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On the rationalization of the circle

Author: Carles Casacuberta
Journal: Proc. Amer. Math. Soc. 118 (1993), 995-1000
MSC: Primary 55P60; Secondary 20J05
MathSciNet review: 1176479
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Abstract: We give an example showing that, for a nilpotent group $ G$ and a set of primes $ P$, the $ P$-localization homomorphism $ l:G \to {G_P}$ need not induce an isomorphism in cohomology with arbitrary (twisted) $ {{\mathbf{Z}}_P}$-module coefficients. From this fact we infer that, in the pointed homotopy category of connected CW-complexes, the inclusion of the subcategory of spaces whose higher homotopy groups are $ {{\mathbf{Z}}_P}$-modules and whose fundamental group is uniquely $ {P'}$-radicable does not admit a left adjoint.

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  • [1] A. K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133-150. MR 0380779 (52:1676)
  • [2] -, Constructions of factorization systems in categories, J. Pure Appl. Algebra 9 (1977), 207-220. MR 0478159 (57:17648)
  • [3] C. Casacuberta and G. Peschke, Localizing with respect to self maps of the circle, Trans. Amer. Math. Soc. (to appear). MR 1123451 (93k:55012)
  • [4] C. Casacuberta, G. Peschke, and M. Pfenniger, Sur la localisation dans les catégories avec une application à la théorie de l'homotopie, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 207-210. MR 1046906 (91i:55014)
  • [5] A. Deleanu and P. Hilton, On Postnikov-true families of complexes and the Adams completion, Fund. Math. 106 (1980), 53-65. MR 585546 (82a:55014)
  • [6] W. Dicks, Groups, trees and projective modules, Lecture Notes in Math., vol. 790, Springer-Verlag, Berlin and New York, 1980. MR 584790 (82j:20079)
  • [7] W. Dicks and M. J. Dunwoody, Groups acting on graphs, Cambridge Stud. Adv. Math., vol. 17, Cambridge Univ. Press, Cambridge and New York, 1989. MR 1001965 (91b:20001)
  • [8] E. Dror Farjoun, Homotopy localization and $ {v_1}$-periodic spaces, Algebraic Topology; Homotopy and Group Cohomology, Lecture Notes in Math., vol. 1509, Springer-Verlag, Berlin and New York, 1992, pp. 104-113. MR 1185964 (93k:55013)
  • [9] S. Gitler, Cohomology operations with local coefficients, Amer. J. Math. 85 (1963), 156-188. MR 0158398 (28:1621)
  • [10] B. Gray, Spaces of the same $ n$-type for all $ n$, Topology 5 (1966), 241-243. MR 0196743 (33:4929)
  • [11] K. W. Gruenberg, Cohomological topics in group theory, Lecture Notes in Math., vol. 143, Springer-Verlag, Berlin and New York, 1970. MR 0279200 (43:4923)
  • [12] P. Hilton, G. Mislin, and J. Roitberg, Localization of nilpotent groups and spaces, North-Holland Math. Stud., vol. 15, North-Holland, Amsterdam, 1975. MR 0478146 (57:17635)
  • [13] D. F. Holt, The cohomological dimensions of locally finite groups, J. London Math. Soc. (2) 24 (1981), 129-134. MR 623679 (83a:20064)
  • [14] C. A. McGibbon and J. M. Møller, On spaces with the same $ n$-type for all $ n$, Topology 31 (1992), 177-201. MR 1153244 (92m:55008)
  • [15] D. Passman, The algebraic structure of group rings, Wiley, New York, 1977. MR 470211 (81d:16001)
  • [16] A. Reynol, $ P$-localization of some classes of groups, Ph.D. thesis, Universidade de São Paulo, Brazil, 1987.
  • [17] P. Ribenboim, Torsion et localisation de groupes arbitraires, Séminaire d'algèbre Paul Dubreil, Lecture Notes in Math., vol. 740, Springer-Verlag, Berlin and New York, 1979, pp. 444-456. MR 563516 (83h:20029)
  • [18] G. W. Whitehead, Elements of homotopy theory, Graduate Texts in Math., vol. 61, Springer-Verlag, New York, 1978. MR 516508 (80b:55001)

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