On the Bott periodicity, $\mathcal{A}$-annihilated classes in $H_*QX$, and the stable symmetric hit problem

Abstract

We provide a characterisation of $\mathcal{A}$-annihilated generators in the homology ring $H_*(QX;\mathbb{Z}/2)$ and $H_*(Q(X_+);\mathbb{Z}/2)$ when $X$ is some path connected space. We also introduce a method to construct such classes. We comment on the application of this result to illustrate how to use the infinite loop space structure on $\mathbb{Z}\times BO$, provided by the Bott periodicity can be used to obtain some information on the (stable) symmetric hit problem of Wood and Janfada. Our methods seem to allow much straightforward calculations. The numerical conditions of our Theorem 3 look very similar to the `spikes' considered by Wood \cite{Wood-Ioa} and Janfada-Wood \cite{JanfadaWood} as well as Janfada \cite{Janfada-P(3)}.