The Dyer-Lashof algebra and the hit problems
This paper is motivated by the hit problem and its `symmetric' variant at the prime \(2\), i.e., the study of the indecomposables \[ \mathbb{F}_2 \otimes_{\mathcal{A}} H^* (BO(1)^{\times k} ) \quad \mbox{ and } \quad \mathbb{F}_2 \otimes_{\mathcal{A}} H^* (BO(k) ) \] for the action of the Steenrod algebra on the mod \(2\) cohomology of the respective classifying spaces, for \(k \in\mathbb{N}\). The author works with homology, so that indecomposables are replaced by \(\mathcal{A}\)-annihilated elements. The basic idea is to work with the infinite loop space \(\mathbb{Z} \times BO\) (as in [\textit{D. J. Pengelley} and \textit{F. Williams}, Algebr. Geom. Topol. 3, 1119--1138 (2003; Zbl 1057.55004)], for example) and also to use the unit map \(QS^0 \rightarrow \mathbb{Z} \times BO\). This allows the Pontrjagin product and the action of the Dyer-Lashof algebra to be exploited. The author uses information on the \(\mathcal{A}\)-annihilated classes of \(H_* (QS^0)\) due to Curtis and to Wellington [\textit{R. J. Wellington}, The unstable Adams spectral sequence for free iterated loop spaces. Providence, RI: American Mathematical Society (AMS) (1982; Zbl 0488.55006)], and the action of the Dyer-Lashof algebra on \(H_* (\mathbb{Z} \times BO)\) given by Priddy [\textit{S. Priddy}, Q. J. Math., Oxf. II. Ser. 26, 179--193 (1975; Zbl 0311.55013)]. The main results give families of non-zero \(\mathcal{A}\)-annihilated classes in \(H_*(\mathbb{Z} \times BO)\), whence classes for the appropriate \(BO (k)\). For instance, a criterion is given on a sequence \(I\) of positive integers for the class \(Q^I e_0 \in H_* (\mathbb{Z} \times BO)\) to be non-zero and \(\mathcal{A}\)-annihilated; this family is sparse. Likewise, for \(t\) a positive integer, the author constructs an explicit non-zero \(\mathcal{A}\)-annihilated class in \(H_{2^t-1} (\mathbb{Z} \times BO)\); using suitable `spike' products of these gives further classes. The author then proposes using the Becker-Gottlieb transfer associated to \(B O(1)^{\times k} \rightarrow BO (k)\) as a method to construct \(\mathcal{A}\)-annihilated classes in \(H_* (BO(1)^{\times k} )\) using the above results. An Appendix (credited to Seyyed Mohammad Ali HasanZadeh and based on joint work with the author) analyses (small length) sequences \(I\) that arise in the work of Curtis and Wellington. It is not completely clear how these results relate to existing ones in the literature or to what extent such methods will allow progress on the (symmetric) hit problem.
scientific article; zbMATH DE number 7382066
scientific article