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$ v\sb 1$-periodic homotopy groups of exceptional Lie groups: torsion-free cases


Authors: Martin Bendersky, Donald M. Davis and Mamoru Mimura
Journal: Trans. Amer. Math. Soc. 333 (1992), 115-135
MSC: Primary 57T20
DOI: https://doi.org/10.1090/S0002-9947-1992-1116310-9
MathSciNet review: 1116310
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Abstract: The $ {v_1}$-periodic homotopy groups $ v_1^{ - 1}{\pi _ {\ast} }(X;p)$ are computed explicitly for all pairs $ (X,p)$, where $ X$ is an exceptional Lie group whose integral homology has no $ p$-torsion. This yields new lower bounds for $ p$-exponents of actual homotopy groups of these spaces. Delicate calculations with the unstable Novikov spectral sequence are required in the proof.


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  • [1] J. F. Adams, On the groups $ J(X)$. IV, Topology 5 (1966), 21-71. MR 0198470 (33:6628)
  • [2] M. Bendersky, Unstable towers in the odd primary homotopy groups of spheres, Trans. Amer. Math. Soc. 287 (1985), 529-542. MR 768724 (86g:55018)
  • [3] -, Some calculations in the unstable Adams-Novikov spectral sequence, Publ. Res. Inst. Math. Sci. 16 (1980), 739-766. MR 602467 (82f:55028)
  • [4] -, The derived functors of the primitives for $ B{P_ {\ast} }(\Omega {S^{2n + 1}})$, Trans. Amer. Math. Soc. 276 (1983), 599-619. MR 688964 (85f:55008a)
  • [5] -, The $ {v_1}$-periodic unstable Novikov spectral sequence, Topology (to appear).
  • [6] -, The $ BP$ Hopf invariant, Amer. J. Math. 108 (1986), 1037-1058. MR 859769 (88d:55015)
  • [7] M. Bendersky, E. B. Curtis, and H. R. Miller, The unstable Adams spectral sequence for a generalized homology, Topology 17 (1978), 229-248. MR 508887 (80g:55035)
  • [8] M. Bendersky and D. M. Davis, $ 2$-primary $ {v_1}$-periodic homotopy groups of $ SU(n)$, Amer. J. Math. (to appear). MR 1165351 (93g:55018)
  • [9] -, Unstable $ BP$-homology and desuspensions, Amer. J. Math. 107 (1985), 833-852. MR 796905 (87h:55002)
  • [10] A. Borel, Sous-groupes commutatifs et torsion des groupes compact connexes, Tôhoku J. Math. 13 (1961), 216-240. MR 0147579 (26:5094)
  • [11] 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), 549-565. MR 554384 (81c:55021)
  • [12] M. Crabb and K. Knapp, The Hurewicz map on stunted complex projective spaces, Amer. J. Math. 110 (1988), 783-809. MR 961495 (90d:55024)
  • [13] D. M. Davis, $ {v_1}$-periodic homotopy groups of $ SU(n)$ at an odd prime, J. London Math. Soc. (to appear).
  • [14] D. M. Davis and M. Mahowald, Three contributions to the homotopy theory of the exceptional Lie groups $ {G_2}$ and $ {F_4}$, J. Math. Soc. Japan (to appear). MR 1126144 (92g:57048)
  • [15] -, $ {v_1}$-periodicity in the unstable Adams spectral sequence, Math. Z. 204 (1990), 319-339. MR 1107466 (92e:55018)
  • [16] -, Some remarks on $ {v_1}$-periodic homotopy groups, Proc. Adams Symposium, London Math. Soc. Lecture Notes (to appear).
  • [17] I. M. James, On Lie groups and their homotopy groups, Proc. Cambridge Phil. Soc. 55 (1959), 244-247. MR 0124051 (23:A1371)
  • [18] A. T. Lundell, A divisibility property for Stirling numbers, J. Number Theory 10 (1978), 35-54. MR 0460135 (57:131)
  • [19] M. Mahowald, The image of $ J$ in the $ EHP$ sequence, Ann. of Math. (2) 116 (1982), 65-112. MR 662118 (83i:55019)
  • [20] H. R. Miller and D. C. Ravenel, Morava stabilizer algebras and the localization of Novikov's $ {E_2}$-term, Duke Math. J. 44 (1977), 433-447. MR 0458410 (56:16613)
  • [21] M. Mimura, G. Nishida, and H. Toda, $ Mod\;p$ decomposition of compact Lie groups, Publ. Res. Inst. Math. Sci. 13 (1977), 627-680. MR 0478187 (57:17675)
  • [22] M. Mimura and H. Toda, Homotopy groups of $ SU(3)$, $ SU(4)$, and $ Sp(2)$, J. Math. Kyoto Univ. 3 (1964), 217-250. MR 0169242 (29:6495a)
  • [23] -, Cohomology operations and the homotopy of compact Lie groups. I, Topology 9 (1970), 317-336. MR 0266237 (42:1144)
  • [24] D. C. Ravenel, Complex cobordism and the stable homotopy groups of spheres, Academic Press, 1986. MR 860042 (87j:55003)
  • [25] P. Selick, Moore conjectures, Lecture Notes in Math., vol. 1318, Springer-Verlag, 1988, pp. 219-227. MR 952582 (90c:55014)
  • [26] R. D. Thompson, The $ {v_1}$-periodic homotopy groups of an unstable sphere at odd primes, Trans. Amer. Math. Soc. 319 (1990), 535-560. MR 1010890 (90j:55021)
  • [27] C. Wilkerson, Self-maps of classifying spaces, Lecture Notes in Math., vol. 418, Springer-Verlag, 1974, pp. 150-157. MR 0383444 (52:4325)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1116310-9
Keywords: Exceptional Lie groups, periodic homotopy groups, unstable Novikov spectral sequence
Article copyright: © Copyright 1992 American Mathematical Society

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