The odd primary $H$-structure of low rank Lie groups and its application to exponents
HTML articles powered by AMS MathViewer
- by Stephen D. Theriault PDF
- Trans. Amer. Math. Soc. 359 (2007), 4511-4535 Request permission
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
- Raoul Bott, A note on the Samelson product in the classical groups, Comment. Math. Helv. 34 (1960), 249β256. MR 123330, DOI 10.1007/BF02565939
- F. R. Cohen, A course in some aspects of classical homotopy theory, Algebraic topology (Seattle, Wash., 1985) Lecture Notes in Math., vol. 1286, Springer, Berlin, 1987, pp.Β 1β92. MR 922923, DOI 10.1007/BFb0078738
- F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. (2) 110 (1979), no.Β 3, 549β565. MR 554384, DOI 10.2307/1971238
- F. R. Cohen and J. A. Neisendorfer, A construction of $p$-local $H$-spaces, Algebraic topology, Aarhus 1982 (Aarhus, 1982) Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp.Β 351β359. MR 764588, DOI 10.1007/BFb0075576
- G. Cooke, J. R. Harper, and A. Zabrodsky, Torsion free $\textrm {mod}\,p$ $H$-spaces of low rank, Topology 18 (1979), no.Β 4, 349β359. MR 551016, DOI 10.1016/0040-9383(79)90025-9
- D.M. Davis and S.D. Theriault, Odd-primary homotopy exponents of simple compact Lie groups, submitted.
- Brayton Gray, On Todaβs fibrations, Math. Proc. Cambridge Philos. Soc. 97 (1985), no.Β 2, 289β298. MR 771822, DOI 10.1017/S0305004100062836
- Bruno Harris, On the homotopy groups of the classical groups, Ann. of Math. (2) 74 (1961), 407β413. MR 131278, DOI 10.2307/1970240
- Kouyemon Iriye and Akira Kono, Mod $p$ retracts of $G$-product spaces, Math. Z. 190 (1985), no.Β 3, 357β363. MR 806893, DOI 10.1007/BF01215135
- I. M. James, Reduced product spaces, Ann. of Math. (2) 62 (1955), 170β197. MR 73181, DOI 10.2307/2007107
- Ioan James and Emery Thomas, Homotopy-abelian topological groups, Topology 1 (1962), 237β240. MR 149483, DOI 10.1016/0040-9383(62)90105-2
- C. A. McGibbon, Homotopy commutativity in localized groups, Amer. J. Math. 106 (1984), no.Β 3, 665β687. MR 745146, DOI 10.2307/2374290
- Mamoru Mimura, Goro Nishida, and Hirosi Toda, Localization of $\textrm {CW}$-complexes and its applications, J. Math. Soc. Japan 23 (1971), 593β624. MR 295347, DOI 10.2969/jmsj/02340593
- Mamoru Mimura, Goro Nishida, and Hirosi Toda, $\textrm {Mod}\ p$ decomposition of compact Lie groups, Publ. Res. Inst. Math. Sci. 13 (1977/78), no.Β 3, 627β680. MR 0478187, DOI 10.2977/prims/1195189602
- Mamoru Mimura and Hirosi Toda, Cohomology operations and homotopy of compact Lie groups. I, Topology 9 (1970), 317β336. MR 266237, DOI 10.1016/0040-9383(70)90056-X
- J. A. Neisendorfer and P. S. Selick, Some examples of spaces with or without exponents, Current trends in algebraic topology, Part 1 (London, Ont., 1981) CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, pp.Β 343β357. MR 686124
- Paul Selick, Odd primary torsion in $\pi _{k}(S^{3})$, Topology 17 (1978), no.Β 4, 407β412. MR 516219, DOI 10.1016/0040-9383(78)90007-1
- P.S. Selick, Space exponents for loop spaces of spheres, Stable and unstable homotopy theory, Fields Inst. Commun.
- W. Specht, Die linearen Beziehungen zwischen hΓΆheren Kommutatoren, Math. Z. 51 (1948), 367β376 (German). MR 28301, DOI 10.1007/BF01181601
- Stephen D. Theriault, The $H$-structure of low-rank torsion free $H$-spaces, Q. J. Math. 56 (2005), no.Β 3, 403β415. MR 2161254, DOI 10.1093/qmath/hah050
- Hirosi Toda, On iterated suspensions. I, II, J. Math. Kyoto Univ. 5 (1966), 87β142, 209β250. MR 210130, DOI 10.1215/kjm/1250524559
- Franz Wever, Operatoren in Lieschen Ringen, J. Reine Angew. Math. 187 (1949), 44β55 (German). MR 34397, DOI 10.1515/crll.1950.187.44
Additional Information
- Stephen D. Theriault
- Affiliation: Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
- MR Author ID: 652604
- Email: s.theriault@maths.abdn.ac.uk
- Received by editor(s): October 18, 2005
- Published electronically: April 17, 2007
- © Copyright 2007 American Mathematical Society
- 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
- MathSciNet review: 2309196