∎

11institutetext: D-ặng Võ Phúc 22institutetext: Faculty of Education Studies
University of Khanh Hoa, Nha Trang, Khanh Hoa, Vietnam
22email<EMAIL_ADDRESS>

# On modules over the mod 2 Steenrod algebra and hit problems

D-ặng Võ Phúc

###### Abstract

Let us consider the prime field of two elements,
$\mathbb{F}_{2}\equiv\mathbb{Z}_{2}.$ 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
$m\geq 5.$

It is our intent in this work is of studying the hit problem of five
variables. More precisely, we develop our previous work [Commun. Korean Math.
Soc. 35 (2020), 371-399] 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. 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 to describe the modular
representations of the general linear group of rank 5 over $\mathbb{F}_{2}.$
As a corollary, the cohomological ”transfer”, defined by William Singer [Math.
Z. 202 (1989), 493-523], is an isomorphism in 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.

Mathematics Subject Classification (2010) 13A50 $\cdot$ 55Q45 $\cdot$ 55S10
$\cdot$ 55S05 $\cdot$ 55T15 $\cdot$ 55R12

###### Keywords:

Adams spectral sequencesSteenrod algebra Hit problemAlgebraic transfer

##  1 Introduction

Let $\mathcal{O}^{S}(i,\mathbb{F}_{2},\mathbb{F}_{2})$ denote the set of all
stable cohomology operations of degree $i,$ with coefficient in the prime
field $\mathbb{F}_{2}.$ Then, the $\mathbb{F}_{2}$-algebra
$\mathscr{A}:=\bigoplus_{i\geq
0}\mathcal{O}^{S}(i,\mathbb{F}_{2},\mathbb{F}_{2})$ is called the mod 2
Steenrod algebra. In other words, the algebra $\mathscr{A}$ is the algebra of
stable operations on the mod 2 cohomology. In J.M , Milnor observed that this
algebra is also a graded connected cocommutative Hopf algebra over
$\mathbb{F}_{2}.$ In some cases, the resulting $\mathscr{A}$-module structure
on $H^{*}(X,\mathbb{F}_{2})$ provides additional information about CW-
complexes $X;$ for instance (see section three for a detailed proof), the CW-
complexes $\mathbb{C}P^{4}/\mathbb{C}P^{2}$ and
$\mathbb{S}^{6}\vee\mathbb{S}^{8}$ have cohomology rings that agree as a
graded commutative $\mathbb{F}_{2}$-algebras, but are different as a module
over $\mathscr{A}.$ Afterwards, the Steenrod algebra is widely studied by
mathematicians whose interests range from algebraic topology and homotopy
theory to manifold theory, combinatorics, representation theory, and more. It
is well-known that the $\mathbb{F}_{2}$-cohomology of the Eilenberg-MacLan
complex $K(\mathbb{F}_{2},1)$ is isomorphic to $\mathbb{F}_{2}[x],$ the
polynomial ring of degree $1$ in one variable. Hence, based upon the Künneth
formula for cohomology, we have an isomorphism of $\mathbb{F}_{2}$-algebras

$\mathcal{P}_{m}:=H^{*}((K(\mathbb{F}_{2},1))^{\times
m},\mathbb{F}_{2})\cong\underset{\text{ $m$
times}}{\underbrace{\mathbb{F}_{2}[x_{1}]\otimes_{\mathbb{F}_{2}}\mathbb{F}_{2}[x_{2}]\otimes_{\mathbb{F}_{2}}\cdots\otimes_{\mathbb{F}_{2}}\mathbb{F}_{2}[x_{m}]}}\cong\mathbb{F}_{2}[x_{1},\ldots,x_{m}],$

where $x_{i}\in H^{1}((K(\mathbb{F}_{2},1))^{\times m},\mathbb{F}_{2})$ for
every $i.$ Since $\mathcal{P}_{m}$ is the cohomology of a CW-complex, it is
equipped with a structure of unstable module over $\mathscr{A}.$ It has been
known (see also S.E ) that $\mathscr{A}$ is spanned by the Steenrod squares
$Sq^{i}$ of degree $i$ for $i\geq 0$ and that the action of $\mathscr{A}$ on
$\mathcal{P}_{m}$ is depicted as follows:

$\begin{array}[]{ll}Sq^{i}(x_{t})&=\left\\{\begin{array}[]{lll}x_{t}&\mbox{if}&i=0,\\\
x_{t}^{2}&\mbox{if}&i=1,\ \ \mbox{({the instability condition})},\\\
0&\mbox{if}&i>1,\end{array}\right.\\\ Sq^{i}(FG)&=\sum_{0\leq\alpha\leq
i}Sq^{\alpha}(F)Sq^{i-\alpha}(G),\ \mbox{for all $F,\ G\in\mathcal{P}_{m}$}\ \
(\mbox{{the Cartan formula})}.\end{array}$

It is to be noted that since $Sq^{\deg(F)}(F)=F^{2}$ for any
$F\in\mathcal{P}_{m}$, the polynomial ring $\mathcal{P}_{m}$ is also an
unstable $\mathscr{A}$-algebra. Letting $GL_{m}:=GL(m,\mathbb{F}_{2})$ for the
general linear group of degree $m$ over $\mathbb{F}_{2}.$ This $GL_{m}$ when
$m\geq 2,$ which can be generated by two elements (see Waterhouse Waterhouse
), acts on $\mathcal{P}_{m}$ by matrix substitution. So, in addition to
$\mathscr{A}$-module structure, $\mathcal{P}_{m}$ is also a (right)
$\mathbb{F}_{2}GL_{m}$-module. The classical ”hit problem” for the algebra
$\mathscr{A}$, which is concerned with seeking a minimal set of
$\mathscr{A}$-generators for $\mathcal{P}_{m}$, has been initiated in a
variety of contexts by Peterson F.P , Priddy S.P , Singer W.S2 , and Wood R.W
. Structure of modules over $\mathscr{A}$ and hit problems are currently one
of the central subjects in Algebraic topology and has a great deal of
intensively studied by many authors like Brunetti and collaborators Brunetti1
; Brunetti2 , Crabb-Hubbuck C.H , Inoue M.I1 ; M.I2 , Janfada-Wood J.W ; J.W2
, Janfada Janfada1 ; Janfada2 , Kameko M.K , Mothebe-Uys M.M , Mothebe M.M2 ,
Pengelley-William P.W , the present author and N. Sum P.S1 ; P.S2 ; D.P1 ;
D.P2 ; D.P3 ; D.P6 ; D.P7 ; N.S1 ; N.S2 ; N.S3 , Walker-Wood W.W ; W.W2 , etc.
As it is known, when $\mathbb{F}_{2}$ is an $\mathscr{A}$-module concentrated
in degree 0, solving the hit problem is to determine an $\mathbb{F}_{2}$-basis
for the space of indecomposables, or ”unhit” elements, $Q^{\otimes
m}:=\mathbb{F}_{2}\otimes_{\mathscr{A}}\mathcal{P}_{m}=\mathcal{P}_{m}/\overline{\mathscr{A}}\mathcal{P}_{m}$
where $\overline{\mathscr{A}}$ is the positive degree part of $\mathscr{A}$.
It is well-known that the action of $GL_{m}$ and the action of $\mathscr{A}$
on $\mathcal{P}_{m}$ commute. So, there is an induced action of $GL_{m}$ on
$Q^{\otimes m}.$ The structure of $Q^{\otimes m}$ has been treated for $m\leq
4$ by Peterson F.P , Kameko M.K  and Sum N.S1 . The general case is an
interesting open problem. Most notably, the study of this space plays a vital
role in describing the $E^{2}$-term of the Adams spectral sequence (Adams SS),
${\rm Ext}_{\mathscr{A}}^{m,m+*}(\mathbb{F}_{2},\mathbb{F}_{2})$ via the
$m$-th Singer cohomological ”transfer” W.S1 . This transfer is a linear map

$Tr^{\mathscr{A}}_{m}:(\mathbb{F}_{2}\otimes_{GL_{m}}P_{\mathscr{A}}((\mathcal{P}_{m})^{*}))_{n}\to{\rm
Ext}_{\mathscr{A}}^{m,m+n}(\mathbb{F}_{2},\mathbb{F}_{2})=H^{m,m+n}(\mathscr{A},\mathbb{F}_{2}),$

from the subspace of all $\overline{\mathscr{A}}$-annihilated elements to the
$E^{2}$-term of the Adams SS. Here
$(\mathcal{P}_{m})^{*}=H_{*}((K(\mathbb{F}_{2},1))^{\times m},\mathbb{F}_{2})$
and $\mathbb{F}_{2}\otimes_{GL_{m}}P_{\mathscr{A}}((\mathcal{P}_{m})^{*})$ are
the dual of $\mathcal{P}_{m}$ and $(Q^{\otimes m})^{GL_{m}},$ respectively,
where $(Q^{\otimes m})^{GL_{m}}$ denotes the space of $GL_{m}$-invariants. A
natural question arises: Why do we need to calculate the Adams $E^{2}$-term?
The answer is that it is involved in determining the stable homotopy groups of
spheres. These groups are pretty fundamental and interesting. Nevertheless,
they are also not fully-understood subjects yet. Therefore, the clarification
of these problems is an important task of Algebraic topology. It has been
shown (see J.B , W.S1 ) that the algebraic transfer is highly nontrivial, more
precisely, that $Tr^{\mathscr{A}}_{m}$ is an isomorphism for $0<m<4$ and that
the ”total” transfer $\bigoplus_{m\geq 0}Tr^{\mathscr{A}}_{m}:\bigoplus_{m\geq
0}(\mathbb{F}_{2}\otimes_{GL_{m}}P_{\mathscr{A}}((\mathcal{P}_{m})^{*}))_{n}\to\bigoplus_{m\geq
0}{\rm Ext}_{\mathscr{A}}^{m,m+n}(\mathbb{F}_{2},\mathbb{F}_{2})$ is a
homomorphism of bigraded algebras with respect to the product by concatenation
in the domain and the usual Yoneda product for the Ext group. Minami’s works
N.M1 ; N.M2  have shown the usefulness of the Singer transfer and the hit
problem for surveying the Kervaire invariant one problem. This problem, which
is a long standing open topic in Algebraic topology, asks when there are
framed manifolds with Kervaire invariant one. (Note that a framing on a closed
smooth manifold $M^{n}$ is a trivialization of the normal bundle $\nu(M,i)$ of
some smooth embedding $i:M\hookrightarrow\mathbb{R}^{n+*}.$ Here $\nu(M,i)$ is
defined to be a quotient of the pullback of the tangent bundle of
$\mathbb{R}^{n+*}$ by the sub-bundle given by the tangent bundle of $M.$ So,
$\nu(M,i)$ is an $*$-dimensional real vector bundle over $M^{n}.$ For more
details, we refer the reader to Snaith .) Framed manifolds of Kervaire
invariant one have been constructed in dimension $2^{k}-2$ for $2\leq k\leq
6.$ In 2016, by using mod 8 equivariant homotopy theory, Hill, Hopkins, and
Ravenel claimed in their surprising work Hill  that the Kervaire invariant is
0 in dimension $2^{k}-2$ for $k\geq 8.$ Up to present, it remains undetermined
for $k=7$ (or dimension $126$) and this has the status of a hypothesis by
Snaith Snaith .

Return to Singer’s transfer, in higher homological degrees, the works B.H.H ,
Ha , Hung , T.N2 , and H.Q  determined completely the image of
$Tr_{4}^{\mathscr{A}}$. The authors show that $Tr_{4}^{\mathscr{A}}$ contains
all the elements of the families $\\{d_{t}|\,t\geq 0\\},\ \\{e_{t}|\,t\geq
0\\}$, $\\{f_{t}|\,t\geq 0\\}$, and $\\{p_{t}|\,t\geq 0\\}$, but none from the
families $\\{g_{t+1}|\,t\geq 0\\}$, $\\{D_{3}(t)|\,t\geq 0\\}$, and
$\\{p^{\prime}_{t}|\,t\geq 0\\}$. Remarkably, since the family
$\\{g_{t+1}|\,t\geq 0\\}\not\subset{\rm Im}(Tr_{4}^{\mathscr{A}}),$ a question
of Minami N.M2  concerning the so-called new doomsday conjecture refuted. In
Hung , Hưng indicated that $Tr_{4}^{\mathscr{A}}$ is not an isomorphism in
infinitely many degrees. In particular, from preliminary calculations in W.S1
, Singer proposed the following.

###### Conjecture 1.1

The transfer homomorphism is a monomorphism in every rank $m>0.$

We have seen above that $Tr_{m}^{\mathscr{A}}$ is an isomorphism for $m<4,$
and so the conjecture holds in these ranks $m.$ Our recent work D.P14  has
shown that it is also true for $m=4,$ but the answer to the general case
remains a mystery, even in the case of $m=5$ with the help of a computer
algebra. It is known, in ranks $\leq 4$, the calculations of Singer W.S1 , Hà
Ha , and Nam T.N2  tell us that the non-zero elements $h_{t}\in{\rm
Ext}_{\mathscr{A}}^{1,2^{t}}(\mathbb{F}_{2},\mathbb{F}_{2}),\,e_{t}\in{\rm
Ext}_{\mathscr{A}}^{4,2^{t+4}+2^{t+2}+2^{t}}(\mathbb{F}_{2},\mathbb{F}_{2}),\,f_{t}\in{\rm
Ext}_{\mathscr{A}}^{4,2^{t+4}+2^{t+2}+2^{t+1}}(\mathbb{F}_{2},\mathbb{F}_{2}),$
for all $t\geq 0,$ are detected by the cohomological transfer. In rank 5,
based on invariant theory, Singer W.S1  gives an explicit element in ${\rm
Ext}_{\mathscr{A}}^{5,5+9}(\mathbb{F}_{2},\mathbb{F}_{2}),$ namely $Ph_{1}$,
that is not detected by $Tr_{5}^{\mathscr{A}}.$ In general, direct calculating
the value of $Tr_{m}^{\mathscr{A}}$ on any non-zero element is difficult.
Moreover, there is no general rule for that, and so, each computation is
important on its own. By this and the above results, in the present text, we
would like to investigate the family $\\{h_{t}f_{t}=h_{t+1}e_{t}\in{\rm
Ext}_{\mathscr{A}}^{5,5+(23.2^{t}-5)}(\mathbb{F}_{2},\mathbb{F}_{2})|\,t\geq
0\\},$ and Singer’s conjecture for $m=5$ in degree
$5(2^{t}-1)+18.2^{t}=23.2^{t}-5$ with $t=0.$ To do this, we use a basis of the
indecomposables $Q^{\otimes 5}$ in degree $18=5(2^{0}-1)+18.2^{0}$, which is
given by our previous work D.P2  (see Proposition 2.7 below). In addition, the
main goal of this work is to also compute explicitly the dimension of
$Q^{\otimes 5}$ in degree $5(2^{t}-1)+18.2^{t}$ for the cases $t\geq 1.$ Then,
Singer’s conjecture for $m=5$ and these degrees will be discussed at the end
of section two. We hope that our results would be necessary to formulate
general solutions.

##  2 Statement of results

Some notes. Throughout this paper, let us write

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt(\mathcal{P}_{m})_{n}&:=\langle\\{f\in\mathcal{P}_{m}|\,\mbox{$f$ is a
homogeneous polynomial of degree $n$}\\}\rangle,\\\ Q^{\otimes
m}_{n}&:=\langle\\{[f]\in Q^{\otimes
m}|\,f\in(\mathcal{P}_{m})_{n}\\}\rangle,\end{array}$

which are $\mathbb{F}_{2}GL_{m}$-submodules of $\mathcal{P}_{m}$ and
$Q^{\otimes m},$ respectively. So $\mathcal{P}_{m}=\bigoplus_{n\geq
0}(\mathcal{P}_{m})_{n}$ and $Q^{\otimes m}=\bigoplus_{n\geq 0}Q^{\otimes
m}_{n}.$ Recall that to solve the hit problem of three variables, Kameko M.K
constructed a $\mathbb{F}_{2}GL_{m}$-modules epimorphism:

$\begin{array}[]{ll}(\widetilde{Sq^{0}_{*}})_{(m,m+2n)}:Q^{\otimes
m}_{m+2n}&\longrightarrow Q^{\otimes m}_{n}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\mbox{[}\prod_{1\leq j\leq
m}x_{j}^{a_{j}}\mbox{]}&\longmapsto\left\\{\begin{array}[]{ll}\mbox{[}\prod_{1\leq
j\leq m}x_{j}^{\frac{a_{j}-1}{2}}\mbox{]}&\text{if $a_{j}$ odd,
$j=1,2,\ldots,m$},\\\ 0&\text{otherwise},\end{array}\right.\end{array}$

which induces the homomorphism $\widetilde{Sq^{0}_{*}}:(Q^{\otimes
m}_{m+2n})^{GL_{m}}\to(Q^{\otimes m}_{n})^{GL_{m}}.$ Since $\mathscr{A}$ is a
cocommutative Hopf algebra, there exists the squaring operations $Sq^{i}:{\rm
Ext}_{\mathscr{A}}^{m,m+n}(\mathbb{F}_{2},\mathbb{F}_{2})\to{\rm
Ext}_{\mathscr{A}}^{m+i,2m+2n}(\mathbb{F}_{2},\mathbb{F}_{2}),$ which share
most of the properties with $Sq^{i}$ on the cohomology of spaces (see May ),
but the classical $Sq^{0}$ is not the identity in general. Remarkably, this
$Sq^{0}$ commutes with the dual of $\widetilde{Sq^{0}_{*}}$ through the Singer
transfer (see J.B , N.M2 ). The reader who is familiar with Kameko’s
$(\widetilde{Sq^{0}_{*}})_{(m,m+2n)}$ will probably agree that this map is
very useful in solving the hit problem. Indeed, Kameko M.K  showed that if
$m=\xi(n)={\rm min}\\{\gamma\in\mathbb{N}:\ n=\sum_{1\leq
i\leq\gamma}(2^{d_{i}}-1),\,d_{i}>0,\forall i,\,1\leq i\leq\gamma\\},$ then
$(\widetilde{Sq^{0}_{*}})_{(m,m+2n)}$ is an isomorphism of
$\mathbb{F}_{2}GL_{m}$-modules. This statement and Wood’s work R.W  together
are sufficient to determine $Q^{\otimes m}_{n}$ in each degree $n$ of the
special ”generic” form $n=r(2^{t}-1)+d.2^{t},$ whenever $0<\xi(d)<r<m,$ and
$t\geq 0$ (see also D.P6 ).

As we mentioned at the beginning, the hit problem was completely solved for
$m\leq 4.$ Very little information is known for $m=5$ and degrees $n$ given
above. At least, it is surveyed by the present writer D.P6  for
$(r,d,t)\in\\{(5,18,0),(5,8,t)\\}.$ We now extend for the case
$(r,d,t)=(5,18,t),$ in which $t$ an arbitrary non-negative integer. We start
with a useful remark.

###### Remark 2.1

It can be easily seen that
$5(2^{t}-1)+18.2^{t}=2^{t+4}+2^{t+2}+2^{t+1}+2^{t-1}+2^{t-1}-5,$ and so
$\xi(5(2^{t}-1)+18.2^{t})=5$ for any $t>1.$ This implies that the iterated
Kameko map

$((\widetilde{Sq^{0}_{*}})_{(5,5(2^{t}-1)+18.2^{t})})^{t-1}:Q^{\otimes
5}_{5(2^{t}-1)+18.2^{t}}\to Q^{\otimes 5}_{5(2^{1}-1)+18.2^{1}}$

is an isomorphism, for all $t\geq 1,$ and therefore, it is enough to determine
$Q^{\otimes 5}_{5(2^{t}-1)+18.2^{t}}$ for $t\in\\{0,1\\}$. The case $t=0$ has
explicitly been computed by us in D.P3 . When $t=1,$ because Kameko’s
homomorphism

$(\widetilde{Sq^{0}_{*}})_{(5,5(2^{1}-1)+18.2^{1})}:Q^{\otimes
5}_{5(2^{1}-1)+18.2^{1}}\to Q^{\otimes 5}_{5(2^{0}-1)+18.2^{0}}$

is an epimorphism, we have an isomorphism

$Q^{\otimes 5}_{5(2^{1}-1)+18.2^{1}}\cong{\rm
Ker}((\widetilde{Sq^{0}_{*}})_{(5,5(2^{1}-1)+18.2^{1})})\bigoplus Q^{\otimes
5}_{5(2^{0}-1)+18.2^{0}}.$

The space $Q^{\otimes 5}_{5(2^{0}-1)+18.2^{0}}$ is known by our previous work
D.P3 . Thus, we need compute the kernel of
$(\widetilde{Sq^{0}_{*}})_{(5,5(2^{1}-1)+18.2^{1})}.$ For this, our approach
can be summarized as follows:

  * (i)

A mononial in $\mathcal{P}_{5}$ is assigned a weight vector $\omega$ of degree
$5(2^{1}-1)+18.2^{1}$, which stems from the binary expansion of the exponents
of the monomial. The space of indecomposable elements ${\rm
Ker}((\widetilde{Sq^{0}_{*}})_{(5,5(2^{1}-1)+18.2^{1})})$ is then decomposed
into a direct sum of $(Q_{5(2^{1}-1)+18.2^{1}}^{\otimes 5})^{0}$ and the
subspaces $(Q^{\otimes 5})^{\omega^{>0}}$ indexed by the weight vectors
$\omega.$ Here $[F]_{\omega}=[G]_{\omega}$ in
$(Q_{5(2^{1}-1)+18.2^{1}}^{\otimes 5})^{\omega}$ if the polynomial $F-G$ is
hit, modulo a sum of monomials of weight vectors less than $\omega.$ Basing
the previous results by Peterson F.P , Kameko M.K , Sum N.S1 , and by us D.P6
, one can easily determine $(Q_{5(2^{1}-1)+18.2^{1}}^{\otimes 5})^{0}.$

  * (ii)

The monomials in a given degree are lexicographically ordered first by weight
vectors and then by exponent vectors. This leads to the concept of admissible
monomial; more explicitly, a monomial is admissible if, modulo hit elements,
it is not equal to a sum of monomials of smaller orders. The space
$(Q_{5(2^{1}-1)+18.2^{1}}^{\otimes 5})^{\omega^{>0}}$ above is easily seen to
be isomorphic to the space generated by admissible monomials of the weight
vector $\omega.$

  * (iii)

In a given (small) degree, we first list all possible weight vectors of an
admissible monomial. This is done by first using a criterion of Singer W.S1
on the hit monomials, and then combining with the results by Kameko M.K  and
Sum N.S1  of the form ”$XZ^{2^{r}}$ (or $ZY^{2^{t}}$) admissible implying $Z$
admissible, under some mild conditions”.

  * (iv)

In a given weight vector, we claim the (strict) inadmissibility of some
explicit monomials. The proof is given for a typical monomial in each case by
explicit computations. Finally, a direct calculation using Theorems 3.2, 3.3,
and some homomorphisms in section three, we obtain a basis of
$(Q_{5(2^{1}-1)+18.2^{1}}^{\otimes 5})^{\omega^{>0}}.$ This approach is much
less computational and it can be applied for all certain degrees and all
variables $m.$ Moreover, the MAGMA computer algebra Magma  has been used for
verifying the results.

Before going into detail and proceeding to the main results, let us provide
some basic concepts. Of course, we assume that the reader is not familiar with
the basics of hit problems.

Weight vector and exponent vector. Let
$\omega=(\omega_{1},\omega_{2},\ldots,\omega_{t},\ldots)$ be a sequence of
non-negative integers. We say that $\omega$ is a weight vector, if
$\omega_{t}=0$ for $t\gg 0.$ Then, we also define $\deg(\omega)=\sum_{t\geq
1}2^{t-1}\omega_{t}.$ Let $X=x_{1}^{u_{1}}x_{2}^{u_{2}}\ldots x_{m}^{u_{m}}$
be a mononial in $\mathcal{P}_{m},$ define two sequences associated with $X$
by

$\omega(X):=(\omega_{1}(X),\omega_{2}(X),\ldots,\omega_{t}(X),\ldots),\ \
u(X):=(u_{1},u_{2},\ldots,u_{m}),$

where $\omega_{t}(X)=\sum_{1\leq j\leq m}\alpha_{t-1}(u_{j})$ in which
$\alpha_{t}(n)$ denotes the $t$-th coefficients in dyadic expansion of a
positive integer $n.$ They are called the weight vector and the exponent
vector of $X,$ respectively. We use the convention that the sets of all the
weight vectors and the exponent vectors are given the left lexicographical
order.

Linear order on $\mathcal{P}_{m}$. Assume that
$X=x_{1}^{u_{1}}x_{2}^{u_{2}}\ldots x_{m}^{u_{m}}$ and
$Y=x_{1}^{v_{1}}x_{2}^{v_{2}}\ldots x_{m}^{v_{m}}$ are the monomials of the
same degree in $\mathcal{P}_{m}.$ 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 $u(X)<v(Y).$

Equivalence relations on $\mathcal{P}_{m}$. For a weight vector $\omega,$ we
denote two subspaces associated with $\omega$ by

$\begin{array}[]{ll}\mathcal{P}^{\leq\omega}_{m}=\langle\\{X\in\mathcal{P}_{m}|\,\deg(X)=\deg(\omega),\
\omega(X)\leq\omega\\}\rangle,\\\
\mathcal{P}^{<\omega}_{m}=\langle\\{X\in\mathcal{P}_{m}|\,\deg(X)=\deg(\omega),\
\omega(X)<\omega\\}\rangle.\end{array}$

Let $F$ and $G$ be the homogeneous polynomials in $\mathcal{P}_{m}$ such that
$\deg(F)=\deg(G).$ We say that

  1. (i)

$F\equiv G$ if and only if
$(F-G)\in\overline{\mathscr{A}}\mathcal{P}_{m}=\sum_{i\geq 0}{\rm
Im}(Sq^{2^{i}}).$ Specifically, if $F\equiv 0,$ then $F$ is hit (or
$\mathscr{A}$-decomposable), i.e., $F$ can be written in the form $\sum_{i\geq
0}Sq^{2^{i}}(F_{i})$ for some $F_{i}\in\mathcal{P}_{m}$;

  2. (ii)

$F\equiv_{\omega}G$ if and only if $F,\,G\in\mathcal{P}^{\leq\omega}_{m}$ and
$(F-G)\in((\overline{\mathscr{A}}\mathcal{P}_{m}\cap\mathcal{P}_{m}^{\leq\omega})+\mathcal{P}_{m}^{<\omega}).$

It is not difficult to show that the binary relations ”$\equiv$” and
”$\equiv_{\omega}$” are equivalence ones. So, one defines the quotient space

$(Q^{\otimes
m})^{\omega}=\mathcal{P}_{m}^{\leq\omega}/((\overline{\mathscr{A}}\mathcal{P}_{m}\cap\mathcal{P}_{m}^{\leq\omega})+\mathcal{P}_{m}^{<\omega}).$

Moreover, due to Sum N.S3 , $(Q^{\otimes m})^{\omega}$ is also an
$\mathbb{F}_{2}GL_{m}$-module.

Admissible monomial and inadmissible monomial. A monomial
$X\in\mathcal{P}_{m}$ is said to be inadmissible if there exist monomials
$Y_{1},Y_{2},\ldots,Y_{k}$ such that $Y_{j}<X$ for $1\leq j\leq k$ and
$X\equiv\sum_{1\leq j\leq k}Y_{j}.$ Then, $X$ is said to be admissible if it
is not inadmissible.

Thus, with the above definitions in hand, it is straightforward to see that
the set of all the admissible monomials of degree $n$ in $\mathcal{P}_{m}$ is
a minimal set of $\mathscr{A}$-generators for $\mathcal{P}_{m}$ in degree $n.$
So, $Q_{n}^{\otimes m}$ is a $\mathbb{F}_{2}$-vector space with a basis
consisting of all the classes represent by the admissible monomials of degree
$n$ in $\mathcal{P}_{m}.$ Further, as stated in D.P2 , the dimension of
$Q_{n}^{\otimes m}$ can be represented as the sum of the dimensions
$(Q^{\otimes m})^{\omega}$ such that $\deg(\omega)=n.$ For later convenience,
we need to set some notation. Let $\mathcal{P}^{0}_{m}$ and
$\mathcal{P}^{>0}_{m}$ denote the $\mathscr{A}$-submodules of
$\mathcal{P}_{m}$ spanned all the monomials $\prod_{1\leq j\leq
m}x_{j}^{t_{j}}$ such that $\prod_{1\leq j\leq m}t_{j}=0,$ and $\prod_{1\leq
j\leq m}t_{j}>0,$ respectively. Let us write $(Q^{\otimes
m})^{0}:=\mathbb{F}_{2}\otimes_{\mathscr{A}}\mathcal{P}^{0}_{m},\ \mbox{and}\
(Q^{\otimes
m})^{>0}:=\mathbb{F}_{2}\otimes_{\mathscr{A}}\mathcal{P}^{>0}_{m},$ from which
one has that $Q^{\otimes m}=(Q^{\otimes m})^{0}\,\bigoplus\,(Q^{\otimes
m})^{>0}.$ For a polynomial $F\in\mathcal{P}_{m},$ we denote by $[F]$ the
classes in $Q^{\otimes m}$ represented by $F.$ If $\omega$ is a weight vector
and $F\in\mathcal{P}_{m}^{\leq\omega},$ then denote by $[F]_{\omega}$ the
classes in $(Q^{\otimes m})^{\omega}$ represented by $F.$ For a subset
$\mathscr{C}\subset\mathcal{P}_{m},$ we also write $|\mathscr{C}|$ for the
cardinal of $\mathscr{C}$ and put
$[\mathscr{C}]=\\{[F]\,:\,F\in\mathscr{C}\\}.$ If $\mathscr{C}\subset
P_{m}^{\leq\omega},$ then put
$[\mathscr{C}]_{\omega}=\\{[F]_{\omega}\,:\,F\in\mathscr{C}\\}.$ Let us denote
by $\mathscr{C}^{\otimes m}_{n}$ the set of all admissible monomials of degree
$n$ in $\mathcal{P}_{m},$ and let $\omega$ be a weight vector of degree $n.$
By setting

$\begin{array}[]{ll}(\mathscr{C}^{\otimes
m}_{n})^{\omega}:=\mathscr{C}^{\otimes
m}_{n}\cap\mathcal{P}_{m}^{\leq\omega},\ \ (\mathscr{C}^{\otimes
m}_{n})^{\omega^{0}}:=(\mathscr{C}^{\otimes
m}_{n})^{\omega}\cap\mathcal{P}^{0}_{m},\ \ (\mathscr{C}^{\otimes
m}_{n})^{\omega^{>0}}:=(\mathscr{C}^{\otimes
m}_{n})^{\omega}\cap\mathcal{P}^{>0}_{m},\\\ (Q_{n}^{\otimes
m})^{\omega^{0}}:=(Q^{\otimes m})^{\omega}\cap(Q^{\otimes m}_{n})^{0},\ \
(Q_{n}^{\otimes m})^{\omega^{>0}}:=(Q^{\otimes m})^{\omega}\cap(Q^{\otimes
m}_{n})^{>0},\end{array}$

then the sets $[(\mathscr{C}^{\otimes
m}_{n})^{\omega}]_{\omega},\,[(\mathscr{C}^{\otimes
m}_{n})^{\omega^{0}}]_{\omega}$ and $[(\mathscr{C}^{\otimes
m}_{n})^{\omega^{>0}}]_{\omega}$ are the bases of the $\mathbb{F}_{2}$-vector
spaces $(Q_{n}^{\otimes m})^{\omega},\ (Q_{n}^{\otimes m})^{\omega^{0}}$ and
$(Q_{n}^{\otimes m})^{\omega^{>0}},$ respectively.

Main results and applications. Let us now return to our study of the kernel of
the Kameko homomorphism $(\widetilde{Sq^{0}_{*}})_{(5,5(2^{1}-1)+18.2^{1})}$
and state our main results in greater detail. Firstly, by direct calculations
using the results by Kameko M.K , Singer W.S1 , Sum N.S1 , and Tín N.T , we
obtain the following, which is one of our main results and is crucial for an
application on the dimension of $Q^{\otimes 6}.$

###### Theorem 2.2

We have an isomorphism

$\mbox{\rm
Ker}(\widetilde{Sq_{*}^{0}})_{(5,5(2^{1}-1)+18.2^{1})}\cong(Q_{5(2^{1}-1)+18.2^{1}}^{\otimes
5})^{0}\bigoplus(Q_{5(2^{1}-1)+18.2^{1}}^{\otimes
5})^{\widetilde{\omega}^{>0}},$

where $\widetilde{\omega}=(3,3,2,1,1)$ is the weight vector of the degree
$5(2^{1}-1)+18.2^{1}.$

###### Remark 2.3

We are given in D.P6  that $(Q_{n}^{\otimes 5})^{0}\cong\bigoplus_{1\leq s\leq
4}\bigoplus_{\ell(\mathcal{J})=s}(Q_{n}^{\otimes\,\mathcal{J}})^{>0},$ where

$Q^{\otimes\mathcal{J}}=\langle[x_{j_{1}}^{t_{1}}x_{j_{2}}^{t_{2}}\ldots
x_{j_{s}}^{t_{s}}]\;|\;t_{i}\in\mathbb{N},\,i=1,2,\ldots,s\\}\rangle\subset
Q^{\otimes 5}$

with $\mathcal{J}=(j_{1},j_{2},\ldots,j_{s}),$ $1\leq j_{1}<\ldots<j_{s}\leq
5$, $1\leq s\leq 4,$ and $\ell(\mathcal{J}):=s$ denotes the length of
$\mathcal{J}.$ This implies that $\dim((Q_{n}^{\otimes 5})^{0}))=\sum_{1\leq
s\leq 4}\binom{5}{s}\dim((Q_{n}^{\otimes\,s})^{>0}),\ \mbox{for all $n\geq
0.$}$ On the other side, since $\xi(5(2^{1}-1)+18.2^{1})=3,$ by Peterson F.P
and Wood R.W , the spaces $Q_{5(2^{1}-1)+18.2^{1}}^{\otimes 1}$ and
$Q_{5(2^{1}-1)+18.2^{1}}^{\otimes 2}$ are trivial. Moreover, following Kameko
M.K  and Sum N.S1 , we have seen that $(Q_{5(2^{1}-1)+18.2^{1}}^{\otimes
3})^{>0}$ is $15$-dimensional and that $(Q_{5(2^{1}-1)+18.2^{1}}^{\otimes
4})^{>0}$ is $165$-dimensional. Therefore, we may conclude that

$\dim((Q^{\otimes
5}_{5(2^{1}-1)+18.2^{1}})^{0}=15.\binom{5}{3}+165.\binom{5}{4}=975.$

Next, due to Remarks 2.1, 2.3, and to Theorem 2.2, the space $Q^{\otimes
5}_{5(2^{1}-1)+18.2^{1}}$ will be determined by computing
$(Q_{5(2^{1}-1)+18.2^{1}}^{\otimes 5})^{\omega^{>0}}.$ To accomplish this, we
use the method described above to explicitly indicate all the admissible
monomials in the set $(\mathscr{C}^{\otimes
5}_{5(2^{1}-1)+18.2^{1}})^{\widetilde{\omega}^{>0}}.$ As a result, it reads as
follows.

###### Theorem 2.4

There exist exactly $925$ admissible monomials of degree $5(2^{1}-1)+18.2^{1}$
in $\mathcal{P}_{5}^{>0}$ such that their weight vectors are
$\widetilde{\omega}.$ Consequently, $(Q_{5(2^{1}-1)+18.2^{1}}^{\otimes
5})^{\widetilde{\omega}^{>0}}$ has dimension $925.$

This theorem, together with the fact that $Q_{5(2^{t}-1)+18.2^{t}}^{\otimes
5}=(Q_{5(2^{t}-1)+18.2^{t}}^{\otimes
5})^{0}\,\bigoplus\,(Q_{5(2^{t}-1)+18.2^{t}}^{\otimes 5})^{>0},$ yields an
immediate corollary that

###### Corollary 2.5

The space $Q_{5(2^{t}-1)+18.2^{t}}^{\otimes 5}$ is $730$-dimensional if $t=0,$
and is $2630$-dimensional if $t\geq 1.$

As applications, one would also be interested in applying results and
techniques of hit problems into the cases of higher ranks $m$ of $Q^{\otimes
m}$ and the modular representations of the general linear groups (see also the
relevant discussions in literatures J.B , N.M1 ; N.M2 , T.N2 , W.W ; W.W2 ).
Two applications below of the contributions of this paper are also not beyond
this target.

First application: the dimension of $Q^{\otimes 6}.$ The hit problem of six
variables has been not yet known. Using Corollary 2.5 for the case $t\geq 1$
and a result in Sum N.S1 , we state that

###### Theorem 2.6

With the generic degree $5(2^{t+4}-1)+41.2^{t+4},$ where $t$ an arbitrary
positive integer, then the $\mathbb{F}_{2}$-vector space $Q^{\otimes 6}$ has
dimension $165690$ in this degree.

Observing from Corollary 2.5 and Theorem 2.6, the readers can notice that the
dimensions of $Q^{\otimes 5}$ and $Q^{\otimes 6}$ in degrees given are very
large. So, a general approach to hit problems, other than providing a mononial
basis of the vector space $Q_{n}^{\otimes m},$ is to find upper/lower bounds
on the dimension of this space. However, in this work, we have not studied
this side of the problem and it is our concern the next time. It is remarkable
that, we have Kameko’s conjecture M.K  on an upper bound for the dimension of
$Q_{n}^{\otimes m},$ but unfortunately, it was refuted for $m\geq 5$ by the
brilliant work of Sum N.S .

Second application: the behavior of the fifth Singer transfer. We adopt
Corollary 2.5 for $t=0,$ together with a fact of the Adams $E^{2}$-term, ${\rm
Ext}_{\mathscr{A}}^{5,5+*}(\mathbb{F}_{2},\mathbb{F}_{2})$, to obtain
information about the behavior of Singer’s cohomological transfer in the
bidegree $(5,5+(5(2^{0}-1)+18.2^{0}))$. More precisely, it is known, the
calculations of Lin W.L , and Chen T.C  imply that ${\rm
Ext}_{\mathscr{A}}^{5,5+(5(2^{t}-1)+18.2^{t})}(\mathbb{F}_{2},\mathbb{F}_{2})=\langle
h_{t}f_{t}\rangle$ and $h_{t}f_{t}=h_{t+1}e_{t}\neq 0$ for all $t\geq 0.$ So,
to determine the transfer map in the above bidegree, we shall compute the
dimension of (the domain of the fifth transfer)
$(\mathbb{F}_{2}\otimes_{GL_{5}}P_{\mathscr{A}}((\mathcal{P}_{5})^{*}))_{5(2^{0}-1)+18.2^{0}}$
by using a mononial basis of $Q^{\otimes 5}_{5(2^{0}-1)+18.2^{0}}.$ (We
emphasize that computing the domain of $Tr_{m}^{\mathscr{A}}$ in each degree
$n$ is very difficult, particularly for values of $m$ as large as $m=5.$ The
understanding of special cases should be a helpful step toward the solution of
the general problem. Moreover, we believe, in principle, that our method could
lead to a full analysis of
$\mathbb{F}_{2}\otimes_{GL_{m}}P_{\mathscr{A}}((\mathcal{P}_{m})^{*})$ in each
$m$ and degree $n>0$, as long as nice decompositions of the space of
$GL_{m}$-invariants of $Q^{\otimes m}$ in degrees given. However, the
difficulty of such a task must be monumental, as $Q^{\otimes m}$ becomes much
larger and harder to understand with increasing $m.$) Details for this
application are as follows. It may need to be recalled that by the previous
discussions D.P3 , we get the technical result below.

###### Proposition 2.7

The following hold:

  1. i)

If $Y\in\mathscr{C}_{5(2^{0}-1)+18.2^{0}}^{\otimes 5},$ then
$\overline{\omega}:=\omega(Y)$ is one of the following sequences:

$\overline{\omega}_{[1]}:=(2,2,1,1),\ \ \overline{\omega}_{[2]}:=(2,2,3),\ \
\overline{\omega}_{[3]}:=(2,4,2),$ $\overline{\omega}_{[4]}:=(4,1,1,1),\ \
\overline{\omega}_{[5]}:=(4,1,3),\ \ \overline{\omega}_{[6]}:=(4,3,2).$

  2. ii)

$|(\mathscr{C}^{\otimes
5}_{5(2^{0}-1)+18.2^{0}})^{\overline{\omega}_{[k]}}|=\left\\{\begin{array}[]{ll}300&\mbox{{\rm
if}}\ k=1,\\\ 15&\mbox{{\rm if}}\ k=2,5,\\\ 10&\mbox{{\rm if}}\ k=3,\\\
110&\mbox{{\rm if}}\ k=4,\\\ 280&\mbox{{\rm if}}\ k=6.\end{array}\right.$

One should note that $|(\mathscr{C}^{\otimes
5}_{5(2^{0}-1)+18.2^{0}})^{\overline{\omega}_{[k]}}|=|(\mathscr{C}^{\otimes
5}_{5(2^{0}-1)+18.2^{0}})^{\overline{\omega}_{[k]}^{>0}}|$ for $k=2,3$, and
that $|(\mathscr{C}^{\otimes
5}_{5(2^{0}-1)+18.2^{0}})^{\overline{\omega}_{[2]}^{0}}|=0=|(\mathscr{C}^{\otimes
5}_{5(2^{0}-1)+18.2^{0}})^{\overline{\omega}_{[3]}^{0}}|.$ Moreover,
$\dim(Q^{\otimes 5}_{5(2^{0}-1)+18.2^{0}})=\sum_{1\leq k\leq
6}|(\mathscr{C}^{\otimes
5}_{5(2^{0}-1)+18.2^{0}})^{\overline{\omega}_{[k]}}|=730.$ Next, applying
these results, we explicitly compute the subspaces of $GL_{5}$-invariants
$((Q_{5(2^{0}-1)+18.2^{0}}^{\otimes 5})^{\overline{\omega}_{[k]}})^{GL_{5}},$
for $1\leq k\leq 6,$ and obtain

###### Theorem 2.8

The following assertions are true:

  1. i)

$((Q_{5(2^{0}-1)+18.2^{0}}^{\otimes 5})^{\overline{\omega}_{[k]}})^{GL_{5}}=0$
with $k\in\\{1,2,3,5,6\\}.$

  2. ii)

$((Q_{5(2^{0}-1)+18.2^{0}}^{\otimes
5})^{\overline{\omega}_{[4]}})^{GL_{5}}=\langle[\Re^{\prime}_{4}]_{\overline{\omega}_{[4]}}\rangle,$
where

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\Re^{\prime}_{4}&=x_{1}x_{2}x_{3}x_{4}x_{5}^{14}+x_{1}x_{2}x_{3}x_{4}^{14}x_{5}+x_{1}x_{2}x_{3}^{14}x_{4}x_{5}+x_{1}x_{2}^{3}x_{3}x_{4}x_{5}^{12}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}x_{2}^{3}x_{3}x_{4}^{12}x_{5}+x_{1}x_{2}^{3}x_{3}^{12}x_{4}x_{5}+x_{1}^{3}x_{2}x_{3}x_{4}x_{5}^{12}+x_{1}^{3}x_{2}x_{3}x_{4}^{12}x_{5}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{3}x_{2}x_{3}^{12}x_{4}x_{5}+x_{1}^{3}x_{2}^{5}x_{3}x_{4}x_{5}^{8}+x_{1}^{3}x_{2}^{5}x_{3}x_{4}^{8}x_{5}+x_{1}^{3}x_{2}^{5}x_{3}^{8}x_{4}x_{5}.\end{array}$

Now, because
$(\mathbb{F}_{2}\otimes_{GL_{5}}P_{\mathscr{A}}((\mathcal{P}_{5})^{*}))_{5.(2^{0}-1)+18.2^{0}}$
is isomorphic to $(Q^{\otimes 5}_{5.(2^{0}-1)+18.2^{0}})^{GL_{5}},$ by Theorem
2.8, we have the following estimate:

$\begin{array}[]{ll}\dim(\mathbb{F}_{2}\otimes_{GL_{5}}P_{\mathscr{A}}((\mathcal{P}_{5})^{*}))_{5.(2^{0}-1)+18.2^{0}}&=\dim(Q^{\otimes
5}_{5.(2^{0}-1)+18.2^{0}})^{GL_{5}}\\\ &\leq\sum_{1\leq k\leq
6}\dim((Q_{5(2^{0}-1)+18.2^{0}}^{\otimes
5})^{\overline{\omega}_{[k]}})^{GL_{5}}\leq 1.\end{array}$

On the other side, as shown in section one, $\\{h_{t}|\,t\geq 0\\}\subset{\rm
Im}(Tr^{\mathscr{A}}_{1}),$ and $\\{f_{t}|\,t\geq 0\\}\subset{\rm
Im}(Tr^{\mathscr{A}}_{4}).$ Combining this with the fact that the total
transfer $\bigoplus_{m\geq 0}Tr_{m}^{\mathscr{A}}$ is a homomorphism of
algebras, it may be concluded that the non-zero element $h_{t}f_{t}\in{\rm
Ext}_{\mathscr{A}}^{5,23.2^{t}}(\mathbb{F}_{2},\mathbb{F}_{2})$ is in the
image of $Tr^{\mathscr{A}}_{5}$ for all $t\geq 0.$ This could be directly
proved as in Appendix. This statement implies that

$\dim(\mathbb{F}_{2}\otimes_{GL_{5}}P_{\mathscr{A}}((\mathcal{P}_{5})^{*}))_{5.(2^{0}-1)+18.2^{0}}\geq
1,$

and therefore
$(\mathbb{F}_{2}\otimes_{GL_{5}}P_{\mathscr{A}}((\mathcal{P}_{5})^{*}))_{5.(2^{0}-1)+18.2^{0}}$
is one-dimensional. As a consequence, we immediately obtain

###### Corollary 2.9

The cohomological transfer

$Tr^{\mathscr{A}}_{5}:(\mathbb{F}_{2}\otimes_{GL_{5}}P_{\mathscr{A}}((\mathcal{P}_{5})^{*}))_{5(2^{0}-1)+18.2^{0}}\to{\rm
Ext}_{\mathscr{A}}^{5,5+5(2^{0}-1)+18.2^{0}}(\mathbb{F}_{2},\mathbb{F}_{2})$

is an isomorphism. Consequently, Conjecture 1.1 holds in the rank 5 case and
the degree $5(2^{0}-1)+18.2^{0}.$

Comments and open issues. From the above results, it would be interesting to
see that $Q^{\otimes 5}$ is $730$-dimensional in degree $5(2^{0}-1)+18.2^{0},$
but the space of $GL_{5}$-coinvariants of it in this degree is only one-
dimensional. In general, it is quite efficient in using the results of the hit
problem of five variables to study
$\mathbb{F}_{2}\otimes_{GL_{5}}P_{\mathscr{A}}((\mathcal{P}_{5})^{*}).$ This
provides a valuable method for verifying Singer’s open conjecture on the fifth
algebraic transfer. We now close the introduction by discussing about
Conjecture 1.1 in the rank 5 case and the internal degree
$n_{t}:=5(2^{t}-1)+18.2^{t}$ for all $t\geq 1.$ Let us note again that the
iterated Kameko homomorphism
$((\widetilde{Sq^{0}_{*}})_{(5,n_{t})})^{t-1}:Q^{\otimes 5}_{n_{t}}\to
Q^{\otimes 5}_{n_{1}}$ is an $\mathbb{F}_{2}GL_{5}$-module isomorphism for all
$t\geq 1.$ So, from a fact of ${\rm
Ext}_{\mathscr{A}}^{5,5+n_{1}}(\mathbb{F}_{2},\mathbb{F}_{2})$, to check
Singer’s conjecture in the above degree, we need only determine
$GL_{5}$-coinvariants of $Q^{\otimes 5}_{n_{t}}$ for $t=1.$ We must recall
that Kameko’s map $(\widetilde{Sq^{0}_{*}})_{(5,n_{1})}:Q^{\otimes
5}_{n_{1}}\to Q^{\otimes 5}_{n_{0}}$ is an epimorphism of $GL_{5}$-modules. On
the other side, as shown before, the non-zero element $h_{1}f_{1}\in{\rm
Ext}_{\mathscr{A}}^{5,5+n_{1}}(\mathbb{F}_{2},\mathbb{F}_{2})$ is detected by
the fifth transfer. From these data and Theorem 2.8, one has an estimate

$0\leq\dim((\mathbb{F}_{2}\otimes_{GL_{5}}P_{\mathscr{A}}((\mathcal{P}_{5})^{*}))_{n_{1}})-1\leq\dim({\rm
Ker}(\widetilde{Sq^{0}_{*}})_{(5,n_{1})})^{GL_{5}}.$

Moreover, basing the proof of Theorem 2.8 together with a few simple
arguments, it follows that the elements in
$(\mathbb{F}_{2}\otimes_{GL_{5}}P_{\mathscr{A}}((\mathcal{P}_{5})^{*}))_{n_{1}}$
are dual to the classes

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&\gamma[x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{29}+x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{4}^{29}x_{5}^{3}+x_{1}^{3}x_{2}^{3}x_{3}^{29}x_{4}^{3}x_{5}^{3}+x_{1}^{3}x_{2}^{7}x_{3}^{3}x_{4}^{3}x_{5}^{25}+x_{1}^{3}x_{2}^{7}x_{3}^{3}x_{4}^{25}x_{5}^{3}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{3}x_{2}^{7}x_{3}^{25}x_{4}^{3}x_{5}^{3}+x_{1}^{7}x_{2}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{25}+x_{1}^{7}x_{2}^{3}x_{3}^{3}x_{4}^{25}x_{5}^{3}+x_{1}^{7}x_{2}^{3}x_{3}^{25}x_{4}^{3}x_{5}^{3}+x_{1}^{7}x_{2}^{11}x_{3}^{3}x_{4}^{3}x_{5}^{17}\\\
&\quad+x_{1}^{7}x_{2}^{11}x_{3}^{3}x_{4}^{17}x_{5}^{3}+x_{1}^{7}x_{2}^{11}x_{3}^{17}x_{4}^{3}x_{5}^{3}]+[\zeta],\end{array}$

where $\gamma\in\mathbb{F}_{2},$ and
$[\zeta]\in\text{Ker}(\widetilde{Sq^{0}_{*}})_{(5,n_{1})}.$ It could be
noticed that calculating explicitly these elements is not easy. However, in
view of our previous works D.P2 ; D.P6 , and motivated by the above
computations, we have the following prediction.

###### Conjecture 2.10

For each $t\geq 1,$ the space of $GL_{5}$-invariants elements of ${\rm
Ker}(\widetilde{Sq^{0}_{*}})_{(5,n_{t})}$ is trivial. Consequently, the
coinvariant
$(\mathbb{F}_{2}\otimes_{GL_{5}}P_{\mathscr{A}}((\mathcal{P}_{5})^{*}))_{n_{t}}$
is 1-dimensional.

Since $h_{t}f_{t}\in{\rm Im}(Tr_{5}^{\mathscr{A}}),$ for all $t\geq 0,$ if
Conjecture 2.10 is true, then $Tr_{5}^{\mathscr{A}}$ is also isomorphism when
acting on the coinvariant
$(\mathbb{F}_{2}\otimes_{GL_{5}}P_{\mathscr{A}}((\mathcal{P}_{5})^{*}))_{n_{t}}$
for $t\geq 1,$ and so, Conjecture 1.1 holds in bidegree $(5,5+n_{t})$. We also
wish that our predictions are correct. If not, Singer’s conjecture will be
disproved. We leave these issues as future research. At the same time, we also
appreciate that some readers may have an interest in solving them.

Overview. Let us give a brief outline of the contents of this paper. Section
three contains a brief review of Steenrod squares and some useful linear
transformations. The dimensions of the polynomial algebras $\mathcal{P}_{5}$
and $\mathcal{P}_{6}$ in the generic degrees $n_{t}=5(2^{t}-1)+18.2^{t}$ and
$5(2^{t+4}-1)+n_{1}.2^{t+4}$ are respectively obtained in section four by
proving Theorems 2.2, 2.4, and 2.6. Section five is to present the proof of
Theorem 2.8. In the remainder of the text, we give a direct proof of an event
claimed above that the non-zero elements $h_{t}f_{t}\in{\rm
Ext}_{\mathscr{A}}^{5,23.2^{t}}(\mathbb{F}_{2},\mathbb{F}_{2})$ are detected
by $Tr_{5}^{\mathscr{A}}$. The proof is based on a representation in the
lambda algebra of the fifth Singer transfer. Finally, we describe the set
$(\mathscr{C}_{n_{1}}^{\otimes 5})^{\widetilde{\omega}^{>0}}$ and list some
the admissible monomials in $\mathscr{C}_{n_{0}}^{\otimes 5}$ and the strictly
inadmissible monomials in ($\mathcal{P}_{5}^{>0})_{n_{1}}.$

###### Acknowledgment

The author would like to give my deepest sincere thanks to Professor N. Sum
(Quy Nhon University, Sai Gon University), my thesis advisor, for meaningful
discussions. I am very grateful to Professor W. Singer for many enlightening
e-mail exchanges.

##  3 The Necessary Preliminaries

This section begins with a few words on the Steenrod algebra over
$\mathbb{F}_{2}$ and ends with a brief sketch of some homomorphisms in N.S1 .
At the same time, we prove some elementary results that will be used in the
rest of this text.

###  3.1 Steenrod squares and their properties

The mod 2 Steenrod algebra $\mathscr{A}$ was defined by Cartan Cartan  to be
the algebra of stable cohomology operations for mod 2 cohomology. This algebra
is generated by the Steenrod squares $Sq^{i}:H^{n}(X,\mathbb{F}_{2})\to
H^{n+i}(X,\mathbb{F}_{2}),$ for $i\geq 0,$ where $H^{n}(X,\mathbb{F}_{2})$
denotes the $n$-th singular cohomology group of a topological space $X$ with
coefficient over $\mathbb{F}_{2}.$ Steenrod and Epstein S.E  showed that these
squares are characterized by the following 5 axioms:

  1. (i)

$Sq^{i}$ is an additive homomorphism and is natural with respect to any
$f:X\to Y.$ So $f^{*}(Sq^{i}(x))=Sq^{i}(f^{*}(x)).$

  2. (ii)

$Sq^{0}$ is the identity homomorphism.

  3. (iii)

$Sq^{i}(x)=x\smile x$ for all $x\in H^{i}(X,\mathbb{F}_{2})$ where $\smile$
denotes the cup product in the graded cohomology ring
$H^{*}(X,\mathbb{F}_{2}).$

  4. (iv)

If $i>\deg(x),$ then $Sq^{i}(x)=0.$

  5. (v)

Cartan’s formula: $Sq^{n}(x\smile y)=\sum_{i+j=n}Sq^{i}(x)\smile Sq^{j}(y).$

In addition, Steenrod squares have the following properties:

$\bullet$ $Sq^{1}$ is the Bockstein homomorphism of the coefficient sequence:
$0\to\mathbb{Z}/2\to\mathbb{Z}/4\to\mathbb{Z}/2\to 0.$

$\bullet$ $Sq^{i}$ commutes with the connecting morphism of the long exact
sequence in cohomology. In particular, it commutes with respect to suspension
$H^{n}(X,\mathbb{F}_{2})\cong H^{n+1}(\Sigma X,\mathbb{F}_{2}).$

$\bullet$ They satisfy the Adem relations: $Sq^{i}Sq^{j}=\sum_{0\leq
t\leq[i/2]}\binom{j-t-1}{i-2t}Sq^{i+j-t}Sq^{t},\,\,0<i<2j,$ where the binomial
coefficients are to be interpreted mod 2. These relations, which were
conjectured by Wu Wu  and established by Adem Adem , allow one to write an
arbitrary composition of Steenrod squares as a sum of Serre-Cartan basis
elements.

Note that the structure of the cohomology $H^{*}(X,\mathbb{F}_{2})$ is not
only as graded commutative $\mathbb{F}_{2}$-algebra, but also as an
$\mathscr{A}$-module. In many cases, the $\mathscr{A}$-module structure on
$H^{*}(X,\mathbb{F}_{2})$ provides additional information on $X.$

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