A note on the modular representation on the $\mathbb Z/2$-homology groups of the fourth power of real projective space and its application

Abstract

One knows that, the connected graded ring $P^{\otimes h}= \mathbb Z/2[t_1, \ldots, t_h]= \{P_n^{\otimes h}\}_{n\geq 0},$ which is graded by the degree of the homogeneous terms $P^{\otimes h}_n$ of degree $n$ in $h$ generators with the degree of each $t_i$ being one, admits a left action of $\mathcal A$ as well as a right action of the general linear group $GL_h.$ A central problem of homotopy theory is to determine the structure of the space of $GL_h$-coinvariants, $\mathbb Z/2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}_n]^{*}.$ Solving this problem is very difficult and still open for $h\geq 4.$ In this Note, our intent is of studying the dimension of $\mathbb Z/2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}_n]^{*}$ for the case $h = 4$ and the "generic" degrees $n$ of the form $n_{k, r, s} = k(2^{s} - 1) + r.2^{s},$ where $k,\, r,\, s$ are positive integers. Applying the results, we investigate the behaviour of the Singer cohomological "transfer" of rank $4.$ Singer's transfer is a homomorphism from a certain subquotient of the divided power algebra $\Gamma(a_1^{(1)}, \ldots, a_h^{(1)})$ to mod-2 cohomology groups ${\rm Ext}_{\mathcal A}^{h, h+*}(\mathbb Z/2, \mathbb Z/2)$ of the algebra $\mathcal A.$ This homomorphism is useful for depicting the Ext groups. Additionally, in higher ranks, by using the results on $\mathcal A$-generators for $P^{\otimes 5}$ and $P^{\otimes 6},$ we show in Appendix that the transfer of rank 5 is an isomorphism in som certain degrees of the form $n_{k, r, s}$, and that the transfer of rank 6 does not detect the non-zero elements $h_2^{2}g_1 = h_4Ph_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_{6, 10, 1}}(\mathbb Z/2, \mathbb Z/2)$, and $D_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_{6, 10, 2}}(\mathbb Z/2, \mathbb Z/2).$ Besides, we also probe the behavior of the Singer transfer of ranks 7 and 8 in internal degrees $\leq 15.$