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

TL;DR

This paper investigates the structure of mod-2 homology groups of the fourth power of real projective space, focusing on the dimension of certain coinvariant spaces and the behavior of the Singer transfer in various ranks and degrees.

Contribution

It provides new results on the dimension of GL_h-coinvariants for h=4 in generic degrees and analyzes the Singer transfer's behavior across ranks 4 to 8 in specific degrees.

Findings

Dimension of coinvariant spaces for h=4 in generic degrees determined.

The transfer of rank 5 is an isomorphism in certain degrees.

Transfer of rank 6 does not detect some non-zero Ext elements.

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 $\leq 15.$