The $v_ 1$-periodic homotopy groups of an unstable sphere at odd primes

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.