Homology, Homotopy and Applications

Volume 23 (2021)

Number 1

Unstable algebras over an operad

Pages: 119 – 141

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n1.a8


Sacha Ikonicoff (Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université de Paris, France)


The aim of this article is to define and study a notion of unstable algebra over an operad that generalises the classical notion of unstable algebra over the Steenrod algebra. For this study we focus on the case of characteristic $2$. We define $\star$-unstable $\mathcal{P}$-algebras, where $\mathcal{P}$ is an operad and $\star$ is a commutative binary operation in $\mathcal{P}$. We then build a functor that takes an unstable module $M$ to the free $\star$-unstable $\mathcal{P}$-algebra generated by $M$. Under some hypotheses on $\star$ and on $M$, we identify this unstable algebra as a free $\mathcal{P}$-algebra. Finally, we give some examples of this result, and we show how to use our main theorem to obtain a new construction of the unstable modules studied by Carlsson, Brown–Gitler, and Campbell–Selick, that takes into account their internal product.


Steenrod algebra, operad, unstable module, unstable algebra

2010 Mathematics Subject Classification

17A30, 18D50, 55S10

Received 7 February 2020

Received revised 7 April 2020

Accepted 11 April 2020

Published 26 August 2020