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

M. Behrens (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

M. Hill (Department of Mathematics, University of Virginia, Charlottesville, Va., U.S.A.)

M.J. Hopkins (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

M. Mahowald (Department of Mathematics, Northwestern University, Evanston, Ilinois, U.S.A.)

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.

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Published 1 January 2008