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

We write $BV_h$ for the classifying space of the elementary Abelian 2-group $V_h$ of rank $h,$ which is homotopy equivalent to the cartesian product of $h$ copies of $\\mathbb RP^{\\infty}.$ Its cohomology with $\\mathbb Z/2$-coefficients can be identified with the graded unstable algebra $P^{\\otimes h} = \\mathbb Z/2[t_1, \\ldots, t_h]= \\bigoplus_{n\\geq 0}P^{\\otimes h}_n$ over the Steenrod ring $\\mathcal A$, where grading is 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. Let $GL_h$ be the usual general linear group of rank $h$ over $\\mathbb Z/2.$ The algebra $P^{\\otimes h}$ admits a left action of $\\mathcal A$ as well as a right action of $GL_h.$ A central problem of homotopy theory is to determine the structure of the space of $GL_h$-coinvariants, $\\mathbb Z/2\\otimes_{GL_h}{\\rm Ann}_{\\overline{\\mathcal A}}H_n(BV_h; \\mathbb Z/2) ,$ where ${\\rm Ann}_{\\overline{\\mathcal A}}H_n(BV_h; \\mathbb Z/2) ={\\rm Ann}_{\\overline{\\mathcal A}}[P^{\\otimes h}_n]^{*}$ denotes the space of primitive homology classes, considered as a representation of $GL_h$ for all $n.$ 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 $k(2^{s} - 1) + r.2^{s},$ where $k,\\, r,\\, s$ are positive integers. Applying the results, we investigate the behaviour of the Singer cohomological transfer of rank $4$, which is a homomorphism from a certain subquotient of the divided power algebra $\\Gamma(a_1^{(1)}, \\ldots, a_4^{(1)})$ to mod-2 cohomology groups ${\\rm Ext}_{\\mathcal A}^{4, 4+n}(\\mathbb Z/2, \\mathbb Z/2)$ of the algebra $\\mathcal A.$ Singer s algebraic transfer is one of the relatively efficient tools in determining the cohomology of the Steenrod algebra.