Subtle characteristic classes for Spin-torsors

Extending (Smirnov and Vishik, Subtle Characteristic Classes, arXiv:1401.6661), we obtain a complete description of the motivic cohomology with $${{\,\mathrm{\mathbb {Z}}\,}}/2$$ Z / 2 -coefficients of the Nisnevich classifying space of the spin group $$Spin_n$$ S p i n n associated to the standard split quadratic form. This provides us with very simple relations among subtle Stiefel–Whitney classes in the motivic cohomology of Čech simplicial schemes associated to quadratic forms from $$I^3$$ I 3 , which are closely related to $$Spin_n$$ S p i n n -torsors over the point. These relations come from the action of the motivic Steenrod algebra on the second subtle Stiefel–Whitney class. Moreover, exploiting the relation between $$Spin_7$$ S p i n 7 and $$G_2$$ G 2 , we describe completely the motivic cohomology ring of the Nisnevich classifying space of $$G_2$$ G 2 . The result in topology was obtained by Quillen (Math Ann 194:197–212, 1971).

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

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 unresolved for $h\geq 4.$ The aim of this Note 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.

On generators of the unstable $\mathscr A$-module $H^{*}((K(\mathbb F_2, 1))^{\times 5})$ in a generic degree and applications

Let us consider the prime field of two elements, $\mathbb F_2.$ It is well-known that the classical "hit problem" for a module over the mod 2 Steenrod algebra $\mathscr A$ is an interesting and important open problem of Algebraic topology, which asks a minimal set of generators for the polynomial algebra $\mathcal P_m:=\mathbb F_2[x_1, x_2, \ldots, x_m]$, regarded as a connected unstable $\mathscr A$-module on $m$ variables $x_1, \ldots, x_m,$ each of degree 1. The algebra $\mathcal P_m$ is the $\mathbb F_2$-cohomology of the product of $m$ copies of the Eilenberg-MacLan complex $K(\mathbb F_2, 1).$ Although the hit problem has been thoroughly studied for more than 3 decades, solving it remains a mystery for $m\geq 5.$ The aim of this work is of studying the hit problem of five variables. More precisely, we develop our previous work \cite{D.P3} on the hit problem for $\mathscr A$-module $\mathcal P_5$ in a degree of the generic form $n_t:=5(2^t-1) + 18.2^t,$ for any non-negative integer $t.$ An efficient approach to solve this problem had been presented. Moreover, we provide an algorithm in MAGMA for verifying the results and studying the hit problem in general. As an consequence, the calculations confirmed Sum's conjecture \cite{N.S2} for the relationship between the minimal sets of $\mathscr A$-generators of the polynomial algebras $\mathcal P_{m-1}$ and $\mathcal P_{m}$ in the case $m=5$ and degree $n_t.$ Two applications of this study are to determine the dimension of $\mathcal P_6$ in the generic degree $5(2^{t+4}-1) + n_1.2^{t+4}$ for all $t > 0$ and describe the modular representations of the general linear group of rank 5 over $\mathbb F_2.$ As a corollary, the cohomological "transfer", defined by W. Singer \cite{W.S1}, is an isomorphism at the bidegree $(5, 5+n_0).$ Singer's transfer is one of the relatively efficient tools to approach the structure of mod-2 cohomology of the Steenrod algebra.

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

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 $\mathbb Z_2$-graded vector 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 the "cohits" space $\mathbb Z_2\otimes_{\mathcal A_2} P_t$ 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 \mbox{$n_s=5(2^{s}-1) + 42.2^{s}$} for every non-negative integer $s.$ As a result, it confirms Sum's conjecture \cite{N.S2} 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 $n_s$. Based on Kameko's map \cite{M.K} and a previous result by Sum \cite{N.S1}, we obtain a inductive formula for the dimension of $\mathbb Z_2\otimes_{\mathcal A_2} P_t$ in a generic degree given. As an application, 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 \cite{W.S1} is an isomorphism in bidegree $(5, 5+n_s)$.

The "hit" problem of five variables in the generic degree and its application

Let $P_s:= \mathbb F_2[x_1,x_2,\ldots ,x_s]$ be the graded polynomial algebra over the prime field of two elements, $\mathbb F_2$, in $s$ variables $x_1, x_2, \ldots , x_s$, each of degree one. This algebra is considered as a graded module over the mod-2 Steenrod algebra, $\mathscr {A}$. We are interested in the "hit" problem of finding a minimal set of generators for $\mathscr A$-module $P_s.$ This problem is unresolved for every $s\geqslant 5.$ In this paper, we study the hit problem of five variables in a generic degree, from which we investigate Singer's conjecture [Math. Z. 202 (1989), 493-523] for the transfer homomorphism of rank $5$ in degrees given. This gives an efficient method to study the algebraic transfer and it is different from the ones of Singer.

On Peterson's open problem and representations of the general linear groups

Fix $\mathbb Z/2$ is the prime field of two elements and write $\mathcal A_2$ for the mod $2$ Steenrod algebra. Denote by $GL_d:= GL(d, \mathbb Z/2)$ the general linear group of rank $d$ over $\mathbb Z/2$ and by $\mathscr P_d$ the polynomial algebra $\mathbb Z/2[x_1, x_2, \ldots, x_d],$ which is viewed as a connected unstable $\mathcal A_2$-module on $d$ generators of degree one. We study the Peterson "hit problem" of finding the minimal set of $\mathcal A_2$-generators for $\mathscr P_d.$ It is equivalent to determining a $\mathbb Z/2$-basis for the space of "cohits"$$Q\mathscr P_d := \mathbb Z/2\otimes_{\mathcal A_2} \mathscr P_d \cong \mathscr P_d/\mathcal A_2^+\mathscr P_d.$$ This $Q\mathscr P_d$ is considered as a form modular representation of $GL_d$ over $\mathbb Z/2.$ The problem for $d= 5$ is not yet completely solved, and unknown in general. In this work, we give an explicit solution to the hit problem of five variables in the generic degree $n = r(2^t -1) + 2^ts$ with $r = d = 5,\ s =8$ and $t$ an arbitrary non-negative integer. An application of this study to the cases $t = 0$ and $t = 1$ shows that the Singer algebraic transfer is an isomorphism in the bidegrees $(5, 5+(13.2^{0} - 5))$ and $(5, 5+(13.2^{1} - 5)).$ Moreover, the result when $t\geq 2$ was also discussed. Here, the Singer transfer of rank $d$ is a $\mathbb Z/2$-algebra homomorphism from $GL_d$-coinvariants of certain subspaces of $Q\mathscr P_d$ to the cohomology groups of the Steenrod algebra, ${\rm Ext}_{\mathcal A_2}^{d, d+*}(\mathbb Z/2, \mathbb Z/2).$ It is one of the useful tools for studying mysterious Ext groups and the Kervaire invariant one problem.

On the lambda algebra and Singer's cohomological transfer

Write $\mathbb A$ for the 2-primary Steenrod algebra, which is the algebra of stable natural endomorphisms of the mod 2 cohomology functor on topological spaces. Working at the prime 2, computing the cohomology of $\mathbb A$ is an important problem of Algebraic topology, because it is the initial page of the Adams spectral sequence converging to stable homotopy groups of the spheres. A relatively efficient tool to describe this cohomology is the Singer algebraic transfer of rank $n$ in \cite{Singer}, which passes from a certain subquotient of a divided power algebra to the cohomology of $\mathbb A.$ Singer predicted that this transfer is a monomorphism, but this remains open for $n\geq 4.$ This short note is to verify the conjecture in the ranks 4 and 5 and some generic degrees.

Structure of the space of $GL_4(\mathbb Z_2)$-coinvariants $\mathbb Z_2\otimes_{GL_4(\mathbb Z_2)} PH_*(\mathbb Z_2^4, \mathbb Z_2)$ in some generic degrees and its application

Let $A$ denote the Steenrod algebra at the prime 2 and let $k = \mathbb Z_2.$ An open problem of homotopy theory is to determine a minimal set of $A$-generators for the polynomial ring $P_q = k[x_1, \ldots, x_q] = H^{*}(k^{q}, k)$ on $q$ generators $x_1, \ldots, x_q$ with $|x_i|= 1.$ Equivalently, one can write down explicitly a basis for the graded vector space $Q^{\otimes q} := k\otimes_{A} P_q$ in each non-negative degree $n.$ This is the content of the classical "hit problem" of Frank Peterson. Based on this problem, we are interested in the $q$-th algebraic transfer $Tr_q^{A}$ of W. Singer \cite{W.S1}, which is one of the useful tools for describing mod-2 cohomology of the algebra $A.$ This transfer is a linear map from the space of $GL_q(k)$-coinvariant $k\otimes _{GL_q(k)} P((P_q)_n^{*})$ of $Q^{\otimes q}$ to the $k$-cohomology group of the Steenrod algebra, ${\rm Ext}_{A}^{q, q+n}(k, k).$ Here $GL_q(k)$ is the general linear group of degree $q$ over the field $k,$ and $P((P_q)_n^{*})$ is the primitive part of $(P_q)^{*}_n$ under the action of $A.$ The present paper is to investigate this algebraic transfer for the cohomological degree $q = 4.$ More specifically, basing the techniques of the hit problem of four variables, we explicitly determine the structure of $k\otimes _{GL_4(k)} P((P_4)_{n}^{*})$ in some generic degrees $n.$ Applying these results and a representation of the rank 4 transfer over the lambda algebra, we show that $Tr_4^{A}$ is an isomorphism in respective degrees. Also, we give some conjectures on the dimensions of $k\otimes_{GL_q(k)} ((P_4)_n^{*})$ for the remaining degrees $n.$ As a consequence, Singer's conjecture for the algebraic transfer is true in the rank 4 case. This study and our previous results \cite{D.P11, D.P12} have been provided a panorama of the behavior of $Tr_4^{A}.$