Structure of the space of $G{L}_4({\mathbb{Z}}_2)$-coinvariants ${\mathbb{Z}}_2{\otimes }_{G{L}_4({\mathbb{Z}}_2)}P{H}_{\ast }({\mathbb{Z}}_2^4,{\mathbb{Z}}_2)$ in some generic degrees and its application

Abstract

Let $A$ denote the Steenrod algebra at the prime 2 and let $k={\mathbb{Z}}_2.$ An open problem of homotopy theory is to determine a minimal set of $A$-generators for the polynomial ring $P_q=k[x_1,\dots ,x_q]=H^{\ast }(k^q,k)$ on $q$ generators $x_1,\dots ,x_q$ with $|x_i|=1.$ Equivalently, one can write down explicitly a basis for the graded vector space $Q^{\otimes q}:=k{\otimes }_A{P}_q$ in each non-negative degree $n.$ This is the content of the classical "hit problem" in literature [30]. Based on this problem, we are interested in the $q$-th cohomological transfer $T{r}_q^A$ of Singer [39], which is one of the useful tools for describing mod-2 cohomology of the algebra $A.$ This transfer is a linear map from the space of $G{L}_q(k)$-coinvariant $k{\otimes }_{G{L}_q(k)}P((P_q{)}_n^{\ast })$ of $Q^{\otimes q}$ to the $k$-cohomology group of the Steenrod algebra, ${\mathrm{Ext}}_A^{q,q+n}(k,k).$ Here $G{L}_q(k)$ is the general linear group of degree $q$ over the field $k,$ and $P((P_q{)}_n^{\ast })$ is the primitive part of $(P_q{)}_n^{\ast }$ under the action of $A.$ Singer conjectured that $T{r}_q^A$ is a monomorphism, but this remains unanswered for all $q\ge 4.$ The present paper is to devoted to the investigation of this conjecture for the rank 4 case. More specifically, basing the techniques of the hit problem of four variables, we explicitly determine the structure of $k{\otimes }_{G{L}_4(k)}P((P_4{)}_n^{\ast })$ in some generic degrees $n.$ Applying these results and a representation of $T{r}_4^A$ over the lambda algebra, we notice that Singer's conjecture is true for the rank 4 transfer in those degrees $n$. Also, we give some conjectures on the dimensions of $k{\otimes }_{G{L}_q(k)}((P_4{)}_n^{\ast })$ for the remaining degrees $n.$ As a consequence, Singer's conjecture holds for $T{r}_4^A.$ This study and our previous results have been provided a panorama of the behavior of the fourth cohomological transfer.

34 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:2105.05738, arXiv:2106.14606; substantial text overlap with arXiv:1412.1709 by other author

References in corpus (4)

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• On the dimension of $H^{\ast }(({\mathbb{Z}}_2{)}^{\times t},{\mathbb{Z}}_2)$ as a module over Steenrod ring
• The Lambda algebra and Sq_0
• On the lambda algebra and Singer's cohomological transfer