ĐặngVõ Phúc Faculty of Education Studies, University of Khanh Hoa, Vietnam
Primary cohomology operationsSteenrod algebraPeterson hit problemActions of
groups on commutative ringsAlgebraic transfer

# On the dimension of the ”cohits” space
$\mathbb{Z}_{2}\otimes_{\mathcal{A}_{2}}H^{*}((\mathbb{R}P(\infty))^{\times
t},\mathbb{Z}_{2})$
and some applications

###### Abstract

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
”cohits” 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
$\mathbb{Z}_{2}\otimes_{\mathcal{A}_{2}}P_{t}$ 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
$n_{s}=5(2^{s}-1)+42.2^{s}$ for every non-negative integer $s.$ As a result,
it confirms Sum’s conjecture [14] 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 $n_{s}$. As applications, 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 [11] is an isomorphism in bidegree $(5,5+n_{s})$.

##  1 Introduction

Recall that if $R$ is a commutative ring, then there is an $E_{\infty}$ -ring
spectrum $HR$ which represents cohomology with coeffcients in $R.$ Consider
$R=\mathbb{Z}_{2},$ the prime field of two elements, then as well known,
$H^{*}(H\mathbb{Z}_{2},\mathbb{Z}_{2})=\mathcal{A}_{2}$ is the Steenrod
algebra of $\mathbb{Z}_{2}$-cohomology operations. This is the graded homotopy
of the $H\mathbb{Z}_{2}$ -module Map($H\mathbb{Z}_{2},H\mathbb{Z}_{2}$), with
(noncommutative) multiplication coming from a composition of cohomology
operations. The algebra $\mathcal{A}_{2}$ is known to be a Hopf algebra (i.e.,
it is a bialgebra over the filed $\mathbb{Z}_{2}$ that has an associative
multiplication and a coassociative comultiplication and is equipped with a
counit, a unit and an antipode). As an algebra over $\mathbb{Z}_{2},$
$\mathcal{A}_{2}$ is generated by the Steenrod squaring operations $Sq^{i}$ of
degree $i$ for $i\geq 0.$ These squares act on $H^{*}(X,\mathbb{Z}_{2})$ for
any spectrum $X.$ Let $X=\mathbb{R}P(\infty)$ be the infinite real projective
space. Then, it has been shown that
$H^{*}(\mathbb{R}P(\infty),\mathbb{Z}_{2})\cong\mathbb{Z}_{2}[x]$ and that the
(left) action of $\mathcal{A}_{2}$ on $\mathbb{Z}_{2}[x]$ can be described by
the rule $Sq^{i}(x^{a})=\binom{a}{i}x^{a+i}.$ This action is deduced by
induction from the case $a=1$ by using the comultiplication on
$\mathcal{A}_{2}.$ Also based on the comultiplication, one can explicitly
depicte the action of $\mathcal{A}_{2}$ on
$H^{*}((\mathbb{R}P(\infty))^{\times t},\mathbb{Z}_{2})\cong
P_{t}:=\mathbb{Z}_{2}[x_{1},\ldots,x_{t}]$. So, as well known, $P_{t}$ is an
unstable left $\mathcal{A}_{2}$-module. It should be noted that
$\mathcal{A}_{2}$ is not an unstable module over itself in general since
$Sq^{1}Sq^{0}=Sq^{1}\neq 0.$ The $\mathbb{Z}_{2}$-algebra $P_{t}$ is a
connected $\mathbb{Z}$-graded algebra. That is,
$P_{t}=\mathbb{Z}_{2}\bigoplus(\bigoplus_{n>0}(P_{t})_{n}),$ where
$(P_{t})_{n}$ is the vector space of homogeneous polynomials of degree $n$. It
is natural to ask: What is a minimal set of generators for $P_{t}$ as an
$\mathcal{A}_{2}$-module? When we fix the degrees $n$ and the number of
variables $t$, answering that question is the same as the problem of
elementary linear algebra of finding the dimension of the quotient space,
$(QP_{t})_{n}:=(\mathbb{Z}_{2}\otimes_{\mathcal{A}_{2}}P_{t})_{n}\cong(P_{t})_{n}/\sum_{i>0}{\rm
Im}(Sq^{i})$. Here $\sum_{i>0}{\rm Im}(Sq^{i})=(P_{t})_{n}\cap{\rm
Ker}(\epsilon)P_{t},$ in which $\epsilon$ denotes the counit
$\mathcal{A}_{2}\to\mathbb{Z}_{2},$ and the Steenrod squaring operations
$Sq^{i}:(P_{t})_{n-i}\to(P_{t})_{n}$ are defined, and their basic properties,
such as Cartan’s formula, are established. People have also shown that
$(QP_{t})_{n}$ has a module structure over the group algebra
$\mathbb{Z}_{2}GL_{t}$ of the usual general linear group $GL_{t}$. The above
problem, which has first appeared in the work of Peterson [5], is usually
called the hit problem for the Steenrod algebra. It is currently famously
difficult and is of great interest to many authors. (See Kameko [2], Mothebe-
Uys [4], the present author [7, 8, 9], Singer [12], Sum [13, 14], Wood [16],
and others.)

A qualitative approach to the hit problem is that one gives conditions on
elements of $(P_{t})_{n}$ to show that they go to zero in $(QP_{t})_{n},$
i.e., belong to $\sum_{i>0}{\rm Im}(Sq^{i}).$ By this, let denote $\alpha(n)$
the number of $1$’s in dyadic expansion of a non-negative integer $n.$ It is
important to consider the arithmetic function $\mu:\mathbb{N}\to\mathbb{N},$
which is defined by $\mu(0)=0$ and $\mu(n)=\mbox{min}\\{\bullet\in\mathbb{N}:\
\alpha(n+\bullet)\leq\bullet\\}.$ Then, Peterson’s conjecture [5], which has
been established by Wood [16], points out that $(QP_{t})_{n}=0$ when
$\mu(n)>t.$ We are interested in a relatively efficient tool often used in
studying hit problems. It is the $\mathbb{Z}_{2}GL_{t}$-modules homomorphism
$\widetilde{Sq^{0}_{*}}:(QP_{t})_{2n+t}\to(QP_{t})_{n},$ determined by
$\widetilde{Sq^{0}_{*}}([f])=g$ if $f=x_{1}\ldots x_{t}g^{2},$ and
$\widetilde{Sq^{0}_{*}}([f])=0$ otherwise. This linear map, given by the
Kameko thesis [2], is related to the above $\mu$-function through the
following technical result.

######  Theorem 1.1 (see Kameko [2]).

If $\mu(2n+t)=t,$ then $\widetilde{Sq^{0}_{*}}$ is an isomorphism.

By straightforward calculations using this theorem and the Peterson
conjecture, it follows that the structure of $(QP_{t})_{n}$ can be determined
in each degree $n$ of the so-called the ”generic” form:

(1) $n=r(2^{s}-1)+m.2^{s},$

whenever $0\leq\mu(m)<r<t,$ and $s\geq 0.$ The cases where $t\leq 4$ are known
by Peterson [5], Kameko [2] and Sum [13]. For $t=5$, it has been partially
studied by us [6, 7, 8, 9, 14] and others. The problem is not yet solved in
general. The present Note is motivated by the previous results and we here
continue this line of investigation by considering $(QP_{t})_{n}$ in degrees
$n$ of the form (1). More precisely, when $t=5,$ we explicitly determine the
structure of $(QP_{5})_{n}$ for $n_{s}:=n=5(2^{s}-1)+42.2^{s}$. As a result,
it confirms Sum’s conjecture [14] for the relationship between the minimal
sets of $\mathcal{A}_{2}$-generators of the polynomial rings $P_{t-1}$ and
$P_{t}$ in the case $t=5$ and degree $n_{s}.$ An efficient approach to solve
the hit problem of five variables had been provided. As applications, we
obtain the dimension of $(QP_{6})_{n}$ in degree $5(2^{s+5}-1)+n_{0}.2^{s+5}$
for all $s\geq 0$, and show that Singer’s conjecture [11] for the fifth
cohomological transfer holds in degrees $n_{s}.$

This paper is an announcement whose details will appear elsewhere.

##  2 The Necessary Preliminaries

In order to formulate the main results of this text and for the convenience of
the reader, let us describe some preliminary material.

######  Definition 2.1 (Weight vector and exponent vector).

We say that a sequence of non-negative integers
$\omega=(\omega_{1},\omega_{2},\ldots,\omega_{i},\ldots)$ is a weight vector,
if $\omega_{i}=0,$ for $i\gg 0.$ Then, one defines $\deg(\omega)=\sum_{i\geq
1}2^{i-1}\omega_{i},$ With a natural number $n,$ let denote $\alpha_{j}(n)$
the $j$-th coefficients in dyadic expansion of $n,$ then
$\alpha(n)=\sum_{j\geq 0}\alpha_{j}(n),$ and $n=\sum_{j\geq
0}\alpha_{j}(n)2^{j},$ where $\alpha_{j}(n)\in\\{0,1\\}$ for all $j\geq 0.$
For a monomial $x=x_{1}^{a_{1}}x_{2}^{a_{2}}\ldots x_{t}^{a_{t}}\in P_{t},$
let us consider two sequences
$\omega(x):=(\omega_{1}(x),\ldots,\omega_{i}(x),\ldots)$ and
$(a_{1},a_{2},\ldots,a_{t}),$ where $\omega_{i}(x)=\sum_{1\leq j\leq
t}\alpha_{i-1}(a_{j}),$ for every $i.$ They are respectively called the weight
vector and the exponent vector of $x.$ By convention, the sets of all the
weight vectors and the exponent vectors are given the left lexicographical
order.

######  Definition 2.2 (Linear order on $P_{t}$).

Let $x=x_{1}^{a_{1}}x_{2}^{a_{2}}\ldots x_{t}^{a_{t}}$ and
$y=x_{1}^{b_{1}}x_{2}^{b_{2}}\ldots x_{t}^{b_{t}}$ be the monomials in
$(P_{t})_{n}.$ We write $a,\,b$ for the exponent vectors of $x$ and $y,$
respectively. We say that $x<y$ if and only if one of the following holds:

  1. (i)

$\omega(x)<\omega(y);$

  2. (ii)

$\omega(x)=\omega(y)$ and $a<b.$

######  Definition 2.3 (Equivalence relations on $P_{t}$).

For a weight vector $\omega$ of degree $n,$ let us denote two subspaces
associated with $\omega$ by
$\begin{array}[]{ll}(P_{t})_{n}^{\omega}&=\langle\\{x\in(P_{t})_{n}|\,\deg(x)=\deg(\omega),\
\omega(x)\leq\omega\\}\rangle,\\\
(P_{t})_{n}^{<\omega}&=\langle\\{x\in(P_{t})_{n}|\,\deg(x)=\deg(\omega),\
\omega(x)<\omega\\}\rangle.\end{array}$
Let $f$ and $g$ be two homogeneous polynomials in $(P_{t})_{n},$ we define the
binary relations ”$\sim$” and ”$\sim_{\omega}$” on $(P_{t})_{n}$:

  1. (i)

$f\sim g$ if and only if $(f-g)\in\sum_{i>0}{\rm Im}(Sq^{i})$. (Note that
since working mod 2, $f-g=f+g.$)

  2. (ii)

$f\sim_{\omega}g$ if and only if $f,\,g\in(P_{t})_{n}^{\omega}$ and
$(f-g)\in(((P_{t})_{n}^{\omega}\cap\sum_{i>0}{\rm
Im}(Sq^{i}))+(P_{t})_{n}^{<\omega}).$
In particular, if $f\sim 0$ (resp. $f\sim_{\omega}0$), then we say that $f$ is
hit (resp. $\omega$-hit). It is straightforward to see that the above binary
relations are equivalence ones. So, one has a quotient space

$(Q_{n}^{\otimes
t})^{\omega}:=(P_{t})_{n}^{\omega}/(((P_{t})_{n}^{\omega})\cap\sum_{i>0}{\rm
Im}(Sq^{i}))+(P_{t})_{n}^{<\omega}),$

from which it is not difficult to check the isomorphism
$(QP_{t})_{n}\cong\bigoplus_{\deg(\omega)=n}(Q_{n}^{\otimes t})^{\omega}$ (see
[7]).

######  Definition 2.4 (Admissible monomial and inadmissible monomial).

A monomial $x$ in $(P_{t})_{n}$ is said to be inadmissible if there exist
monomials $y_{1},y_{2},\ldots,y_{k}$ in $(P_{t})_{n}$ such that $y_{j}<x$ for
all $j,\ 1\leq j\leq k,$ and $x\sim(y_{1}+y_{2}+\cdots+y_{k}).$ Then, we say
that $x$ is admissible, if it is not inadmissible.

The following important theorem, which is due to Kameko [2], can be viewed as
a criterion in the calculation of the inadmissible monomials.

######  Theorem 2.5.

Let $x$ and $z$ be monomials in $(P_{t})_{n}.$ For an integer $r>0,$ assume
that there exists an index $i>r$ such that $\omega_{i}(x)=0$, from which if
$z$ is inadmissible, then $xz^{2^{r}}$ is, too.

From the above data, we have seen that $(QP_{t})_{n}$ is an
$\mathbb{Z}_{2}$-vector space with a basis consisting of all the classes
represent by the admissible monomials in $(P_{t})_{n}.$

######  Definition 2.6 (Spike monomial).

A monomial $z=\prod_{1\leq j\leq t}x_{j}^{b_{j}}$ in $(P_{t})_{n}$ is called a
spike if every exponent $b_{j}$ is of the form $2^{\beta_{j}}-1.$ In
particular, if the exponents $\beta_{j}$ can be arranged to satisfy
$\beta_{1}>\beta_{2}>\ldots>\beta_{r-1}\geq\beta_{r}\geq 1,$ where only the
last two smallest exponents can be equal, and $\beta_{j}=0$ for $r<j\leq t,$
then $z$ is called a minimal spike.

######  Theorem 2.7 (see Phúc-Sum [6]).

All the spikes in $(P_{t})_{n}$ are admissible and their weight vectors are
weakly decreasing. Furthermore, if a weight vector
$\omega=(\omega_{1},\omega_{2},\ldots)$ is weakly decreasing and
$\omega_{1}\leq t,$ then there is a spike $z\in(P_{t})_{n}$ such that
$\omega(z)=\omega.$

######  Theorem 2.8 (see Singer [12]).

Suppose that $x\in(P_{t})_{n}$ and $\mu(n)\leq t.$ Let $z$ be the minimal
spike in $(P_{t})_{n}.$ If $\omega(x)<\omega(z),$ then $x\sim 0.$

For $1\leq l\leq t,$ let $\mathsf{q}_{(l,\,t)}:P_{t-1}\to P_{t}$ be an
$\mathcal{A}_{2}$-homomorphism, which is depicted by

$\mathsf{q}_{(l,\,t)}(x_{j})=\left\\{\begin{array}[]{ll}{x_{j}}&\text{if
}\;1\leq j\leq l-1,\\\ x_{j+1}&\text{if}\;l\leq j\leq t-1.\end{array}\right.$

We assume throughout this text that $\mathscr{C}^{\otimes t}_{n}$ is the set
of all admissible monomials in $(P_{t})_{n}.$ The following technicality is
very useful for our calculations in the next section.

######  Theorem 2.9 (see Mothebe-Uys [4]).

Let $l,d$ be positive integers such that $1\leq l\leq t.$ If
$x\in\mathscr{C}^{\otimes(t-1)}_{n},$ then
$x_{l}^{2^{d}-1}\mathsf{q}_{(l,\,t)}(x)\in\mathscr{C}^{\otimes
t}_{n+2^{d}-1}.$

We set
$\mathcal{N}_{t}:=\\{(l,\mathscr{L})\;|\;\mathscr{L}=(l_{1},l_{2},\ldots,l_{r})\\},$
where $1\leq l<l_{1}<l_{2}<\ldots<l_{r}\leq t$ and $0\leq r\leq t-1.$ By
convention, $\mathscr{L}=\emptyset$ if $r=0.$ Let $r=\ell(\mathscr{L})$ be the
length of $\mathscr{L}$. For each the pair
$(l,\mathscr{L})\in\mathcal{N}_{t},$ and $1\leq u<r,$ let us denote by
$x_{(\mathscr{L},\,u)}:=x_{l_{u}}^{2^{r-1}+2^{r-2}+\,\cdots\,+2^{r-u}}\prod_{u<d\leq
r}x_{l_{d}}^{2^{r-d}},$ where $x_{(\emptyset,1)}=1.$ Sum [13] defines an
$\mathbb{Z}_{2}$-linear function
$\psi_{(l,\mathscr{L})}:P_{t-1}\longrightarrow P_{t},$ which is determined by

$\begin{array}[]{ll}&\psi_{(l,\mathscr{L})}(\prod_{1\leq j\leq
t-1}x_{j}^{a_{j}})\\\ &=(x_{l}^{2^{r}-1}\mathsf{q}_{(l,\,t)}(\prod_{1\leq
s\leq t-1}x_{s}^{a_{s}}))/x_{(\mathscr{L},\,u)},\end{array}$

if there exists $u$ such that:

(2) $\begin{array}[]{ll}a_{l_{1}-1}+1=\cdots=a_{l_{(u-1)}-1}+1=2^{r},\\\
a_{l_{u}-1}+1>2^{r},\\\ \alpha_{r-d}(a_{l_{u}-1})-1=0,\,1\leq d\leq u,\\\
\alpha_{r-d}(a_{l_{d}-1})-1=0,\,\ u+1\leq d\leq r,\end{array}$

and $\psi_{(l,\mathscr{L})}(\prod_{1\leq j\leq t-1}x_{j}^{a_{j}})=0$
otherwise. One has the following observation.

######  Remark 2.10.

If $\mathscr{L}=\emptyset,$ then $\psi_{(l,\mathscr{L})}=\mathsf{q}_{(l,\,t)}$
for $1\leq l\leq t.$ It is in fact not hard to show that if
$\psi_{(l,\mathscr{L})}(\prod_{1\leq s\leq t-1}x_{s}^{a_{s}})\neq 0,$ then
$\omega(\psi_{(l,\mathscr{L})}(\prod_{1\leq s\leq
t-1}x_{s}^{a_{s}}))=\omega(\prod_{1\leq s\leq t-1}x_{s}^{a_{s}}).$
Alternatively, $\psi_{(l,\mathscr{L})}$ is not the homomorphism of
$\mathcal{A}_{2}$-modules in general. As an illustrated example, we consider
$t=4,$ $\mathscr{L}=(2,3,4)\neq\emptyset,$ and the monomial
$x_{1}^{12}x_{2}^{6}x_{3}^{9}\in(P_{3})_{27},$ then, straightforward
calculations show that
$\begin{array}[]{ll}\psi_{(1,\mathscr{L})}(x_{1}^{12}x_{2}^{6}x_{3}^{9})&=\dfrac{x_{1}^{2^{3}-1}\mathsf{q}_{(1,4)}(x_{1}^{12}x_{2}^{6}x_{3}^{9})}{x_{(\mathscr{L},\,1)}}\\\
&=\dfrac{x_{1}^{7}x_{2}^{12}x_{3}^{6}x_{4}^{9}}{x_{2}^{4}x_{3}^{2}x_{4}}=x_{1}^{7}x_{2}^{8}x_{3}^{4}x_{4}^{8}\in(P_{4})_{27},\end{array}$
and
$\omega(x_{1}^{12}x_{2}^{6}x_{3}^{9})=(1,1,2,2)=\omega(\psi_{(1,\mathscr{L})}(x_{1}^{12}x_{2}^{6}x_{3}^{9})).$
So, using the Steenrod squares $Sq^{2},$ one gets
$\begin{array}[]{ll}&Sq^{2}(\psi_{(1,\mathscr{L})}(x_{1}^{12}x_{2}^{6}x_{3}^{9}))=x_{1}^{9}x_{2}^{8}x_{3}^{4}x_{4}^{8}\\\
&\quad\quad\quad\quad\quad\quad\neq
x_{1}^{7}x_{2}^{8}x_{3}^{6}x_{4}^{8}=\psi_{(1,\mathscr{L})}(Sq^{2}(x_{1}^{12}x_{2}^{6}x_{3}^{9})).\end{array}$

For a subset $\mathscr{C}\subset(P_{t-1})_{n},$ let us consider the sets:
$\begin{array}[]{ll}\widetilde{\Phi^{0}}(\mathscr{C})&=\bigcup\limits_{1\leq
l\leq t}\psi_{(l,\emptyset)}(\mathscr{C})=\bigcup_{1\leq l\leq
t}\mathsf{q}_{(l,t)}(\mathscr{C}),\\\
\widetilde{\Phi^{>0}}(\mathscr{C})&=\bigcup\limits_{(l;\mathscr{L})\in\mathcal{N}_{t},\;1\leq\ell(\mathscr{L})\leq
t-1}(\psi_{(l,\mathscr{L})}(\mathscr{C})\setminus(P_{t})_{n}^{0}),\end{array}$
where

$(P_{t})_{n}^{0}=\langle\\{\prod_{1\leq i\leq
t}x_{i}^{a_{i}}\in(P_{t})_{n}:\,\prod_{1\leq i\leq t}a_{i}=0\\}\rangle$

is an $\mathbb{Z}_{2}$-subspace of $(P_{t})_{n}.$ For later convenience, let
us denote by
$\widetilde{\Phi_{*}}(\mathscr{C}):=\widetilde{\Phi^{0}}(\mathscr{C})\bigcup\widetilde{\Phi^{>0}}(\mathscr{C}).$
Let $\omega$ be a weight vector of degree $n$ and let $(\mathscr{C}^{\otimes
t}_{n})^{\omega}:=\mathscr{C}^{\otimes t}_{n}\cap(P_{t})_{n}^{\omega}.$ In
[14], Sum sets up the following conjecture, which plays a crucial role in
studying the minimal set of $\mathcal{A}_{2}$-generators for $P_{t}$ in each
positive degree.

######  Conjecture 2.11.

Under the notations chosen, for each $(l,\mathscr{L})\in\mathcal{N}_{t},$ and
$1\leq r=\ell(\mathscr{L})\leq t-1,$ if $x=\prod_{1\leq j\leq
t-1}x_{j}^{a_{j}}\in\mathscr{C}^{\otimes(t-1)}_{n},$ and there exist $u,\
1\leq u\leq r,$ which satisfies (2), then
$\psi_{(l,\mathscr{L})}(x)\in\mathscr{C}^{\otimes t}_{n}.$ Moreover, if
$\omega$ is a weight vector of degree $n$, then
$\widetilde{\Phi_{*}}((\mathscr{C}^{\otimes(t-1)}_{n})^{\omega})\subseteq(\mathscr{C}^{\otimes
t}_{n})^{\omega}.$

Let us consider the set

$\mathcal{U}(t,n):=\\{\rho\in\mathbb{N}\ :\ \alpha(n-(2^{\rho}-1)+t-1)\leq
t-1\\}.$

For each $\rho\in\mathcal{U}(t,n),$ suppose that the monomial $y=\prod_{1\leq
i\leq t-1}x_{i}^{a_{i}}\in\mathscr{C}^{\otimes(t-1)}_{n-(2^{\rho}-1)}.$ We
denote
$\mathscr{C}(t,n):=\bigcup\limits_{\rho\in\mathcal{U}(t,n)}\langle\\{x_{1}^{a_{1}}\ldots
x_{j-1}^{a_{j-1}}x_{j}^{2^{\rho}-1}x_{j+1}^{a_{j+1}}\ldots
x_{t}^{a_{t-1}}\in(P_{t})_{n}\ :\ 1\leq j\leq t\\}\rangle$ and write

$(\mathscr{C}(t,n))^{\omega}=\\{x\in\mathscr{C}(t,n):\,\omega(x)=\omega\\}.$

Combining Conjecture 2.11 with a result in [4], we propose the following.

######  Conjecture 2.12.

For each $(l,\mathscr{L})\in\mathcal{N}_{t},$ if
$x\in(\mathscr{C}_{n}^{\otimes(t-1)})^{\omega}$ and

$\mbox{max}\bigg{\\{}r=\ell(\mathscr{L})\leq t-1:\psi_{(l,\mathscr{L})}(x)\neq
0\bigg{\\}}=|\mathcal{U}(t,n)|,$

then

$(\widetilde{\Phi_{*}}((\mathscr{C}^{\otimes(t-1)}_{n})^{\omega})\cup(\mathscr{C}(t,n))^{\omega})=(\mathscr{C}^{\otimes
t}_{n})^{\omega}.$

To close this section, we recall a result of Sum [13] on the inductive formula
for the dimension of $(QP_{t})_{n}.$

######  Theorem 2.13 (see Sum [13]).

Consider the degree $n$ of the form (1) with $r=t-1,$ and $s,m$ positive
integers such that $1\leq t-3\leq\mu(m)\leq t-2,$ and
$\mu(m)=\alpha(m+\mu(m)).$ Then for each $s\geq t-1,$ we have
$\dim(QP_{t})_{n}=(2^{t}-1)\dim(QP_{t-1})_{m},$

By using iteration of the Kameko homomorphism $\widetilde{Sq^{0}_{*}}$ and the
proof of Theorem 2.13, one has an equivalent statement of Theorem 2.13 that
will be applied in the sequel.

######  Theorem 2.14.

Let us consider the generic degrees of the form (1):
$n_{s}:=(t-1)(2^{s}-1)+m.2^{s}.$ Suppose there is a integer $\zeta$ such that
$0\leq\zeta<s$ and $1\leq
t-3\leq\mu(n_{\zeta})=\alpha(n_{\zeta}+\mu(n_{\zeta}))\leq t-2.$ Then, for
each $s\geq\zeta,$ we have

$\begin{array}[]{ll}&\dim(QP_{t})_{(t-1)(2^{s-\zeta+t-1}-1)+n_{\zeta}2^{s-\zeta+t-1}}\\\
&=(2^{t}-1)\dim(QP_{t-1})_{n_{s}}.\end{array}$

##  3 Statement of results

We now survey our main results in this Note. Let us consider the generic
degree of the form (1) with $r=t=5,$ and $m=42.$ Then, we have
$n_{s}:=5(2^{s}-1)+42.2^{s}$ with $s\geq 0.$ To solve the hit problem of five
variables in this degree, our main idea is to combine the homomorphisms
$\psi_{(l,\mathscr{L})}$ and Theorem 2.9 above. This approach, which is quite
effective, helps us to reduce many computations. This is shown specifically in
Theorem 3.3 below.

######  Notation 3.1.

In what follows:
$\begin{array}[]{ll}(P_{t})_{n_{s}}^{>0}&=\langle\\{\prod_{1\leq i\leq
t}x_{i}^{a_{i}}\in(P_{t})_{n_{s}}:\,\prod_{1\leq i\leq
t}a_{i}>0\\}\rangle\\\\[1.99168pt]
(QP_{t})_{n_{s}}^{0}&=\mathbb{Z}_{2}\otimes_{\mathcal{A}_{2}}(P_{t})_{n_{s}}^{0},(QP_{t})_{n_{s}}^{>0}=\mathbb{Z}_{2}\otimes_{\mathcal{A}_{2}}(P_{t})_{n_{s}}^{>0}.\end{array}$
Clearly, $(P_{t})_{n_{s}}^{>0}$ is an $\mathbb{Z}_{2}$-subspace of
$(P_{t})_{n},$ and
$(QP_{t})_{n_{s}}\cong(QP_{t})_{n_{s}}^{0}\,\bigoplus\,(QP_{t})_{n_{s}}^{>0}.$

For a polynomial $f\in(P_{t})_{n_{s}},$ we denote by $[f]$ the classes in
$(QP_{t})_{n_{s}}$ represented by $f.$ If $\omega$ is a weight vector of
degree $n_{s},$ and $f\in(P_{t})_{n_{s}}^{\omega},$ then denote by
$[f]_{\omega}$ the classes in $(QP_{t})_{n_{s}}^{\omega}$ represented by $f.$
For a subset $\mathscr{C}\subset(P_{t})_{n_{s}},$ denote by
$[\mathscr{C}]=\\{[f]\,:\,f\in\mathscr{C}\\}.$ If
$\mathscr{C}\subset(P_{t})_{n_{s}}^{\omega},$ then denote by
$[\mathscr{C}]_{\omega}=\\{[f]_{\omega}\,:\,f\in\mathscr{C}\\}.$ It should be
noted that if $\omega$ is a weight vector of mimimal spike, then
$[\mathscr{C}]_{\omega}=[\mathscr{C}].$ We put

$\begin{array}[]{ll}(\mathscr{C}^{\otimes
t}_{n_{s}})^{\omega^{0}}&:=(\mathscr{C}^{\otimes
t}_{n_{s}})^{\omega}\cap(P_{t})_{n_{s}}^{0},\\\\[1.99168pt]
(\mathscr{C}^{\otimes t}_{n_{s}})^{\omega^{>0}}&:=(\mathscr{C}^{\otimes
t}_{n_{s}})^{\omega}\cap(P_{t})_{n_{s}}^{>0},\\\\[1.99168pt]
(Q_{n_{s}}^{\otimes
t})^{\omega^{0}}&:=(QP_{t})_{n_{s}}^{\omega}\cap(QP_{t})_{n_{s}}^{0},\\\\[1.99168pt]
(Q_{n_{s}}^{\otimes
t})^{\omega^{>0}}&:=(QP_{t})_{n_{s}}^{\omega}\cap(QP_{t})_{n_{s}}^{>0}.\end{array}$

Then, $[(\mathscr{C}^{\otimes t}_{n_{s}})^{\omega}]_{\omega},$
$[(\mathscr{C}^{\otimes t}_{n_{s}})^{\omega^{0}}]_{\omega}$ and
$[(\mathscr{C}^{\otimes t}_{n_{s}})^{\omega^{>0}}]_{\omega}$ are respectively
the bases of the vector spaces $(Q_{n_{s}}^{\otimes t})^{\omega},\
(Q_{n_{s}}^{\otimes t})^{\omega^{0}},$ and $(Q_{n_{s}}^{\otimes
t})^{\omega^{>0}}.$

For a start, we have the following important remark.

######  Remark 3.2.

Noticing that $\mu(n_{s})=5$ for all $s>0.$ This, together with Theorem 1.1,
implies that the map

(3) $(\widetilde{Sq^{0}_{*}})^{s}:(QP_{5})_{n_{s}}\to(QP_{5})_{n_{0}}$

is an isomorphism of $\mathbb{Z}_{2}$-vector spaces for any $s\geq 0$, from
which we only need to compute $(QP_{5})_{n_{0}}.$ We have
$n_{0}=5(2^{0}-1)+42.2^{0}=42.$ Then, if $x\in\mathscr{C}^{\otimes 5}_{42},$
then the weight vector $\overline{\omega}:=\omega(x)$ is one of the following
sequences: $\overline{\omega}_{(1)}=(4,3,2,1,1),$
$\overline{\omega}_{(2)}=(4,3,2,3),$ and $\overline{\omega}_{(3)}=(4,3,4,2).$
Indeed, it should be noted that
$z=x_{1}^{31}x_{2}^{7}x_{3}^{3}x_{4}\in(P_{5})_{42}$ is minimal spike, and so,
by Theorem 2.7, $z$ belongs to $\mathscr{C}^{\otimes 5}_{42}.$ Moreover,
because $\omega(z)=(4,3,2,1,1),$ based upon Theorem 2.8, we see that if
$x\in\mathscr{C}^{\otimes 5}_{42},$ then $\omega_{1}(x)=4.$ This means that
$x$ is of the form $x_{i}x_{j}x_{k}x_{l}y^{2}$ with $y\in(P_{5})_{19}$ and
$1\leq i<j<k<l\leq 5.$ Since $x\in\mathscr{C}^{\otimes 5}_{42},$ by Theorem
2.5, we must have $y\in\mathscr{C}^{\otimes 5}_{19}.$ So, the desired
conclusion now follows from a result in [15] that $\omega(y)$ belongs to the
set $\\{(3,2,1,1),\,(3,2,3),\,(3,4,2)\\}.$

From the above remarks, the following is immediate:

(4) $(QP_{5})_{42}\cong(QP_{5})_{42}^{0}\bigoplus\big{(}\bigoplus_{1\leq j\leq
3}(Q^{\otimes 5}_{42})^{\overline{\omega}_{(j)}^{>0}}\big{)}.$

Since $\mu(42)=4,$ $(QP_{4})_{42}\cong(QP_{4})_{19}.$ Then, one can verify
that

$(QP_{4})_{42}=(QP_{4})_{42}^{>0}=(Q_{42}^{\otimes
4})^{\overline{\omega}_{(1)}^{>0}}=(Q_{42}^{\otimes
4})^{\overline{\omega}_{(1)}},$

and so, $\dim(QP_{4})_{42}=\dim(QP_{4})_{19}=140.$ On the other side, it has
been shown (see also [8]) that

$\dim(QP_{5})^{0}=\sum_{1\leq s\leq 4}\binom{5}{s}\dim(QP_{s})_{n}^{>0},$

for all $n\geq 0.$ Using this formula, together with Peterson’s conjecture and
Theorem 1.1, we are forced to conclude that

(5) $\begin{array}[]{ll}\dim(QP_{5})_{42}^{0}&=\dim(Q_{42}^{\otimes
5})^{\overline{\omega}_{(1)}^{0}}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&=140.\binom{5}{4}=700,\\\ (\mathscr{C}_{42}^{\otimes
5})^{\overline{\omega}_{(1)}^{0}}&=\widetilde{\Phi^{0}}((\mathscr{C}^{\otimes
4}_{42})^{\overline{\omega}_{(1)}^{>0}})=\widetilde{\Phi^{0}}(\mathscr{C}^{\otimes
4}_{42}).\end{array}$

For $1\leq l\leq 5,$ we consider the following sets:

$\begin{array}[]{ll}\mathscr{C}_{1}&:=\\{x_{l}\mathsf{q}_{(l,\,5)}(x):\
x\in(\mathscr{C}^{\otimes 4}_{41})^{>0},\\},\\\\[1.99168pt]
\mathscr{C}_{2}&:=\\{x_{l}^{3}\mathsf{q}_{(l,\,5)}(x):\
x\in(\mathscr{C}^{\otimes 4}_{39})^{>0}\\},\\\\[1.99168pt]
\mathscr{C}_{3}&:=\\{x_{l}^{7}\mathsf{q}_{(l,\,5)}(x):\
x\in(\mathscr{C}^{\otimes 4}_{35})^{>0}\\},\\\\[1.99168pt]
\mathscr{C}_{4}&:=\\{x_{l}^{31}\mathsf{q}_{(l,\,5)}(x):\
x\in(\mathscr{C}^{\otimes 4}_{11})^{>0}\\}.\end{array}$

Then, using Theorem 2.9, it can be shown that

$\mathscr{C}^{>0}(\overline{\omega}_{(1)}):=\bigcup_{1\leq i\leq
4}\mathscr{C}_{i}\subseteq(\mathscr{C}^{\otimes
5}_{42})^{\overline{\omega}_{(1)}^{>0}}.$

We put
$E:=\widetilde{\Phi^{>0}}((\mathscr{C}_{42}^{\otimes
4})^{\overline{\omega}_{(1)}})\setminus(\widetilde{\Phi^{>0}}((\mathscr{C}_{42}^{\otimes
4})^{\overline{\omega}_{(1)}})\cap\mathscr{C}^{>0}(\overline{\omega}_{(1)})).$
Then, we have the theorem below, which is our first main result.

######  Theorem 3.3.

The following statements hold:

  * i)

$(\mathscr{C}_{42}^{\otimes
5})^{\overline{\omega}_{(j)}^{>0}}=(\mathscr{C}_{42}^{\otimes
5})^{\overline{\omega}_{(j)}}=\emptyset,$ for $j=2,3.$

  * ii)

$(\mathscr{C}_{42}^{\otimes 5})^{\overline{\omega}_{(1)}^{>0}}=E\bigcup
F\bigcup\mathscr{C}^{>0}(\overline{\omega}_{(1)}),$

where $|E|=542,\ |F|=248,$ and
$|\mathscr{C}^{>0}(\overline{\omega}_{(1)})|=1030.$ Consequently,
$(Q_{42}^{\otimes 5})^{\overline{\omega}_{(1)}^{>0}}$ is $1820$-dimensional.

From (3), (5), and Theorem 3.3, it may be concluded that Conjecture 2.11 holds
for $t=5$ and the generic degree $n_{s}$ for any $s\geq 0.$

As immediate consequences from (3), (4), (5) and Theorem 3.3, we may assert
that

######  Corollary 3.4.

There exist exactly $2520$ admissible monomials of degree $n_{s}$ in $P_{5}$
for all $s\geq 0.$ Consequently, $(QP_{5})_{n_{s}}$ is $2520$-dimensional.

The above computations confirm the dimension of $(QP_{5})_{n_{s}},$ which is
informed in [8] by using the MAGMA computer algebra system.

Next, based on Theorem 2.14 and Corollary 3.4, we have immediately the
following.

######  Corollary 3.5.

When degree $n_{s}$ is as in Theorem 2.14, for $t=6$, $m=42,$ and $s\geq 0,$
we have $\dim(QP_{6})_{n_{s+5}}=(2^{6}-1)\dim(QP_{5})_{n_{s}}=158760.$

Singer’s cohomological transfer. The next contribution of the paper is to
apply Theorem 3.3 into the study of the behavior of the fifth transfer
homomorphism of W. Singer [11]. It may need to be recalled that the group
$GL_{t}$ acts regularly on $P_{t}$ by matrix substitution. Further, the two
actions of $\mathcal{A}_{2}$ and $GL_{t}$ upon $P_{t}$ commute with each
other; hence there is an inherited action of $GL_{t}$ on
$QP_{t}=\mathbb{Z}_{2}\otimes_{\mathcal{A}_{2}}P_{t}.$ Thus, the hit problem
becomes important in studying the modular representation of $GL_{t}.$ This is
of interest since the Singer cohomological ”transfer” [11] relates
$(QP_{t})_{n}$ to $H^{t,t+n}(\mathcal{A}_{2},\mathbb{Z}_{2})={\rm
Ext}_{\mathcal{A}_{2}}^{t,t+n}(\mathbb{Z}_{2},\mathbb{Z}_{2}),$ the
$\mathbb{Z}_{2}$-cohomology groups of $\mathcal{A}_{2},$ and thus to the
stable homotopy groups of spheres. Singer’s transfer is established as
follows. Let us recall first that if $\mathcal{M}$ the category of graded left
$\mathcal{A}_{2}$-modules and degree zero of $\mathcal{A}_{2}$-linear maps,
then for each integer $t,$ the supension functor
$S^{t}:\mathcal{M}\to\mathcal{M}$ is defined by $(S^{t}U)^{n}:=U^{n-t}$ where
$n\in\mathbb{Z}$. The action of $\mathcal{A}_{2}$ on $S^{t}U$ is given by
$\theta(S^{t}u)=S^{t}(\theta u),$ for all $u\in U$ and
$\theta\in\mathcal{A}_{2}.$ Next, consider the polynomial ring
$P_{1}=\mathbb{Z}_{2}[x]$ with $\deg(x)=1,$ the canonical
$\mathcal{A}_{2}$-action on $P_{1}$ is extended to an $\mathcal{A}_{2}$-action
on the ring of finite Laurent series $\mathbb{Z}_{2}[x_{1},x_{1}^{-1}]$. Then,
there exists an $\mathcal{A}_{2}$-submodule
$\overline{P}=\langle\\{x_{1}^{t}|\ t\geq-1\\}\rangle$ of
$\mathbb{Z}_{2}[x_{1},x_{1}^{-1}].$ One has a short-exact sequence $0\to
P_{1}\xrightarrow{q}\overline{P}\xrightarrow{\pi}S^{-1}\mathbb{Z}_{2},$ where
$q$ is the inclusion and $\pi$ is given by $\pi(x_{1}^{t})=0$ if $t\neq-1$ and
$\pi(x_{1}^{-1})=1.$ Writing $e_{1}$ for the corresponding element in ${\rm
Ext}_{\mathcal{A}_{2}}^{1}(S^{-1}\mathbb{Z}_{2},P_{1}).$ Basing the cross, the
Yoneda and the cap products in co(homology) with $\mathbb{Z}_{2}$-coeffcients,
we have a homomorphism

$\begin{array}[]{ll}Tr^{\mathcal{A}_{2}}_{t}:{\rm
Tor}_{t}^{\mathcal{A}_{2}}(\mathbb{Z}_{2},S^{-t}\mathbb{Z}_{2})&\longrightarrow{\rm
Tor}_{0}^{\mathcal{A}_{2}}(\mathbb{Z}_{2},P_{t})=QP_{t}\\\ \hskip
56.9055ptz&\longmapsto e_{t}\cap z,\end{array}$

where $e_{t}=(e_{1}\times P_{t-1})\circ\cdots\circ(e_{1}\times P_{1})\circ
e_{1}$. Its image is a submodule of the space of $GL_{t}$-invariants
$(QP_{t})^{GL_{t}}.$ It is known, the supension $S^{-t}$ induces an
isomorphism

${\rm Tor}_{t,n}^{\mathcal{A}_{2}}(\mathbb{Z}_{2},S^{-t}\mathbb{Z}_{2})\cong
H_{t,t+n}(\mathcal{A}_{2},\mathbb{Z}_{2})$

and so $Tr^{\mathcal{A}_{2}}_{t}$ induces the homomorphism

$Tr^{\mathcal{A}_{2}}_{t}:H_{t,t+n}(\mathcal{A}_{2},\mathbb{Z}_{2})\to(QP_{t})_{n}^{GL_{t}}.$

The dual $(Tr^{\mathcal{A}_{2}}_{t})^{*}:((QP_{t})_{n}^{GL_{t}})^{*}\to
H^{t,t+n}(\mathcal{A}_{2},\mathbb{Z}_{2})$ of $Tr^{\mathcal{A}_{2}}_{t}$ is
called the $t$-th cohomological transfer. In [11], Singer proves that
$(Tr_{t}^{\mathcal{A}_{2}})^{*}$ is an isomorphism for $t=1,2.$ Afterwards,
based on Kameko’s thesis [2], Boardman [1] states that
$(Tr^{\mathcal{A}_{2}}_{3})^{*}$ is also an isomorphism. These works tell us
that Singer’s transfer map is highly non-trivial, and so, it serves as a
reliable tool for depicting the cohomology of the Steenrod ring. In higher
cohomological degrees, Singer [11] gives some calculations to show that the
fourth transfer is an isomorphism up to a range of internal degrees, but the
fifth transfer is not an epimorphism, from which he makes the following
prediction.

######  Conjecture 3.6.

The cohomological transfer is a monomorphism.

This was confirmed for $1\leq t\leq 3$ by Singer himself [11] and Boardman
[1]. Our recent work [10] shows that the conjecture is also true for $t=4.$ So
far it remains open in general. Very little information is known when $t=5.$
Now, based upon an admissible monomial basis for $QP_{5}$ in degree
$5(2^{s}-1)+42.2^{s}$ (see Theorem 3.3), we verify Conjecture 3.6 for $t=5$
and the respective degree. The following theorem is our second main result.

######  Theorem 3.7.

The transfer homomorphism
$(Tr^{\mathcal{A}_{2}}_{5})^{*}:((QP_{5})_{5(2^{s}-1)+42.2^{s}}^{GL_{5}})^{*}\to
H^{5,47.2^{s}}(\mathcal{A}_{2},\mathbb{Z}_{2})$ is a trivial isomorphism, for
all $s\geq 0.$

This result can be proved by explicitly computing the space of
$GL_{5}$-invariants $(QP_{5})_{42}^{GL_{5}}$ combining the fact that
$H^{5,47.2^{s}}(\mathcal{A}_{2},\mathbb{Z}_{2})$ is trivial (see Lin [3]).
Thus, with these data in hand, the reader can see that

######  Corollary 3.8.

Conjecture 2.11 holds for the rank 5 case and degree $5(2^{s}-1)+42.2^{s}$
with $s\geq 0.$

######  Acknowledgments.

The author research is partially funded by the NAFOSTED grant No.
101.04-2017.05.

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