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Transactions of the American Mathematical Society

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On the double suspension homomorphism at odd primes


Authors: J. R. Harper and H. R. Miller
Journal: Trans. Amer. Math. Soc. 273 (1982), 319-331
MSC: Primary 55T15; Secondary 55P40, 55Q25, 55Q45, 55U99
DOI: https://doi.org/10.1090/S0002-9947-1982-0664045-2
MathSciNet review: 664045
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Abstract: We work with the $ {E_1}$-term for spheres and the stable Moore space, given by the $ \Lambda $-algebra at odd primes. Writing $ W(n) = \Lambda (2n + 1)/\Lambda (2n - 1)$ and $ M(0) = {H_ \ast }({S^0}{ \cup _p}{e^1})$, we construct compatible maps $ {f_n} \cdot W(n) \to M(0)\tilde \otimes \Lambda $ and prove the Metastability Theorem: in homology $ {f_n}$ induces an isomorphism for $ \sigma < 2({p^2} - 1)(s - 2) + pqn - 2p - 2$ where $ \sigma =$   stem degree$ $, $ s =$   homological degree  resulting from the bigrading of $ \Lambda $ and $ q = 2p - 2$. There is an operator $ {\upsilon _1}$ corresponding to the Adams stable self-map of the Moore space and $ {\upsilon _1}$ extends to $ W(n)$. A corollary of the Metastability Theorem and the Localization Theorem of the second author is that the map $ {f_n}$ induces an isomorphism on homology after inverting $ {\upsilon _1}$.


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  • [1] J. F. Adams, On the groups $ J(X)$. IV, Topology 5 (1966), 21-71. MR 0198470 (33:6628)
  • [2] A. K. Bousfield et al., The $ \operatorname{mod} {\text{ - }}p$ lower central series and the Adams spectral sequence, Topology 5 (1966), 331-342. MR 0199862 (33:8002)
  • [3] A. K. Bousfield and D. M. Kan, The homotopy spectral sequence of a space with coefficients in a ring, Topology 11 (1977), 79-106. MR 0283801 (44:1031)
  • [4] J. Harper, Rank $ 2\operatorname{mod} 3$ $ H$-spaces, Canad. Math. Soc. Conf. Proc., Current Trends in Algebraic Topology, Western Ontario, 1981.
  • [5] D. C. Johnson, H. R. Miller, W. S. Wilson and R. S. Zahler, Boundary homomorphisms in the generalized Adams spectral sequence and the nontriviality of infinitely many $ {\gamma _t}$ in stable homotopy, Notas de Mat. y Symp., No. 1: Reunion Sobre Teoria de Homotopia, Northwestern Univ., Soc. Mat. Mex., 1975, pp. 47-59. MR 761720
  • [6] A. Liulevicius, Zeroes of the cohomology of the Steenrod algebra, Proc. Amer. Math. Soc. 14 (1963), 972-976. MR 0157383 (28:617)
  • [7] M. Mahowald, On the double suspension homomorphism, Trans. Amer. Math. Soc. 214 (1975), 169-178. MR 0438333 (55:11248)
  • [8] H. R. Miller, A localization theorem in homological algebra, Math. Proc. Cambridge Philos. Soc. 84 (1978), 73-84. MR 0494105 (58:13036)
  • [9] H. R. Miller and C. Wilkerson, Vanishing lines for modules over the Steenrod algebra, J. Pure Appl. Algebra 22 (1981), 293-308. MR 629336 (82m:55024)
  • [10] M. C. Tangora, Some remarks on the lambda algebra, Geometric Applications of Homotopy Theory. II, (Proceedings, Evanston 1977), Lecture Notes in Math., vol. 658, Springer-Verlag, Berlin and New York, 1978, pp. 476-487. MR 513587 (80d:55024)
  • [11] F. R. Cohen, The unstable decomposition of $ {\Omega ^2}{\Sigma ^2}X$ and its applications (to appear).

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DOI: https://doi.org/10.1090/S0002-9947-1982-0664045-2
Article copyright: © Copyright 1982 American Mathematical Society

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