On the existence of a $v^{32}_2$-self map on $M(1,4)$ at the prime 2

摘要

Let $M(1)$ be the mod $2$ Moore spectrum. J.F. Adams proved that $M(1)$ admits a minimal $v_1$-self map $v_1^4 colon Sigma^8 M(1) ightarrow M(1)$. Let $M(1,4)$ be the cofiber of this self-map. The purpose of this paper is to prove that $M(1,4)$ admits a minimal $v_2$-self map of the form $v_2^{32} colon Sigma^{192} M(1,4) ightarrow M(1,4)$. The existence of this map implies the existence of many $192$-periodic families of elements in the stable homotopy groups of spheres.