Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

The odd primary $ H$-structure of low rank Lie groups and its application to exponents


Author: Stephen D. Theriault
Journal: Trans. Amer. Math. Soc. 359 (2007), 4511-4535
MSC (2000): Primary 55P45, 55Q52, 57T20
DOI: https://doi.org/10.1090/S0002-9947-07-04304-8
Published electronically: April 17, 2007
MathSciNet review: 2309196
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A compact, connected, simple Lie group $ G$ localized at an odd prime $ p$ is shown to be homotopy equivalent to a product of homotopy associative, homotopy commutative spaces, provided the rank of $ G$ is low. This holds for $ SU(n)$, for example, if $ n\leq (p-1)(p-3)$. The homotopy equivalence is usually just as spaces, not multiplicative spaces. Nevertheless, the strong multiplicative features of the factors can be used to prove useful properties, which after looping can be transferred multiplicatively to $ \Omega G$. This is applied to prove useful information about the torsion in the homotopy groups of $ G$, including an upper bound on its exponent.


References [Enhancements On Off] (What's this?)

  • [B] R. Bott, A note on the Samelson product in the classical Lie groups, Comment. Math. Helv. 34 (1960), 245-256. MR 0123330 (23:A658)
  • [C] F.R. Cohen, A short course in some aspects of classical homotopy theory, Lecture Notes in Math. 1286, Springer-Verlag (1987), 1-92. MR 922923 (89e:55027)
  • [CMN] F.R. Cohen, J.C. Moore, and J.A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. 110 (1979), 549-565. MR 554384 (81c:55021)
  • [CN] F.R. Cohen and J.A. Neisendorfer, A construction of $ p$-local $ H$-spaces, pp. 351-359. Lecture Notes in Math. Vol. 1051, Springer, Berlin, 1984. MR 764588 (86e:55011)
  • [CHZ] G. Cooke, J. Harper, and A. Zabrodsky, Torsion free mod $ p$ $ H$-spaces of low rank, Topology 18 (1979), 349-359. MR 551016 (80k:55032)
  • [DT] D.M. Davis and S.D. Theriault, Odd-primary homotopy exponents of simple compact Lie groups, submitted.
  • [G] B. Gray, On Toda's fibrations, Math. Proc. Camb. Phil. Soc. 97 (1985), 289-298. MR 771822 (86i:55016)
  • [H] B. Harris, On the homotopy groups of the classical groups, Ann. of Math. 74 (1961), 407-413. MR 0131278 (24:A1130)
  • [IK] K. Iriye and A. Kono, Mod $ p$ retracts of $ G$-product spaces, Math. Z. 190 (1985), 357-363. MR 806893 (88a:55017)
  • [J] I.M. James, Reduced Product Spaces, Ann. of Math. 62 (1955), 170-197. MR 0073181 (17:396b)
  • [JT] I.M. James and E. Thomas, Homotopy-abelian topological groups, Topology 1 (1962), 237-240. MR 0149483 (26:6970)
  • [Mc] C.A. McGibbon, Homotopy commutativity in localized groups, Amer. J. Math 106 (1984), 665-687. MR 745146 (86a:55011)
  • [MNT1] M. Mimura, G. Nishida, and H. Toda, Localization of $ CW$-complexes and its applications, J. Math. Soc. Japan 23 (1971), 593-624. MR 0295347 (45:4413)
  • [MNT2] M. Mimura, G. Nishida, and H. Toda, Mod-$ p$ decomposition of compact Lie groups, Publ. RIMS, Kyoto Univ 13 (1977), 627-680. MR 0478187 (57:17675)
  • [MT] M. Mimura and H. Toda, Cohomology operations and the homotopy of compact Lie groups I, Topology 9 (1970), 317-336. MR 0266237 (42:1144)
  • [NS] J.A. Neisendorfer and P.S. Selick, Some examples of spaces with or without exponents, Current trends in algebraic topology, Part 1, CMS Conf. Proc. 2, Amer. Math. Soc., 1982, 343-357. MR 686124 (84b:55017)
  • [S1] P.S. Selick, Odd primary torsion in $ \pi_{k}(S^{3})$, Topology 17 (1978), 407-412. MR 516219 (80c:55010)
  • [S2] P.S. Selick, Space exponents for loop spaces of spheres, Stable and unstable homotopy theory, Fields Inst. Commun.
  • [Sp] W. Specht, Die linearen Beziehumgen zwischen höheren Kommutatoren, Math. Z. 51 (1948), 367-376. 19, Amer. Math. Soc., 1998, 279-283. MR 0028301 (10:425d)
  • [T] S.D. Theriault, The $ H$-structure of low rank torsion free $ H$-spaces, Quart. J. Math. Oxford 56 (2005), 403-415. MR 2161254
  • [To] H. Toda, On Iterated Suspensions I, J. Math. Kyoto Univ. 5 (1965), 87-142. MR 0210130 (35:1024)
  • [W] F. Wever, Operatoren in Lieschen Ringen, J. Reine Angew. Math. 187 (1949), 44-55. MR 0034397 (11:579i)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 55P45, 55Q52, 57T20

Retrieve articles in all journals with MSC (2000): 55P45, 55Q52, 57T20


Additional Information

Stephen D. Theriault
Affiliation: Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
Email: s.theriault@maths.abdn.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-07-04304-8
Keywords: Lie group, exponent, Whitehead product, $H$-space
Received by editor(s): October 18, 2005
Published electronically: April 17, 2007
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society