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Transactions of the American Mathematical Society

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

 

 

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