Infinite loop spaces and Neisendorfer localization

McGibbon, C.

doi:10.1090/S0002-9939-97-03744-1

Infinite loop spaces and Neisendorfer localization
HTML articles powered by AMS MathViewer

by C. A. McGibbon PDF

Proc. Amer. Math. Soc. 125 (1997), 309-313 Request permission

Abstract:

There is a localization functor $L$ with the property that $L(X)$ is the $p$-completion of $X$ whenever $X$ is a finite dimensional complex. This same functor is shown to have the property that $L(E)$ is contractible whenever $E$ is a connected infinite loop space with a torsion fundamental group. One consequence of this is that many finite dimensional complexes $X$ are uniquely determined, up to $p$-completion, by the homotopy fiber of any map from $X$ into the classifying space $B E$.

References

J. F. Adams, The Kahn-Priddy theorem, Proc. Cambridge Philos. Soc. 73 (1973), 45–55. MR 310878, DOI 10.1017/s0305004100047459
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
Carles Casacuberta, Recent advances in unstable localization, The Hilton Symposium 1993 (Montreal, PQ), CRM Proc. Lecture Notes, vol. 6, Amer. Math. Soc., Providence, RI, 1994, pp. 1–22. MR 1290581, DOI 10.1090/crmp/006/01
Emmanuel Dror Farjoun, Homotopy localization and $v_1$-periodic spaces, Algebraic topology (San Feliu de Guíxols, 1990) Lecture Notes in Math., vol. 1509, Springer, Berlin, 1992, pp. 104–113. MR 1185964, DOI 10.1007/BFb0087504
E. Dror Farjoun, Localizations, fibrations and conic structures, preprint 1992.
E. Dror Farjoun, Cellular spaces, preprint 1993.
E. Dror Farjoun, Cellular inequalities, Contemporary Math. 181 (1995), 159–181.
C. A. McGibbon, A note on Miller’s theorem about maps out of certain classifying spaces, Proc. Amer. Math. Soc., to appear.
Haynes Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. (2) 120 (1984), no. 1, 39–87. MR 750716, DOI 10.2307/2007071
Joseph A. Neisendorfer, Localization and connected covers of finite complexes, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 385–390. MR 1321002, DOI 10.1090/conm/181/02044