论文标题
在“ cohits”空间的维度上
On the dimension of the "cohits" space $\mathbb Z_2\otimes_{\mathcal A_2} H^{*}((\mathbb RP(\infty))^{\times t}, \mathbb Z_2)$ and some applications
论文作者
论文摘要
我们用$ \ mathbb z_2 $表示两个元素的素数，$ p_t = \ mathbb z_2 [x_1，\ ldots，x_t] $ $ t $生成器的polyenmial algebra $ x_1，\ ldots，x_t $，每个$ x_i $是一个。令$ \ MATHCAL A_2 $为$ \ Mathbb Z_2。 $ t \ leq 4的系统研究。$本文专门研究$ \ mathbb z_2 \ otimes _ {\ mathcal a_2} p_t $的结构。更具体地说，我们明确确定$ \ mathbb z_2 \ otimes _ {\ Mathcal a_2} p_5 $ in Leg $ n_s = 5（2^{s} -1） + 42.2^{s} $的最小$ n n _ s}的最新情况，以确认$ s。 $ \ MATHCAL A_2 $ - 代数$ P_ {T-1} $和$ p_ {t} $的生成器，$ t = 5 $和度$ n_s $。作为应用程序，我们获得了$ \ Mathbb Z_2 \ outimes _ {\ Mathcal A_2} p_6 $的通用度$ 5（2^{s+5} -1）+N_0.2^{s+5} $的$ 5（2^{s+5} -1）$ $（5，5+n_s）$。
We denote by $\mathbb Z_2$ the prime field of two elements and by $P_t = \mathbb Z_2[x_1, \ldots, x_t]$ the polynomial algebra of $t$ generators $x_1, \ldots, x_t$ 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 $t\leq 4.$ 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 $s\geq 0,$ and show that the Singer's cohomological transfer [11] is an isomorphism in bidegree $(5, 5+n_s)$.
