The Dyer-Lashof algebra and the hit problems
type: Item
instance of: scholarly article
title: The Dyer-Lashof algebra and the hit problems
zbMATH Open document ID: 1473.55009
author: Yanyan Li
published in: The New York Journal of Mathematics
publication date: 2021-08-16T00:00:00Z
full work available at URL: 27-45.html
review text: 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 trans…
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 55S12
Mathematics Subject Classification ID: 55P47
zbMATH DE Number: 7382066
zbMATH Keywords: hit problem
zbMATH Keywords: symmetric hit problem
zbMATH Keywords: Steenrod algebra
zbMATH Keywords: Dyer-Lashof algebra
zbMATH Keywords: Becker-Gottlieb transfer
zbMATH Keywords: infinite loop space
reviewed by: Geoffrey Powell
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7382066