On the dimension of the "cohits" space ${\mathbb{Z}}_2{\otimes }_{{\mathcal{A}}_2}{H}^{\ast }((\mathbb{R}P(\infty ){)}^{\times t},{\mathbb{Z}}_2)$ and some applications

Abstract

We denote by ${\mathbb{Z}}_2$ the prime field of two elements and by $P_t={\mathbb{Z}}_2[x_1,\dots ,x_t]$ the polynomial algebra of $t$ generators $x_1,\dots ,x_t$ with the degree of each $x_i$ being one. Let ${\mathcal{A}}_2$ be the Steenrod algebra over ${\mathbb{Z}}_2.$ A central problem of homotopy theory is to determine a minimal set of generators for the "cohits" space ${\mathbb{Z}}_2{\otimes }_{{\mathcal{A}}_2}{P}_t.$ This problem, which is called the "hit" problem for Steenrod algebra, has been systematically studied for $t\le 4.$ The present paper is devoted to the investigation of the structure of ${\mathbb{Z}}_2{\otimes }_{{\mathcal{A}}_2}{P}_t$ in some certain "generic" degrees. More specifically, we explicitly determine a monomial basis of ${\mathbb{Z}}_2{\otimes }_{{\mathcal{A}}_2}{P}_5$ in degree $n_s=5(2^s-1)+42.2^s$ for every non-negative integer $s.$ As a result, it confirms Sum's conjecture [14] for a relation between the minimal sets of ${\mathcal{A}}_2$-generators of the algebras $P_{t-1}$ and $P_t$ in the case $t=5$ and degree $n_s$. As applications, we obtain the dimension of ${\mathbb{Z}}_2{\otimes }_{{\mathcal{A}}_2}{P}_6$ in the generic degree $5(2^{s+5}-1)+n_0.2^{s+5}$ for all $s\ge 0,$ and show that the Singer's cohomological transfer [11] is an isomorphism in bidegree $(5,5+n_s)$.

6 pages. This paper is an announcement whose details will appear elsewhere. Comments are welcome! arXiv admin note: text overlap with arXiv:1907.08768