## A continuous localization and completion

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- by Norio Iwase
- Trans. Amer. Math. Soc.
**320**(1990), 77-90 - DOI: https://doi.org/10.1090/S0002-9947-1990-1031978-1
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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 $\mathbf {D}$.## References

- John Frank Adams,
*Infinite loop spaces*, Annals of Mathematics Studies, No. 90, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978. MR**505692** - A. K. Bousfield and D. M. Kan,
*Homotopy limits, completions and localizations*, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR**0365573** - Glen E. Bredon,
*Equivariant cohomology theories*, Lecture Notes in Mathematics, No. 34, Springer-Verlag, Berlin-New York, 1967. MR**0214062** - Albrecht Dold and Renรฉ Thom,
*Quasifaserungen und unendliche symmetrische Produkte*, Ann. of Math. (2)**67**(1958), 239โ281 (German). MR**97062**, DOI 10.2307/1970005 - E. Dror, W. G. Dwyer, and D. M. Kan,
*An arithmetic square for virtually nilpotent spaces*, Illinois J. Math.**21**(1977), no.ย 2, 242โ254. MR**438330** - E. Dror Farjoun and A. Zabrodsky,
*Homotopy equivalence between diagrams of spaces*, J. Pure Appl. Algebra**41**(1986), no.ย 2-3, 169โ182. MR**849903**, DOI 10.1016/0022-4049(86)90108-8 - Samuel Eilenberg and Norman Steenrod,
*Foundations of algebraic topology*, Princeton University Press, Princeton, N.J., 1952. MR**0050886** - A. D. Elmendorf,
*Systems of fixed point sets*, Trans. Amer. Math. Soc.**277**(1983), no.ย 1, 275โ284. MR**690052**, DOI 10.1090/S0002-9947-1983-0690052-0 - Sรถren Illman,
*Equivariant singular homology and cohomology. I*, Mem. Amer. Math. Soc.**1**(1975), no.ย issue 2, 156, ii+74. MR**375286**, DOI 10.1090/memo/0156
โ, - Norio Iwase,
*Certain missing terms in an unstable Adams spectral sequence*, Mem. Fac. Sci. Kyushu Univ. Ser. A**41**(1987), no.ย 2, 97โ113. MR**907598**, DOI 10.2206/kyushumfs.41.97
โ, - Norio Iwase and Mamoru Mimura,
*Higher homotopy associativity*, Algebraic topology (Arcata, CA, 1986) Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp.ย 193โ220. MR**1000378**, DOI 10.1007/BFb0085229 - Takao Matumoto,
*On $G$-$\textrm {CW}$ complexes and a theorem of J. H. C. Whitehead*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**18**(1971), 363โ374. MR**345103** - J. Peter May,
*Simplicial objects in algebraic topology*, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR**0222892** - J. P. May, J. McClure, and G. Triantafillou,
*Equivariant localization*, Bull. London Math. Soc.**14**(1982), no.ย 3, 223โ230. MR**656603**, DOI 10.1112/blms/14.3.223 - J. P. May,
*Equivariant completion*, Bull. London Math. Soc.**14**(1982), no.ย 3, 231โ237. MR**656604**, DOI 10.1112/blms/14.3.231 - J. Peter May,
*Classifying spaces and fibrations*, Mem. Amer. Math. Soc.**1**(1975), no.ย 1, 155, xiii+98. MR**370579**, DOI 10.1090/memo/0155 - J. Milnor,
*On axiomatic homology theory*, Pacific J. Math.**12**(1962), 337โ341. MR**159327** - Mamoru Mimura, Goro Nishida, and Hirosi Toda,
*Localization of $\textrm {CW}$-complexes and its applications*, J. Math. Soc. Japan**23**(1971), 593โ624. MR**295347**, DOI 10.2969/jmsj/02340593 - Graeme Segal,
*Categories and cohomology theories*, Topology**13**(1974), 293โ312. MR**353298**, DOI 10.1016/0040-9383(74)90022-6 - James Dillon Stasheff,
*Homotopy associativity of $H$-spaces. I, II*, Trans. Amer. Math. Soc.**108**(1963), 293โ312. 108 (1963), 275-292; ibid. MR**0158400**, DOI 10.1090/S0002-9947-1963-0158400-5 - N. E. Steenrod,
*A convenient category of topological spaces*, Michigan Math. J.**14**(1967), 133โ152. MR**210075** - Toshio Sumi,
*Localization of $G$-$CW$ complexes at a system of primes*, Osaka J. Math.**25**(1988), no.ย 4, 865โ875. MR**983808**

*Reduction of the transformation group in equivariant CW complexes: Applications to joinwise and suspensionwise skeletal approximation of*$G$

*-mappings*, preprint.

*On the ring structure of*${K^ * }(X{P^n})$, Master Thesis, Kyushu Univ., 1983. (in Japanese)

## Bibliographic Information

- © Copyright 1990 American Mathematical Society
- 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