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A continuous localization and completion


Author: Norio Iwase
Journal: Trans. Amer. Math. Soc. 320 (1990), 77-90
MSC: Primary 55P60; Secondary 55N91, 55P20, 55U40
DOI: https://doi.org/10.1090/S0002-9947-1990-1031978-1
MathSciNet review: 1031978
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Abstract: The main goal of this paper is to construct a localization and completion of Bousfield-Kan type as a continuous functor for a virtually nilpotent CW-complex. Then the localization and completion of an $ {A_n}$-space is given to be an $ {A_n}$-homomorphism between $ {A_n}$-spaces. For any general compact Lie group, this gives a continuous equivariant localization and completion for a virtually nilpotent $ G$-CW-complex. More generally, we have a continuous localization with respect to a system of core rings for a virtually nilpotent $ \mathbf{D}$-CW-complex for a polyhedral category $ \mathbf{D}$.


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  • [1] J. F. Adams, Infinite loop spaces, Ann. of Math. Studies, no. 90, Princeton Univ. Press, Princeton, N.J., 1972. MR 505692 (80d:55001)
  • [2] A. K. Bousfield and D. M. Kan, Homotopy limits, completions, and localizations, Lecture Notes in Math., vol. 304, Springer, Berlin, Heidelberg, and New York, 1972. MR 0365573 (51:1825)
  • [3] G. E. Bredon, Equivariant cohomology theories, Lecture Notes in Math., vol. 34, Springer, Berlin, Heidelberg, and New York, 1979. MR 0214062 (35:4914)
  • [4] A. Dold and R. Thom, Quasifaserungen und unendliche symmetrische produkte, Ann. of Math. (2) 67 (1958), 239-281. MR 0097062 (20:3542)
  • [5] E. W. Dror, G. Dwyer, and D. M. Kan, An arithmetic square for virtually nilpotent spaces, Illinois J. Math. 21 (1977), 242-254. MR 0438330 (55:11246)
  • [6] E. Dror and A. Zabrodsky, Homotopy equivalence between diagrams of spaces, J. Pure Appl. Algebra 41 (1986), 169-182. MR 849903 (87g:55021)
  • [7] S. Eilenberg and N. E. Steenrod, Foundations of algebraic topology, Princeton Univ. Press, Princeton, N.J., 1952. MR 0050886 (14:398b)
  • [8] A. Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 277 (1983), 275-284. MR 690052 (84f:57029)
  • [9] S. Illman, Equivariant singular homology and cohomology. I, Mem. Amer. Math. Soc. No. 156 (1975). MR 0375286 (51:11482)
  • [10] -, Reduction of the transformation group in equivariant CW complexes: Applications to joinwise and suspensionwise skeletal approximation of $ G$-mappings, preprint.
  • [11] N. Iwase, Certain missing terms in an unstable Adams spectral sequence, Mem. Fac. Sci. Kyushu Univ. Ser. A 41 (1987), 97-113. MR 907598 (89i:55003)
  • [12] -, On the ring structure of $ {K^ * }(X{P^n})$, Master Thesis, Kyushu Univ., 1983. (in Japanese)
  • [13] N. Iwase and M. Mimura, Higher homotopy associativity, Proceedings of the Arcata Conference, Lecture Notes in Math., vol. 1370, Springer-Verlag, Berlin, Heidelberg, and New York, 1989. MR 1000378 (90f:55017)
  • [14] T. Matumoto, On $ G$-CW complexes and a theorem of J. H. C. Whitehead, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18 (1971), 363-374. MR 0345103 (49:9842)
  • [15] J. P. May, Simplicial methods in algebraic topology, Van Nostrand, 1967. MR 0222892 (36:5942)
  • [16] J. P. May, J. McClure, and G. Triantafillou, Equivariant localization, Bull. London Math. Soc. 14 (1982), 223-230. MR 656603 (83h:55021)
  • [17] J. P. May, Equivariant completion, Bull. London Math. Soc. 14 (1982), 231-237. MR 656604 (83h:55022)
  • [18] -, Classifying spaces and fibrations, Mem. Amer. Math. Soc. No. 155 (1975). MR 0370579 (51:6806)
  • [19] J. W. Milnor, On axiomatic homology theory, Pacific J. Math. 12 (1962), 337-341. MR 0159327 (28:2544)
  • [20] M. Mimura, G. Nishida, and H. Toda, Localization of CW-complexes and its applications, J. Math. Soc. Japan 23 (1971), 593-624. MR 0295347 (45:4413)
  • [21] G. Segal, Categories and cohomology theories, Topology 13 (1974), 293-312. MR 0353298 (50:5782)
  • [22] J. D. Stasheff, Homotopy associativity of $ H$-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275-292, 293-312. MR 0158400 (28:1623)
  • [23] N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133-152. MR 0210075 (35:970)
  • [24] T. Sumi, Localization of $ G$-CW complexes at a system of primes, Osaka J. Math. 25 (1988), 865-875. MR 983808 (90d:55019)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-1031978-1
Keywords: Localization, completion, continuous functor, higher homotopy associativity, $ G$-space, $ \mathbf{D}$-space
Article copyright: © Copyright 1990 American Mathematical Society

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