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# Homology, Homotopy and Applications

## Volume 10 (2008)

### Number 3

### Proceedings of a Conference in Honor of Douglas C. Ravenel and W. Stephen Wilson

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

Pages: 45 – 84

DOI: https://dx.doi.org/10.4310/HHA.2008.v10.n3.a4

#### Authors

#### Abstract

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

#### Keywords

$v2$-periodicity; stable homotopy

#### 2010 Mathematics Subject Classification

Primary 55Q51. Secondary 55Q40.

Published 1 January 2008