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Transactions of the American Mathematical Society

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The $ v\sb 1$-periodic homotopy groups of an unstable sphere at odd primes


Author: Robert D. Thompson
Journal: Trans. Amer. Math. Soc. 319 (1990), 535-559
MSC: Primary 55Q40
DOI: https://doi.org/10.1090/S0002-9947-1990-1010890-8
MathSciNet review: 1010890
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Abstract: The $ \bmod \;p$ $ {v_1}$-periodic homotopy groups of a space $ X$ are defined by considering the homotopy classes of maps of a Moore space into $ X$ and then inverting the Adams self map. In this paper we compute the $ p$ $ {v_1}$-periodic homotopy groups of an odd dimensional sphere, localized at an odd prime. This is done by showing that these groups are isomorphic to the stable $ \bmod \;p$ $ {v_1}$-periodic homotopy groups of $ B\Sigma _p^{2(p - 1)n}$, the $ 2(p - 1)n$ skeleton of the classifying space for the symmetric group $ {\Sigma _p}$. There is a map $ {\Omega ^{2n + 1}}{S^{2n + 1}} \to {\Omega ^\infty }(J \wedge B\Sigma _p^{2(p - 1)n})$, where $ J$ is a spectrum constructed from connective $ K$-theory, and the image in homotopy is studied.


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DOI: https://doi.org/10.1090/S0002-9947-1990-1010890-8
Article copyright: © Copyright 1990 American Mathematical Society

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