Available in electronic format
Available in print format
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
     

Central extensions and generalized plus-constructions

Author(s): G. Mislin; G. Peschke
Journal: Trans. Amer. Math. Soc. 353 (2001), 585-608.
MSC (1991): Primary 19D06, 55P60, 55Q15
Posted: October 23, 2000
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract:

We describe the effect of homological plus-constructions on the homotopy groups of Eilenberg-MacLane spaces in terms of universal central extensions.


References:

1.
Donald W. Anderson, Localizing $CW$ complexes, Ill. J. Math. 16 (1972), 519-525. MR 45:4401

2.
M. Arkowitz, The loop of a homotopy action, unpublished, 1-7.

3.
A.J. Berrick, An approach to algebraic $K$-theory, Res. Notes Math., no. 56, Pitman, Boston, MA-London, 1982. MR 84g:18028

4.
A.K. Bousfield, Types of acyclicity, J. Pure Appl. Algebra 4 (1974), 293-298. MR 51:4220

5.
-, The localization of spaces with respect to homology, Topology 14 (1975), 133-150. MR 52:1676

6.
-, Homological localization towers for groups and $\Pi$-modules, AMS Memoirs, no. 186, Amer. Math. Soc., Providence, RI, 1977. MR 56:5688

7.
-, On homology equivalences and homology localizations of spaces, American J. Math. 104 (1982), no. 5, 1025-1029. MR 84g:55014

8.
-, Homotopical localizations of spaces, American J. Math. 119 (1997), no. 6, 1321-1354. MR 98m:55009

9.
A.K. Bousfield and D.M. Kan, Homotopy limits, completions and localizations, second corrected printing ed., Lect. N. Math., vol. 304, Springer-Verlag, Berlin/New York, 1987. MR 51:1825 (1st ed.)

10.
C. Casacuberta, Anderson localization from a modern point of view, Contemp. Math. 181 (1995), 35-44. MR 86a:55017

11.
C. Casacuberta and J.L. Rodriguez, On towers approximating homological localizations, J. London Math. Soc. 56 (1997), 645-656. MR 99b:15016

12.
C. Casacuberta, J.L. Rodriguez, and J.Y. Tai, Localizations of abelian Eilenberg-Mac Lane spaces of finite type, preprint (1997).

13.
Albrecht Dold and René Thom, Quasifaserungen und unendliche symmetrische produkte, Annals Math. 67 (1958), 239-281. MR 20:3542

14.
E. Dror Farjoun, Higher homotopies of natural constructions, J. Pure Appl. Algebra 108 (1996), no. 1, 23-34. MR 97c:55020

15.
E. Dror Farjoun and J.H. Smith, Homotopy localization nearly preserves fibrations, Topology 34 (1994), no. 2, 359-375. MR 96c:55014

16.
Emmanuel Dror Farjoun, Homotopy localization and $v_{1}$-periodic spaces, Algebraic Topology - Homotopy and Group Cohomology (Berlin New York) (J. Aguadé, M. Castellet, and F.R. Cohen, eds.), Lect. Notes in Math., no. 1509, Springer-Verlag, 1992, pp. 104-114. MR 93k:55013

17.
-, Localizations, fibrations, and conic structures, preprint (1992).

18.
-, Cellular Spaces, Null Spaces, and Homotopy Localization, Lect. Notes in Math., no. 1622, Springer-Verlag, Berlin New York, 1995. MR 98f:55010

19.
W.G. Dwyer and D.M. Kan, The enveloping ring of a $\Pi$-algebra, Advances in Homotopy Theory (Cambridge), vol. 139, Cambridge UP, 1989, pp. 49-60. MR 91j:55015

20.
J.C. Hausmann and D. Husemoller, Acyclic maps, L'Enseignement Mathématique 25 (1979), 53-75. MR 80k:55044

21.
Philip S. Hirschhorn, Localization, cellularization, and homotopy colimits, Book draft, 1995.

22.
Daniel M. Kan and William P. Thurston, Every connected space has the homology of a $K(\pi,1)$, Topology 15 (1976), no. 3, 253-258. MR 54:1210

23.
Michel A. Kervaire, Multiplicateurs de Schur et $K$-théorie, Essays on Topology and related topics: Mémoires dédiés à Georges de Rham (1903- ) (Berlin New York) (A. Haefliger and R. Narasimhan, eds.), Springer-Verlag, 1970, pp. 212-225. MR 43:321

24.
Willi Meier, Acyclic maps and knot complements, Math. Annalen 243 (1979), no. 3, 247-259. MR 81i:55020

25.
-, Pullback theorems and phantom maps, Quart. J. Math., Ox. Ser. 2 29 (1979), 469-481. MR 80g:55023

26.
G. Mislin, Localization with respect to $K$-theory, J. Pure and Applied Algebra 10 (1977), 201-213. MR 57:7585

27.
Daniel G. Quillen, Cohomology of groups, Actes du Congrès International des Mathématiciens (Nice, 1970) (Paris), vol. 2, Comité d'Organisation du Congrès, Gauthier Villars, 1971, pp. 47-51. MR 58:7627a

28.
-, Higher $K$-theory for categories with exact sequences, New Developments in Topology (Proc. Sympos. Algebraic Topology, Oxford, 1972) (Cambridge, UK) (G.B. Segal, ed.), Lond. Math. Soc. Lec. Notes Ser., no. 11, Cambridge Univ. Press, 1974, pp. 95-103. MR 49:384

29.
Jin-Yen Tai, On generalized plus-constructions of Eilenberg-Mac Lane spaces, preprint (1997), 1-14.

30.
-, Generalized plus-constructions and fundamental groups, J. Pure Appl. Algebra 132 (1998), no. 2, 207-220. MR 99i:55017

31.
-, A version of Zabrodsky's lemma, Proc. AMS 126 (1998), no. 5, 1573-1578. MR 98j:55011

32.
K. Varadarajan, Groups for which Moore spaces $M(\pi,1)$ exist, Annals Math. 84 (1966), 368-371. MR 34:2017

33.
George W. Whitehead, Elements of Homotopy Theory, Graduate T. Math., no. 61, Springer-Verlag, Berlin New York, 1978. MR 80b:55001


Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 19D06, 55P60, 55Q15

Retrieve articles in all Journals with MSC (1991): 19D06, 55P60, 55Q15


Additional Information:

G. Mislin
Affiliation: Departement Mathematik, ETH, 8092 Zürich, Switzerland
Email: mislin@math.ethz.ch

G. Peschke
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Canada T6G 2G1
Email: George.Peschke@UAlberta.Ca

DOI: 10.1090/S0002-9947-00-02727-6
PII: S 0002-9947(00)02727-6
Keywords: Plus-constructions, localization, colocalization, central extension
Received by editor(s): April 14, 1998
Posted: October 23, 2000
Additional Notes: Research partially supported by NSERC of Canada
Copyright of article: Copyright 2000, American Mathematical Society


  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement
Search the AMSPowered by Google