The 𝑣₁-periodic homotopy groups of an unstable sphere at odd primes

Author:

Thompson Robert D.

Abstract

The mod p \bmod \;p v 1 {v_1} -periodic homotopy groups of a space X X are defined by considering the homotopy classes of maps of a Moore space into X X and then inverting the Adams self map. In this paper we compute the p p v 1 {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 mod p \bmod \;p v 1 {v_1} -periodic homotopy groups of B Σ p 2 ( p 1 ) n B\Sigma _p^{2(p - 1)n} , the 2 ( p 1 ) n 2(p - 1)n skeleton of the classifying space for the symmetric group Σ p {\Sigma _p} . There is a map Ω 2 n + 1 S 2 n + 1 Ω ( J B Σ p 2 ( p 1 ) n ) {\Omega ^{2n + 1}}{S^{2n + 1}} \to {\Omega ^\infty }(J \wedge B\Sigma _p^{2(p - 1)n}) , where J J is a spectrum constructed from connective K K -theory, and the image in homotopy is studied.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

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