On the relation between Dyer-Lashof algebra and the hit problems

The aim of this note is to use geometric methods to study the hit problem of Peterson for $H_*\mathbb{R} P^{\times k}$ as well as the symmetric hit problem of Janfada and Wood for $H_*BO(k)$. We continue by exploring the applications of the results of \cite{Zare-symmetric} on the $\mathcal{A}$-annihilated generators of $H_*QX$ to obtain a family of generic `new' examples of $\mathcal{A}$-annihilated in $H_*(\mathbb{Z}\times BO)$ and $H_*BO$, i.e. the case of stable symmetric hit problem, where an essential step is provided by the infinite loop space structure on $BO$ implied by the Bott periodicity. Applying a length filtration allows to state our results in the case of symmetric hit problem for $H_*BO(k)$. Using the Becker-Gottlieb transfer associated to $\mathbb{R} P^{\times k}=BO(1)^{\times k}\to BO(k)$ we are able to restate our results for the classic hit problem of $H_*\mathbb{R} P^{\times k}$. We use the phrase `stable hit problem' to the study of hit problem for $H_*\mathbb{R} P^{\times k}$ for all $k>0$ at once, which allows to use multiplicative structures on which we seem to have taken some new steps after \cite{Ault-Singer}. Our new examples depend on specific numerical conditions of which we have provided an algorithm to construct in \cite{Zare-symmetric}. The methodological outcome is that such conditions also have to taken into account while dealing with counting arguments. The numerical conditions we obtain seem to have not appeared in the literature in this context, although they may include previous ones as examples. Therefore, our work provides an infinite family of new examples and consequently raises the lower bounds obtained previously.