Generators of the cohomology of \(BV_3\)
Let \(P_k=\mathbb{F}_2[x_1,\dots,x_k]\) be a polynomial algebra over the field \(\mathbb{F}_2\) generated by \(x_1,\dots,x_k\). By assigning degree 1 to each \(x_j\), \(P_k\) is regarded as a graded algebra over the ground field \(\mathbb{F}_2\). The mod 2 cohomology ring of the classifying space \(BV_k\) of the elementary abelian 2-group \(V_k\) with rank \(k\), is isomorphic to \(P_k\) as a graded algebra. Through this isomorphism, we may regard \(P_k\) as an \({\mathcal A}\)-module where \({\mathcal A}\) stands for the mod 2 Steenrod algebra. From early days of algebraic topology, topologists have been studying this cohomology ring and by making use of this cohomology ring, topologists have been proving many theorems. But our knowledge of this cohomology ring is not deep enough. For instance, we do not know even the dimension of the vector space \(QP^n_k=(\mathbb{F}_2\otimes_{\mathcal A}P_k)^n\) for \(k=4\). In this paper, we determine the upper bound for the dimension of the above vector space \(QP^n_3\) and give a set of generators in terms of monomials in Theorem 5.2 and we will see that \(\dim QP^n_3\leq 21\). In his thesis [Johns Hopkins Univ. 1990], the author gave the proof for the linear independence of these monomials in \(QP^n_3\). After his thesis was submitted, the lower bound for \(\dim QP^n_3\) is provided by \textit{Mohamed Ali Alghamdi}, \textit{M. C. Crabb} and \textit{J. R. Hubbuck} [Lond. Math. Soc. Lect. Note Ser. 176, 217-234 (1992; Zbl 0752.55012)] and \textit{J. M. Boardman} [Contemp. Math. 146, 49-70 (1993; Zbl 0789.55015)] investigating homology of \(BV_3\) instead of its cohomology. In this paper, we do not give the proof of linear independence of our generators but they actually form a minimal set of generators.
scientific article; zbMATH DE number 1309663