Connected finite loop spaces with maximal tori
HTML articles powered by AMS MathViewer
- by J. M. Møller and D. Notbohm
- Trans. Amer. Math. Soc. 350 (1998), 3483-3504
- DOI: https://doi.org/10.1090/S0002-9947-98-02247-8
- PDF | Request permission
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.References
- J. Frank Adams, Lectures on Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0252560
- J. F. Adams and Z. Mahmud, Maps between classifying spaces, Inv. Math. 35 (1976), 1–41. MR 0423352, DOI 10.1007/BF01390132
- J. Aguadé, C. Broto, and D. Notbohm, Homotopy classification of spaces with interesting cohomology and a conjecture of Cooke. I, Topology 33 (1994), no. 3, 455–492. MR 1286926, DOI 10.1016/0040-9383(94)90023-X
- 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, DOI 10.1007/978-3-540-38117-4
- M. G. Barratt, F. Cohen, B. Gray, M. Mahowald, and W. Richter, Two results on the $2$-local $EHP$ spectral sequence, Proc. Amer. Math. Soc. 123 (1995), no. 4, 1257–1261. MR 1246514, DOI 10.1090/S0002-9939-1995-1246514-4
- W. G. Dwyer and C. W. Wilkerson, The center of a $p$-compact group, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 119–157. MR 1320990, DOI 10.1090/conm/181/02032
- W. Dwyer and A. Zabrodsky, Maps between classifying spaces, Algebraic topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, Springer, Berlin, 1987, pp. 106–119. MR 928826, DOI 10.1007/BFb0083003
- J. H. Gunawardena, J. Lannes, and S. Zarati, Cohomologie des groupes symétriques et application de Quillen, Advances in homotopy theory (Cortona, 1988) London Math. Soc. Lecture Note Ser., vol. 139, Cambridge Univ. Press, Cambridge, 1989, pp. 61–68 (French). MR 1055868, DOI 10.1017/CBO9780511662614.008
- Richard M. Kane, The homology of Hopf spaces, North-Holland Mathematical Library, vol. 40, North-Holland Publishing Co., Amsterdam, 1988. MR 961257
- Saunders MacLane, Homology, 1st ed., Die Grundlehren der mathematischen Wissenschaften, Band 114, Springer-Verlag, Berlin-New York, 1967. MR 0349792
- Jean Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un $p$-groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 135–244 (French). With an appendix by Michel Zisman. MR 1179079, DOI 10.1007/BF02699494
- Jesper Michael Møller, Rational isomorphisms of $p$-compact groups, Topology 35 (1996), no. 1, 201–225. MR 1367281, DOI 10.1016/0040-9383(94)00005-0
- J. M. Møller and D. Notbohm, Centers and finite coverings of finite loop spaces, J. Reine Angew. Math. 456 (1994), 99–133. MR 1301453
- Dietrich Notbohm, Fake Lie groups with maximal tori. IV, Math. Ann. 294 (1992), no. 1, 109–116. MR 1180453, DOI 10.1007/BF01934316
- Dietrich Notbohm, Homotopy uniqueness of classifying spaces of compact connected Lie groups at primes dividing the order of the Weyl group, Topology 33 (1994), no. 2, 271–330. MR 1273786, DOI 10.1016/0040-9383(94)90015-9
- Dietrich Notbohm, On the “classifying space” functor for compact Lie groups, J. London Math. Soc. (2) 52 (1995), no. 1, 185–198. MR 1345725, DOI 10.1112/jlms/52.1.185
- Dietrich Notbohm, Kernels of maps between classifying spaces, Israel J. Math. 87 (1994), no. 1-3, 243–256. MR 1286829, DOI 10.1007/BF02772997
- Erich Rothe, Topological proofs of uniqueness theorems in the theory of differential and integral equations, Bull. Amer. Math. Soc. 45 (1939), 606–613. MR 93, DOI 10.1090/S0002-9904-1939-07048-1
- David L. Rector, Loop structures on the homotopy type of $S^{3}$, Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle, Wash., 1971) Lecture Notes in Math., Vol. 249, Springer, Berlin, 1971, pp. 99–105. MR 0339153
- Clarence Wilkerson, Rational maximal tori, J. Pure Appl. Algebra 4 (1974), 261–272. MR 343264, DOI 10.1016/0022-4049(74)90006-1
- A. Zabrodsky, On phantom maps and a theorem of H. Miller, Israel J. Math. 58 (1987), no. 2, 129–143. MR 901174, DOI 10.1007/BF02785672
Bibliographic Information
- J. M. Møller
- Affiliation: Matematisk Institut, Universitetsparken 5, DK–2100 København Ø, Denmark
- ORCID: 0000-0003-4053-2418
- Email: moller@math.ku.dk
- D. Notbohm
- Affiliation: Mathematisches Institut, Bunsenstr. 3-5, 37073 Göttingen, Germany
- Email: notbohm@cfgauss.uni-math.gwdg.de
- Received by editor(s): July 11, 1995
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 3483-3504
- MSC (1991): Primary 55P35, 55R35
- DOI: https://doi.org/10.1090/S0002-9947-98-02247-8
- MathSciNet review: 1487627