Homology, Homotopy and Applications

Volume 22 (2020)

Number 1

On some sequences of modules over the $\operatorname{mod} p$ Steenrod algebra

Pages: 185 – 202

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n1.a11

Authors

Danila Emelyanov (Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia)

Theodore Popelensky (Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia)

Abstract

In [3] Toda conjectured the exactness of some sequences of modules over the Steenrod algebra $\operatorname{mod} 2$. One of the conjectures was proved by Wall [1]. In this note we show that in general the analogue of the Toda conjecture for $p \gt 2$ does not hold. We give an upper bound on dimensions in which exactness holds. Also we consider examples of exact sequences.

Keywords

Steenrod algebra, reduced power, monomial basis, Toda conjecture

2010 Mathematics Subject Classification

55S05, 55S10

Received 18 June 2018

Received revised 10 March 2019

Accepted 27 May 2019

Published 13 November 2019