Dyer-Lashof代数与命中问题的关系
On the relation between Dyer-Lashof algebra and the hit problems
摘要：
本文的目的是用几何方法研究Peterson对$H_*\mathbb{R} P^{\times k}$的命中问题以及Janfada和Wood对$H_*BO(k)$的对称命中问题.我们继续探索\cite{Zare-symmetric}的结果的应用程序上的$\mathcal{A}$湮灭生成器的$H_*QX$，以获得一个家庭的通用的“新”的例子$\mathcal{A}$湮灭在$H_*(\mathbb{Z}\times BO)$和$H_*BO$，即稳定的对称命中问题的情况下，其中一个重要的步骤是由无限循环空间结构$BO$暗示的Bott周期。应用长度过滤允许状态我们的结果的情况下，对称命中问题的$H_*BO(k)$。使用与$\mathbb{R} P^{\times k}=BO(1)^{\times k}\to BO(k)$相关的Becker-Gottlieb转移，我们能够重申我们对$H_*\mathbb{R} P^{\times k}$的经典命中问题的结果。我们使用短语“稳定命中问题”来研究$H_*\mathbb{R} P^{\times k}$同时对所有$k>0$的命中问题，这允许使用乘法结构，我们似乎在\cite{Ault-Singer}之后采取了一些新的步骤。我们的新例子依赖于特定的数值条件，我们提供了一个算法来构建\cite{Zare-symmetric}。方法论的结果是，这些条件也必须考虑到，同时处理计数参数。我们得到的数值条件似乎没有出现在文献中，在这种情况下，虽然他们可能包括以前的例子。因此，我们的工作提供了一个新的例子无限的家庭，从而提高了以前得到的下限。
英文摘要：
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.
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