Homology, Homotopy and Applications

Volume 22 (2020)

Number 2

On a conjecture of Mahowald on the cohomology of finite sub-Hopf algebras of the Steenrod algebra

Pages: 59 – 72

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n2.a3

Author

Paul Shick (Department of Mathematics and Computer Science, John Carroll University, University Heights, Ohio, U.S.A.)

Abstract

Mahowald’s conjecture arose as part of a program attempting to view chromatic phenomena in stable homotopy theory through the lens of the classical Adams spectral sequence. The conjecture predicts the existence of nonzero classes in the cohomology of the finite sub-Hopf algebras $A(n)$ of the $\operatorname{mod} 2$ Steenrod algebra that correspond to generators in the homotopy rings of certain periodic spectra. The purpose of this note is to present a proof of the conjecture.

Keywords

Mahowald conjecture, periodicity, cohomology, Steenrod algebra

2010 Mathematics Subject Classification

55Q45, 55Q51, 55T15

Received 30 April 2019

Received revised 22 July 2019

Accepted 27 September 2019

Published 25 March 2020