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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Connected finite loop spaces with maximal tori

Authors: J. M. Møller and D. Notbohm
Journal: Trans. Amer. Math. Soc. 350 (1998), 3483-3504
MSC (1991): Primary 55P35, 55R35
MathSciNet review: 1487627
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Abstract: Finite loop spaces are a generalization of compact Lie groups. However, they do not enjoy all of the nice properties of compact Lie groups. For example, having a maximal torus is a quite distinguished property. Actually, an old conjecture, due to Wilkerson, says that every connected finite loop space with a maximal torus is equivalent to a compact connected Lie group. We give some more evidence for this conjecture by showing that the associated action of the Weyl group on the maximal torus always represents the Weyl group as a crystallographic group. We also develop the notion of normalizers of maximal tori for connected finite loop spaces, and prove for a large class of connected finite loop spaces that a connected finite loop space with maximal torus is equivalent to a compact connected Lie group if it has the right normalizer of the maximal torus. Actually, in the cases under consideration the information about the Weyl group is sufficient to give the answer. All this is done by first studying the analogous local problems.

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  • [1] J.F. Adams, Lecturs on Lie groups, W. A. Benjamin, 1969. MR 40:5780
  • [2] J.F. Adams and Z. Mahmud, Maps between classifying spaces, Invent. Math. 35 (1976), 1-41. MR 54:11331
  • [3] J. Aguadé, C. Broto and D. Notbohm, Homotopy classification of some spaces with interesting cohomology and a conjecture of Cooke, Part I, Topology 33 (1994), 455-492. MR 95i:55006
  • [4] A. Bousfield and D. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer Verlag, 1972. MR 51:1825
  • [5] W.G. Dwyer and C.W. Wilkerson, Homotopy fixed point methods for Lie groups and finite loop spaces, Ann. of Math. (2) 139 (1994), 395-442. MR 95e:55018
  • [6] W.G. Dwyer and C.W. Wilkerson, The center of a $p$-compact group, The Cech Contennial, Contemp. Math. 181 (1995), 119-157. MR 96a:55024
  • [7] W.G. Dwyer and A. Zabrodsky, Maps between classifying spaces, Proceedings of the 1986 Barcelona Conference on Algebraic Topology, Lecture Notes in Math. 1298, Springer Verlag, 1988, pp. 109-119. MR 89b:55018
  • [8] J.H. Gunawardena, J. Lannes, and S. Zarati, Cohomlogie des groupes symmétrique et application de Quillen, L.M.S Lectures Note Ser. 139 (1989), 61-68. MR 91d:18013
  • [9] R.M. Kane, The Homology of Hopf Spaces, North-Holland, 1988. MR 90f:55018
  • [10] S. MacLane, Homology, Springer Verlag, Berlin Heidelberg New York, 1975. MR 50:2285
  • [11] J. Lannes, Sur les espaces fonctionnels dont la source est le classifiant d'un $p$-groupe abélien élémentaire, Publ. Math. IHES 75 (1992), 135-244. MR 93j:55019
  • [12] J.M. Møller, Rational isomorphisms between $p$-compact groups, Topology 35 (1996), 201-225. MR 97b:55019
  • [13] J.M. Møller and D.Notbohm, Centers and finite coverings of finite loop spaces, J. reine u. angew. Math. 456 (1994), 99-133. MR 95j:55029
  • [14] D. Notbohm, Fake Lie groups with maximal tori IV, Math. Ann. 294 (1992), 109-116. MR 94e:55025
  • [15] D. Notbohm, Homotopy uniqueness of classifying spaces of compact connected Lie groups at primes dividing the order of the Weyl group, Topology 33 (1994), 271-330. MR 95e:55020
  • [16] D. Notbohm, On the classifying space functor for compact Lie groups, J. London Math. Soc. 52 (1995), 185-198. MR 96g:55019
  • [17] D. Notbohm, Kernels of maps between classifying spaces, Israel J. Math. 87 (1994), 243-256. MR 95k:55032
  • [18] D.Notbohm and L.Smith, Fake Lie groups and maximal tori I, Math. Ann. 288 (1991), 637-661. MR 93k:55012a
  • [19] D.L. Rector, Loop structures on the homotopy type of $S^{3}$, Symposium on Algebraic Topology, Seattle 1971 (P.J. Hilton, ed.), Lecture Notes in Mathematics 249, Springer-Verlag, Berlin-Heidelberg-New York, 1971. MR 49:3916
  • [20] C. Wilkerson, Rational maximal tori, J. Pure Appl. Algebra 4 (1974), 261-272. MR 49:8008
  • [21] A. Zabrodsky, On phantom maps and a theorem of H. Miller, Israel J. Math. 58 (1987), 129-143. MR 88m:55028

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Additional Information

J. M. Møller
Affiliation: Matematisk Institut, Universitetsparken 5, DK–2100 København Ø, Denmark

D. Notbohm
Affiliation: Mathematisches Institut, Bunsenstr. 3-5, 37073 Göttingen, Germany

Keywords: Finite loop space, $p$--compact group, classifying space, maximal torus, normalizer, Weyl group, covering
Received by editor(s): July 11, 1995
Article copyright: © Copyright 1998 American Mathematical Society

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