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 $P^{\otimes h}=\mathbb{Z}/2[t_1,\dots ,t_h]=\{P_n^{\otimes h}{\}}_{n\ge 0},$ which is graded by the degree of the homogeneous terms $P_n^{\otimes h}$ 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 $G{L}_h.$ A central problem of homotopy theory is to determine the structure of the space of $G{L}_h$-coinvariants, $\mathbb{Z}/2{\otimes }_{G{L}_h}{\mathrm{Ann}}_{\overline{\mathcal{A}}}[P_n^{\otimes h}{]}^{\ast }.$ Solving this problem is very difficult and still open for $h\ge 4.$ In this Note, our intent is of studying the dimension of $\mathbb{Z}/2{\otimes }_{G{L}_h}{\mathrm{Ann}}_{\overline{\mathcal{A}}}[P_n^{\otimes h}{]}^{\ast }$ 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 $\Gamma (a_1^{(1)},\dots ,a_h^{(1)})$ to mod-2 cohomology groups ${\mathrm{Ext}}_{\mathcal{A}}^{h,h+\ast }(\mathbb{Z}/2,\mathbb{Z}/2)$ 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 $h_2^2{g}_1=h_{4}P{h}_2\in {\mathrm{Ext}}_{\mathcal{A}}^{6,6+n_{6,10,1}}(\mathbb{Z}/2,\mathbb{Z}/2)$, and $D_2\in {\mathrm{Ext}}_{\mathcal{A}}^{6,6+n_{6,10,2}}(\mathbb{Z}/2,\mathbb{Z}/2).$ Besides, we also probe the behavior of the Singer transfer of ranks 7 and 8 in internal degrees $\le 15.$

51 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:2105.05738; substantial text overlap with arXiv:1412.1709 by other author

References in corpus (3)

• On the dimension of $H^{\ast }(({\mathbb{Z}}_2{)}^{\times t},{\mathbb{Z}}_2)$ as a module over Steenrod ring
• The Lambda algebra and Sq_0
• On the lambda algebra and Singer's cohomological transfer

Cited by in corpus (1)

• On the dimension of $H^{\ast }(({\mathbb{Z}}_2{)}^{\times t},{\mathbb{Z}}_2)$ as a module over Steenrod ring