof of the Adams conjecture.


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

  • [1] J. F. Adams, On the groups $ J(X)\, - \,I$, Topology 2 (1963), 181-195. MR 0159336 (28:2553)
  • [2] G. Allaud, On the classification of fibre spaces. Math. Z. 92 (1966), 110-125. MR 0189035 (32:6462)
  • [3] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math., vol. 304, Springer-Verlag, Berlin and New York, 1972. MR 0365573 (51:1825)
  • [4] R. Bruner, G. Lewis, J. P. May, J. McClure and M. Steinberger, $ {H_\infty }$ ring spaces and their applications, Lecture Notes in Math., Springer-Verlag, Berlin and New York, in preparation.
  • [5] A. Dold, Partitions of unity in the theory of fibrations, Ann. of Math. (2) 78 (1963), 223-225. MR 0155330 (27:5264)
  • [6] -, Halbexact Homotopiefunktoren, Lecture Notes in Math., vol. 12, Springer-Verlag, Berlin and New York, 1966.
  • [7] E. Fadell, The equivalence of fibre spaces and bundles, Bull. Amer. Math. Soc. 66 (1960), 50-53. MR 0112137 (22:2992)
  • [8] H. Glover and G. Mislin, Immersion in the metastable range and 2-localization, Proc. Amer. Math. Soc. 43 (1974), 443-448. MR 0341504 (49:6255)
  • [9] D. H. Gottlieb, The total space of universal fibrations, Pacific J. Math. 46 (1973), 415-417. MR 0331384 (48:9717)
  • [10] P. Hilton, G. Mislin and J. Roitberg, Localization of nilpotent groups and spaces, Mathematics Studies, vol. 15, North-Holland, Amsterdam, 1975. MR 0478146 (57:17635)
  • [11] I. James, The space of bundle maps, Topology 2 (1963), 45-59. MR 0198472 (33:6630)
  • [12] P. J. Kahn, Mixing homotopy types of manifolds, Topology 14 (1975), 203-216. MR 0377937 (51:14106)
  • [13] J. P. May, The geometry of iterated loop spaces, Lecture Notes in Math., vol. 271, Springer-Verlag, Berlin and New York, 1972. MR 0420610 (54:8623b)
  • [14] -, Classifying spaces and fibrations, Mem. Amer. Math. Soc., No. 155, 1975. MR 0370579 (51:6806)
  • [15] -(with contributions by F. Quinn, N. Ray and J. Tornehave), $ {E_\infty }$ ring spaces and $ {E_\infty }$ ring spectra, Lecture Notes in Math., vol. 577, Springer-Verlag, Berlin and New York, 1977. MR 0494077 (58:13008)
  • [16] -, The homotopical foundations of algebraic topology, Academic Press, New York (in preparation).
  • [17] J. Milnor, On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc. 90 (1959), 272-280. MR 0100267 (20:6700)
  • [18] R. Schon, Fibrations over a CWh-base, Proc. Amer. Math. Soc. 62 (1977), 165-166. MR 0431163 (55:4165)
  • [19] J. D. Stasheff, A classification theorem for fibre spaces, Topology 2 (1963), 239-246. MR 0154286 (27:4235)
  • [20] D. Sullivan, The genetics of homotopy theory and the Adams conjecture, Ann. of Math. (2) 100 (1974), 1-79. MR 0442930 (56:1305)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0554323-8
Keywords: Adams conjecture, classifying space, localization, completion, fibre homotopy equivalence, fibrewise join, fibrewise smash product, orientation
Article copyright: © Copyright 1980 American Mathematical Society

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