1. Introduction
Everywhere in the text of this article, we will be working over the field $\mathbb {F}_2\cong \mathbb {Z}/2\mathbb {Z}$
of characteristic 2 and taking (co)homology with coefficients in $\mathbb {F}_2.$
It is well-known that the calculation of the stable homotopy groups of spheres $\pi ^{S}_*(\mathbb {S}^{0})$
is one of the most central and intractable problems in Algebraic topology. Historically, in the 1950s, Serre [Reference Serre24] used his spectral sequence to study this problem. In the late 1950s, Adams [Reference Adams1] constructed his celebrated spectral sequence that converges to $\pi ^{S}_*(\mathbb {S}^{0})$
, completed at prime 2. He claimed that $E_2$
-page of that spectral sequence could be identified with
the bigraded cohomology algebra of the classical, singly-graded Steenrod algebra $\mathscr A$
over $\mathbb {F}_2.$
This cohomology has been explicitly computed by Adem [Reference Adem3] for $q = 1,$
by Adams [Reference Adams2] and Wall [Reference Wall31] for $q = 2$
, by Adams [Reference Adams2] and Wang [Reference Wang32] for $q = 3$
, by Lin [Reference Lin12] for $q = 4$
, by Lin [Reference Lin12] and Chen [Reference Chen6] for $q = 5.$
However, it is still largely mysterious for all $q > 5.$
With an idea that we can study the structure of ${\rm Ext}^{q}_{\mathscr {A}}(\mathbb {F}_2,\, \mathbb {F}_2)$
through the modular invariant theory, in 1989, W. Singer [Reference Singer25] introduced a ‘transfer’ homomorphism of rank $q,$
which passes from coinvariants of a certain representation of the general linear group $GL(q)$
over $\mathbb {F}_2$
to mod-2 cohomology of $\mathscr A$
. It has been shown that this transfer is highly non-trivial (see the works by Boardman [Reference Boardman4], Minami [Reference Minami13], Bruner, Hà and Hu’ng [Reference Bruner, Hà and Hu’ng5], Hu’ng [Reference Hu’ng9], Hà [Reference Hà8], Nam [Reference Nam15], Hu’ng and Quỳnh [Reference Hu’ng and Quỳnh10], Cho’n and Hà [Reference Cho’n and Hà7], Sum [Reference Sum29], the author [Reference Phúc17–Reference Phúc23], and others). In order to better understand it, we offer some related issues. Let us denote by $V^{\oplus q}\cong \prod _{1\leq i\leq q}(\mathbb {Z}/2\mathbb {Z})$
a rank $q$
elementary abelian 2-group, which is considered as $q$
-dimensional vector $\mathbb {F}_2$
-space. It is known, $H^{*}(V^{\oplus q})\cong S(V_*^{\oplus q}),$
the symmetric algebra over the dual space $V_*^{\oplus q} \equiv H^{1}(V^{\oplus q})$
of $V^{\oplus q}.$
Pick $u_1,\, \ldots,\, u_q$
to be a basis of $H^{1}(V^{\oplus q}).$
Then, it has been shown that $\mathcal {P}_q:=H^{*}(V^{\oplus q})\cong \mathbb {F}_2[u_1,\, \ldots,\, u_q],$
the connected $\mathbb {Z}$
-graded polynomial algebra on generators of degree $1,$
equipped with the canonical unstable algebra structure over $\mathscr A.$
By dualizing, $H_*(V^{\oplus q})\cong \Gamma (a_1,\, \ldots,\, a_q),$
the divided power algebra generated by $a_1,\, \ldots,\, a_q,$
each of degree one, where $a_i \equiv a_i^{(1)}$
is dual to $u_i.$
It is to be noted that this algebra and the polynomial algebra $\mathcal {P}_q$
are not in general isomorphic as $\mathbb {F}_2GL(q)$
-modules. Now, let us recall that the algebra $\mathscr A$
consists of the Steenrod squaring operations $Sq^{i}$
for $i\geq 0.$
The operations $Sq^{0}$
and $Sq^{2^{i}},\, i\geq 0,$
constitute a system of multiplicative generators for $\mathscr A$
(see also Walker and Wood [Reference Walker and Wood30]). Emphasizing that these $Sq^{i}$
are the cohomology operations satisfying the naturality property. Moreover, they commute with the suspension maps, and therefore, they are stable. Let $P_{\mathscr A}H_*(V^{\oplus q})\cong {\rm Ext}_{\mathscr A}^{0}(\mathcal {P}_q,\, \mathbb {F}_2)$
be the subspace of $H_*(V^{\oplus q})$
consisting of all elements that are annihilated by all $Sq^{i}$
for $i > 0.$
The group $GL(q)$
acts regularly on $V^{\oplus q}$
and therefore on $\mathcal {P}_q$
and $H_*(V^{\oplus q}).$
This action commutes with that of the algebra $\mathscr A$
and so acts $\mathbb {F}_2 \otimes _{\mathscr A} \mathcal {P}_q$
and $P_{\mathscr A}H_*(V^{\oplus q}).$
Singer [Reference Singer25] constructed a homomorphism from $P_{\mathscr A}H_n(V^{\oplus q})$
to ${\rm Ext}^{q, q+n}_{\mathscr A}(\mathbb {F}_2,\, \mathbb {F}_2)$
, which commutes with two $Sq^{0}$
's on $P_{\mathscr A}H_n(V^{\oplus q})$
and ${\rm Ext}^{q, q+n}_{\mathscr A}(\mathbb {F}_2,\, \mathbb {F}_2)$
(see also [Reference Boardman4, Reference Minami14]). He shows that this map factors through the quotient of its domain's $GL(q)$
-coinvariants to give rise to the so-called algebraic transfer of rank q
In fact, this transfer is induced over the $E_2$
-term of the Adams spectral sequence by the geometrical transfer map $\Sigma ^{\infty }(B(V^{\oplus q})_{+})\longrightarrow \Sigma ^{\infty }(\mathbb {S}^{0})$
between the suspension spectrum in stable homotopy category. Singer [Reference Singer25] demonstrated that the ‘total’ transfer $\{Tr_q(\mathbb {F}_2)\}_{q\geq 0}$
is an algebra homomorphism and that $Tr_q(\mathbb {F}_2)$
is an isomorphism for $q = 1,\, 2.$
Afterwards, Boardman [Reference Boardman4] stated that $Tr_3(\mathbb {F}_2)$
is also an isomorphism. Remarkably, in mostly all the decade 1980s, Singer believed that $Tr_q(\mathbb {F}_2)$
is an isomorphism for all $q.$
However, in the rank 5 case, he himself claimed in [Reference Singer25] that it is not an isomorphism by showing that the indecomposable element $Ph_1\in {\rm Ext}^{5, 14}_{\mathscr A}(\mathbb {F}_2,\, \mathbb {F}_2)$
does not belong to the image of the transfer homomorphism, where $P$
denotes the Adams periodicity operator. Thence, he proposed the following yet-left open.
Conjecture 1.1 [Reference Singer25, Conj. 1.1]
$Tr_q(\mathbb {F}_2)$
is a monomorphism for any $q.$
As shown above, Singer's transfer is an isomorphism in ranks $\leq 3,$
and so, conjecture 1.1 is true in these ranks. The rank 4 case is the subject of this paper. Remarkably, the investigation of the image of the algebraic transfer of rank $4$
was completed by the authors in [Reference Bruner, Hà and Hu’ng5, Reference Hà8, Reference Hu’ng9, Reference Nam15] and [Reference Hu’ng and Quỳnh10]. More precisely, in [Reference Bruner, Hà and Hu’ng5], Bruner, Hà and Hu’ng proved that
Theorem 1.2 [Reference Bruner, Hà and Hu’ng5, Thm. 1.1]
For each $s \geq 1,$
the non-zero element $g_{s}\in {\rm Ext}^{4, 12.2^{s}}_{\mathscr A}(\mathbb {F}_2,\, \mathbb {F}_2)$
is not in the image of $Tr_4(\mathbb {F}_2).$
This result refuted a prediction by Minami in [Reference Minami14] that the localization of the algebraic transfer
given by inverting $Sq^{0}$
is an isomorphism. More explicitly, when $q = 4$
and $n = 12\cdot 2^{s}-4,$
following [Reference Bruner, Hà and Hu’ng5, Cor. 1.2], the localization of the fourth transfer given by inverting $Sq^{0}$
is not an epimorphism. As a continuation of the work [Reference Bruner, Hà and Hu’ng5], Hu’ng proved in [Reference Hu’ng9] that
Theorem 1.3 [Reference Hu’ng9, Thm. 1.9]
Any element in the $Sq^{0}$
-families $\{D_3(s)|\, s\geq 0\}$
and $\{p'_s|\, s\geq 0\}$
does not belong to the image of $Tr_4(\mathbb {F}_2).$
Theorems 1.2 and 1.3 imply that there are infinitely many degrees in which the fourth algebraic transfer is not an isomorphism. Further, those theorems together with the following establish a prediction in Hu’ng [Reference Hu’ng9, Conj. 1.10].
Theorem 1.4 [Reference Hà8, Reference Hu’ng and Quỳnh10, Reference Nam15]
Every element in the following families belongs to the image of the fourth algebraic transfer $Tr_4(\mathbb {F}_2)$
:
(i) $\{d_s|\, s\geq 0\},$
$\{e_s|\, s\geq 0\}$ (see Hà [Reference Hà8, Thm. 1.1]);(ii) $\{f_s|\, s\geq 0\}$
(see Nam [Reference Nam15, Thm. 1.4]);(iii) $\{p_s|\, s\geq 0\}$
(see Hu’ng and Quỳnh [Reference Hu’ng and Quỳnh10, Thm. 1.1]).
Alternatively, since the total transfer $\{Tr_q(\mathbb {F}_2)\}_{q\geq 0}$
is a homomorphism of algebras and $Tr_q(\mathbb {F}_2)$
is an isomorphism for $q = 1,\, 2,\, 3,$
all decomposable elements in ${\rm Ext}^{4}_{\mathscr A}(\mathbb {F}_2,\, \mathbb {F}_2)$
belong to the image of $Tr_4(\mathbb {F}_2).$
Now, based on the results of [Reference Bruner, Hà and Hu’ng5, Reference Hà8–Reference Hu’ng and Quỳnh10, Reference Nam15] on the image of $Tr_4(\mathbb {F}_2),$
Singer's conjecture 1.1 for $Tr_4(\mathbb {F}_2)$
turns out to be equivalent to the following.
Conjecture 1.5 Stated by the referee
Here $n$
is called bad if it equals to the stem of one element in the three families $\{g_s|\, s\geq 1\},$
$\{D_3(s)|\, s\geq 0\}$
and $\{p'_s|\, s\geq 0\},$
whose every element is not in the image of the algebraic transfer of rank four. Otherwise, $n$
is said to be not bad.
Thus, by verifying this conjecture, we will get the answer for conjecture 1.1 on the fourth transfer. This will be presented in the sequel.
The algebraic transfer we are describing is closely related to the hit problem in literature [Reference Peterson16] of determination of a minimal generating set for the $\mathbb {F}_2$
-algebra $\mathcal {P}_q$
, considered as an unstable $\mathscr A$
-module. The reader is familiar with an event that if $\mathbb {F}_2$
is an $\mathscr A$
-module concentrated in degree 0, then solving the hit problem is equivalent to finding a basis consisting of all equivalence classes of homogeneous polynomials for the $\mathbb {Z}$
-graded vector space over the field $\mathbb {F}_2:$
where the homogeneous components $(Q\mathcal {P}_q)_n:=(\mathbb {F}_2 \otimes _{\mathscr A} \mathcal {P}_q)_n$
of degrees $n$
are $\mathbb {F}_2GL(q)$
-submodules of $Q\mathcal {P}_q.$
Usually, one would investigate this tensor product. Its structure was systematically depicted by Peterson [Reference Peterson16] for $q = 1,\, 2$
, by Kameko's thesis [Reference Kameko11] for $q=3$
, and by Sum [Reference Sum27, Reference Sum28] for $q=4.$
So far it has been thoroughly studied for more than three decades by many topologist (see also [Reference Boardman4, Reference Bruner, Hà and Hu’ng5, Reference Kameko11, Reference Phúc17–Reference Phúc19, Reference Phúc21, Reference Phúc23, Reference Sum26–Reference Sum28, Reference Walker and Wood30, Reference Wood33]), but it remains unanswered for $q\geq 5.$
We also emphasize that in general, it is not easy to compute or even estimate the dimension of $Q\mathcal {P}_q$
in each positive degree. Most notably, in his thesis [Reference Kameko11], Kameko conjectured that an upper bound on the dimension of $(Q\mathcal {P}_q)_n$
is the order of the factor group of $GL(q)$
by the Borel subgroup $B_q$
, i.e.,
for any $n\geq 0.$
However, in 2010, the famous work [Reference Sum26] of Sum refuted the above prediction. In order to reduce the process of the calculation of $Q\mathcal {P}_q$
in each certain degree, one considers the arithmetic function $\mu : \mathbb {N}\to \mathbb {N},$
which is defined by
Theorem 1.6 For each non-negative integer $n,$
the following assertions are true:
(i) $(Q\mathcal {P}_q)_n = 0$
if and only if $\mu (n) > q$ (see Peterson's conjecture [Reference Peterson16], Wood [Reference Wood33]);(ii) $(Q\mathcal {P}_q)_n \cong (Q\mathcal {P}_q)_{(n-q)/2}$
if and only if $\mu (n) = q$ (see Kameko's thesis [Reference Kameko11]).
The statement (i) is given by Peterson's conjecture. Peterson himself confirmed it for $q \leq 2.$
Afterwards, Wood proved the general case under a stronger form.
To close this section, we recall the already known results on ${\rm Ext}_{\mathscr A}^{q, *}(\mathbb {F}_2,\, \mathbb {F}_2)$
for $q\leq 4.$
Theorem 1.7 [Reference Adams2, Reference Adem3, Reference Lin12, Reference Wall31, Reference Wang32]
The following hold:
(i) ${\rm Ext}_{\mathscr A}^{1, *}(\mathbb {F}_2,\, \mathbb {F}_2)$
is generated by $h_i$ for $i\geq 0$ (see Adem [Reference Adem3]);(ii) ${\rm Ext}_{\mathscr A}^{2, *}(\mathbb {F}_2,\, \mathbb {F}_2)$
is generated by $h_ih_j$ for $j\geq i\geq 0$ and $j\neq i+1$ (see Adams [Reference Adams2] and Wall [Reference Wall31]);(iii) ${\rm Ext}_{\mathscr A}^{3, *}(\mathbb {F}_2,\, \mathbb {F}_2)$
is generated by $h_ih_jh_{\ell },\, c_t$ for $t\geq 0;$ $\ell \geq j\geq i\geq 0,$ and subject only to the relations $h_ih_{i+1} = 0,\,\ h_ih_{i+2}^{2} = 0$ and $h_i^{3} = h^{2}_{i-1}h_{i+1}$ (see Adams [Reference Adams2] and Wang [Reference Wang32]);(iv) ${\rm Ext}_{\mathscr A}^{4, *}(\mathbb {F}_2,\, \mathbb {F}_2)$
is generated by $h_ih_jh_{\ell }h_m,\, h_uc_v,\, d_t,\, e_t,\, f_t,\, g_{t+1},\, p_t,\, D_3(t),\, p'_t$ for $m\geq \ell \geq j\geq i\geq 0$, $u,\,\ v,\,\ t\geq 0$, and subject to the relations in (iii) together with $h^{2}_ih^{2}_{i+3} = 0,\, h_{v-1}c_v = 0,\, h_{v}c_v = 0,\, h_{v+2}c_v = 0$ and $h_{v+3}c_v = 0$ (see Lin [Reference Lin12]).
2. A solution to Singer's conjecture on the rank 4 transfer
As mentioned above, the goal of this section is to verify Singer's conjecture for the algebraic transfer of rank four. To make this, we prove conjecture 1.5. Firstly, let us recall that the domain of $Tr_4(\mathbb {F}_2)$
is isomorphic to the $GL(4)$
-invariants space $(QP_4)_n^{GL(4)}$
as vector spaces for all $n.$
By this and theorem 1.6, we shall compute the dimension of the domain of $Tr_4(\mathbb {F}_2)$
in the internal degrees $n$
satisfying $\mu (n) < q = 4.$
In these cases, due to Sum [Reference Sum27], $n$
is of the following ‘generic’ forms:
whenever $s,\, t,\, u$
are the positive integers. We are now in a position to present our main result.
Main Theorem Let us consider generic degrees in (2.1). Then, conjecture 1.5 is true in these degrees. Further, $Tr_4(\mathbb {F}_2)$
is an isomorphism in these internal degrees, except item (i) with $(s,\, t)\in \{(6,\, 2),\, (5,\, 3)\},$
item (ii) with $(s,\, t) = (2,\, 3)$
and item (iii) with $(s,\, t)\in \{(4,\, 3),\, (1,\, 7)\}.$
In these items, $Tr_4(\mathbb {F}_2)$
is a monomorphism, but it is not an epimorphism.
The theorem has been proved by Bruner, Hà and Hu’ng [Reference Bruner, Hà and Hu’ng5, Prop. 4.4] for item (iii) with $(s,\, t) = (1,\, 2),$
by Hu’ng [Reference Hu’ng9, Thm. 7.3] for item (iii) with $(s,\, t) \in \{(1,\, 7),\, (4,\, 3)\},$
by Sum [Reference Sum29, Thm. 4.1] for item (i) and by the present author for items (ii), (iii) where $t\neq 3$
(see [Reference Phúc20, Reference Phúc21]) and item (iv) (see [Reference Phúc22]). It is remarkable that in [Reference Phúc20, Reference Phúc21], we have proved the theorem for the cases $(s,\, t) \in \{(1,\, 2),\, (1,\, 7)\}$
by another method. Thus, we need only to prove the theorem for items (ii) and (iii) with $t = 3.$
Note again that the case $(s,\, t) = (4,\, 3)$
has been proved by Hu’ng [Reference Hu’ng9]. However, it will be proved in this paper using other techniques. Before going into detail, the known results will be briefly presented as well for the reader's convenience.
We first discuss the theorem for item (i), which has been proved by Sum [Reference Sum29].
Case $n:= n_{s,\, t} = 2^{s+1} - t,\,\ 1\leq t\leq 3,\, s\geq 1$
. According to Sum [Reference Sum29], the dimension of the domain of $Tr_4(\mathbb {F}_2)$
in degree $n_{s\, t}$
is determined by
On the other side, by theorem 1.7, one gets
We see that $n_{s,\, t}$
is bad for $(s,\, t) = (6,\, 2),\, (5,\, 3)$
and that by the equalities (2.2), (2.3), conjecture 1.5 holds for the degrees $n_{s,\, t}$
for $1\leq t\leq 3$
and arbitrary positive integer $s.$
Notice that the non-zero elements $h_0^{3}h_{s+1},$
$h_0^{2}h_s^{2},$
and $h_0^{2}h_6^{2}$
are decomposable in the fourth cohomology groups ${\rm Ext}^{4, 4+n_{s,\, t}}_{\mathscr A}(\mathbb {F}_2,\, \mathbb {F}_2),$
and so they are in the image of $Tr_4(\mathbb {F}_2).$
It is well-known that by Hu’ng [Reference Hu’ng9], the indecomposable elements $D_3(0)$
and $D_3(1)$
are not in the image of $Tr_4(\mathbb {F}_2)$
(see also theorem 1.3) and that by Hà [Reference Hà8], the indecomposable element $d_0$
is in the image of $Tr_4(\mathbb {F}_2)$
(see also theorem 1.4(i)). These results together with the equalities (2.2) and (2.3) show that $Tr_4(\mathbb {F}_2)$
is an isomorphism in degrees $n_{s,\, t}$
, except the degrees $n_{6, 2}$
and $n_{5, 3}.$
In the degrees $n_{6, 2}$
and $n_{5, 3},$
the fourth transfer is a monomorphism, but it is not an epimorphism, since
Next, we discuss the theorem for items (ii), (iii) where $t\neq 3$
and item (iv). In item (iii), the cases $(s,\, t) = (1,\, 2)$
and $(1,\, 7)$
have been proved by Bruner, Hà and Hu’ng [Reference Bruner, Hà and Hu’ng5] and Hu’ng [Reference Hu’ng9], respectively. The remaining cases have been proved by the present author in [Reference Phúc20–Reference Phúc22]. More precisely, with item (ii), we have the following case.
Case $n:=n_{s,\, t} = 2^{s+t+1} +2^{s+1} - 3,\,\ t\geq 1,\, t\neq 3,\, s\geq 1$
. Our previous works [Reference Phúc20, Reference Phúc21] have shown that
On the other hand, from theorem 1.7, we get
Then, the equalities (2.4) and (2.5) show that $n_{s,\, t}$
is not bad and conjecture 1.5 also holds for the degree $n_{s,\, t}$
whenever $t\geq 1,\, t\neq 3,\,$
and $s\geq 1.$
As is well known, $\{h_j|\, j\geq 0\}\subset {\rm Im}(Tr_1(\mathbb {F}_2))$
(see Singer [Reference Singer25]), $\{c_j|\, j\geq 0\}\subset {\rm Im}(Tr_3(\mathbb {F}_2))$
(see Boardman [Reference Boardman4]), $\{e_j|\, j\geq 0\}\subset {\rm Im}(Tr_4(\mathbb {F}_2))$
(see Hà [Reference Hà8] and theorem 1.4(i)), and the ‘total’ transfer $\{Tr_q(\mathbb {F}_2)\}_{q\geq 0}$
is an algebra homomorphism (see Singer [Reference Singer25]). So, together with the equalities (2.4) and (2.5), we assert that $Tr_4(\mathbb {F}_2)$
is an isomorphism in degrees $n_{s, t}$
for each $t\geq 1,\, t\neq 3$
and all $s\geq 1.$
Next, for item (iii) with $t\neq 3,$
we have the following case.
Case $n:=n_{s,\, t}= 2^{s+t} + 2^{s} - 2,\,\ t\geq 1,\, t\neq 3,\, s\geq 1$
. By theorem 1.7, we obtain
Based upon the results in Bruner, Hà and Hu’ng [Reference Bruner, Hà and Hu’ng5, Prop. 4.4], Hu’ng [Reference Hu’ng9, Thm. 7.3] and our previous papers [Reference Phúc20, Reference Phúc21], the dimension of the domain of $Tr_4(\mathbb {F}_2)$
in degree $n_{s,\, t}$
is determined by
From the equalities (2.6) and (2.7), the only degree $n_{1,\, 7}$
is bad and conjecture 1.5 is true for the degree $n_{s,\, t}$
whenever $t\geq 1,\, t\neq 3$
and $s\geq 1.$
Based on theorems 1.3, 1.4, the equalities (2.6), (2.7) and the facts that $\{h_s|\, s\!\geq\! 0\} \subset {\rm Im}(Tr_1(\mathbb {F}_2))$
and the total algebraic transfer $\{Tr_q(\mathbb {F}_2)\}_{q\geq 0}$
is a homomorphism of algebras (see Singer [Reference Singer25]), we may claim that $Tr_4(\mathbb {F}_2)$
is an isomorphism in degrees $n_{s,\, t}$
for all $t\geq 1,\, t\not \in \{3,\, 7\}$
, and any $s \geq 1.$
When $s = 1$
and $t = 7,$
it has been shown in the proof of theorem 7.3 in Hu’ng [Reference Hu’ng9] that the fourth transfer is a monomorphism, but it is not an epimorphism in the internal degree $n_{1, 7}.$
Case $n:=n_{s,\, u,\, t} = 2^{s+t+u} + 2^{s+t} + 2^{s} - 3,\,\ s\geq 1,\, u\geq 1,\, t\geq 1$
. From theorem 1.7 and our previous work in [Reference Phúc22], we have the following results:
We should note that the remaining cases of $s,\, u,\, t$
where $\dim \mathbb {F}_2 \otimes _{GL(4)}P_{\mathscr A}H_{n_{s,\, u,\, t}} (V^{\oplus 4})= 0 = \dim {\rm Ext}^{4, 4+n_{s,\, u,\, t}}_{\mathscr A}(\mathbb {F}_2,\, \mathbb {F}_2)$
are described as follows: $s = 1$
, $u = 1$
and $t\geq 1$
; $s = 2$
, $u = 1$
and $t\geq 2$
; $s\geq 3$
, $u=1$
and $t\geq 1$
; $s = 1$
, $u = 2$
and $t= 1$
; $s = 1$
, $u\geq 3$
and $t\geq 1$
; $s \geq 3$
, $u\geq 2$
and $t= 1$
. It is straightforward to see that $n_{s,\, u,\, t}$
is not bad for all $s,\, t,\, u$
and that by the equalities (2.8), (2.9), conjecture 1.5 holds in the degree $n_{s,\, u,\, t}$
for every $s\geq 1,\, u\neq 1$
and $t\geq 1.$
Because $\{h_s|\, s\geq 0\}\subset {\rm Im}(Tr_1(\mathbb {F}_2))$
(see Singer [Reference Singer25]) and $\{c_s|\, s\geq 0\}\subset {\rm Im}(Tr_3(\mathbb {F}_2))$
(see Boardman [Reference Boardman4]) and the total transfer $\{Tr_q(\mathbb {F}_2)\}_{q\geq 0}$
is a homomorphism of algebras, by equalities (2.8) and (2.9), we may assert that $Tr_4(\mathbb {F}_2)$
is an isomorphism in the degrees $n_{s,\, u,\, t}$
whenever $s\geq 1,\, u\geq 1,\, t\geq 1.$
We now prove the main theorem for items (ii) and (iii) with $t = 3.$
Note that the method of calculation used is similar to our previous works in [Reference Phúc20–Reference Phúc22].
Proof of main theorem. We first consider item (ii) with $t = 3$
, i.e., degree $n:=n_{s,\, 3}= 2^{s+4} +2^{s+1} - 3$
for all $s\geq 1.$
Let us recall that due to Sum [Reference Sum27], the dimension of $(Q\mathcal {P}_4)_{n_{s,\, 3}}$
is given by the following table:
A monomial basis of $(Q\mathcal {P}_4)_{n_{s,\, 3}}$
is also given in the same paper [Reference Sum27]. Taking this basis, together with a computational technique similar to that of our works in [Reference Phúc21, Reference Phúc22], we obtain that the $GL(4)$
-invariant space $(Q\mathcal {P}_4)^{GL(4)}_{n_{s,\, 3}}$
is trivial if $s = 2$
and is 1-dimensional if $s\neq 2.$
As it is known, $\mathbb {F}_2 \otimes _{GL(4)}P_{\mathscr A}H_{n_{s,\, 3}}(V^{\oplus 4})\cong (Q\mathcal {P}_4)^{GL(4)}_{n_{s,\, 3}},$
the dimensions of the domain of $Tr_4(\mathbb {F}_2)$
in degrees $n_{s,\, 3}$
are determined by
On the other hand, from the result by Lin [Reference Lin12], it follows that
So, by the equalities (2.10) and (2.11), we deduce that $n_{s,\, 3}$
is bad for $s =2$
and that conjecture 1.5 holds for the degrees $n_{s,\, 3} = 2^{s+4} +2^{s+1} - 3$
for $s\geq 1.$
Since $h_0h_s^{2}h_{s+4}\in {\rm Im}(Tr_4(\mathbb {F}_2))$
, by theorems 1.3, 1.4 and the equalities (2.10), (2.11), the fourth transfer
is an isomorphism for $s\neq 2$
, and that $Tr_4(\mathbb {F}_2)$
is a monomorphism, but not an epimorphism for $s = 2.$
Finally, we consider item (iii) with $t = 3$
, i.e., degree $n:=n_{s,\, 3} = 2^{s+3} + 2^{s} - 2$
for all $s\geq 1.$
Notice that $n_{s,\, 3}$
is an even degree, and so, the Kameko map
is an epimorphism of $\mathbb {F}_2GL(4)$
-modules. This leads to an estimate
Here $[({\rm Ker}((\overline {Sq}^{0})_{n_{s,\, 3}}))^{GL(4)}]^{*}$
denotes the dual of $({\rm Ker}((\overline {Sq}^{0})_{n_{s,\, 3}}))^{GL(4)}.$
Following Sum [Reference Sum29] and the equality (2.10), the coinvariant $\mathbb {F}_2 \otimes _{GL(4)}P_{\mathscr A}H_{2^{s+2} + 2^{s-1}-3}(V^{\oplus 4})$
is trivial if $s = 1,\, 4$
and is 1-dimensional otherwise. We need to compute the dimension of the $GL(4)$
-invariant $({\rm Ker}((\overline {Sq}^{0})_{n_{s,\, 3}}))^{GL(4)}$
. By Sum [Reference Sum27, Reference Sum28], the dimension of the kenel of $(\overline {Sq}^{0})_{n_{s,\, 3}}$
is determined as follows:
Using this result and a similar computation as in [Reference Phúc20, Reference Phúc22], we claim that $({\rm Ker}((\overline {Sq}^{0})_{n_{s,\, 3}}))^{GL(4)}$
is trivial if $s = 1,\, 2$
and has dimension $1$
if $s\geq 3.$
From these data, the inequality (2.12) implies that
On the other side, due to Lin [Reference Lin12], we find that
It is known, the non-zero elements $h_1^{3}h_5$
, $h_0^{2}h_sh_{s+3},$
for $s\geq 3,$
and $h_1h_{s-1}^{2}h_{s+3}$
for $s\geq 5$
are detected by the fourth transfer. In fact, this could also be directly proved as our previous works [Reference Phúc20–Reference Phúc22] by using $E_1$
-level of $Tr_4(\mathbb {F}_2)$
. For instance, a direct computation shows that
where the elements $a_1^{(1)}a_2^{(2^{s-1}-1)}a_3^{(2^{s-1}-1)}a_4^{(2^{s+3}-1)}$
belong to $P_{\mathscr A}H_{n_{s,\, 3}}(V^{\oplus 4}),$
while the linear transformation $\psi _4$
viewed as a representation in the lambda algebra of $Tr_4(\mathbb {F}_2)$
and determined as in [Reference Cho’n and Hà7]. The above equality implies that $h_1h_{s-1}^{2}h_{s+3}\in {\rm Im}(Tr_4(\mathbb {F}_2))$
for every $s\geq 5.$
Hence, combining with (2.14) and theorems 1.3, 1.4 gives
Noting that by (2.13) and (2.15), the coinvariant space $\mathbb {F}_2 \otimes _{GL(4)}P_{\mathscr A}H_{n_{4,\, 3}}(V^{\oplus 4})$
is 1-dimensional and has been computed by Hu’ng [Reference Hu’ng9, Thm. 7.3] using other techniques. So, his result showed that conjecture 1.5 holds for the degrees $n_{4,\,3}= 2^{4+3} + 2^{4} - 2.$
It is easy to see that the only degree $n_{4,\, 3}$
is bad. These data together with the inequalities (2.13), (2.14) and (2.15) imply that conjecture 1.5 is true for the degrees $n_{s,\,3}= 2^{s+3} + 2^{s} - 2$
for every positive integer $s.$
It is well-known that by Hu’ng [Reference Hu’ng9], the indecomposable element $p'_1$
is not in the image of $Tr_4(\mathbb {F}_2)$
(see also theorem 1.3) and that by Hu’ng and Quỳnh [Reference Hu’ng and Quỳnh10], the indecomposable element $p_1$
is in the image of $Tr_4(\mathbb {F}_2)$
(see also theorem 1.4(iii)). At the same time, as shown above, the decomposable elements $h_1^{3}h_5$
, $h_0^{2}h_sh_{s+3},$
for $s\geq 3,$
and $h_1h_{s-1}^{2}h_{s+3}$
for $s\geq 5$
are in the image of $Tr_4(\mathbb {F}_2).$
So, combining with the inequalities (2.13), (2.14) and (2.15), we conclude that the fourth transfer
is an isomorphism for $s\neq 4$
, and that $Tr_4(\mathbb {F}_2)$
is a monomorphism, but not an epimorphism for $s = 4.$
The proof of the theorem is complete.
Based on the main theorem and the results in Bruner, Hà and Hu’ng [Reference Bruner, Hà and Hu’ng5], Hu’ng [Reference Hu’ng9], Hà [Reference Hà8], Nam [Reference Nam15], Hu’ng and Quỳnh [Reference Hu’ng and Quỳnh10], we obtain the following corollaries.
Corollary 2.1 Let us consider the following generic degrees:
where $s$
is an arbitrary positive integer. Then, conjecture 1.5 holds true in these degrees and Singer's transfer is an isomorphism in the bidegree $(4,\,4+n_s^{(j)})$
for $1\leq j\leq 4$
and all $s >0.$
It is to be noted that in each degree $n_s^{(j)},$
we do not consider the case $s = 0$
since it has been discussed in the proof of the main theorem. Remark that by the previous works in Hà [Reference Hà8], Nam [Reference Nam15], Hu’ng and Quỳnh [Reference Hu’ng and Quỳnh10], Singer's transfer is an epimorphism in the bidegree $(4,\,4+n_s^{(j)})$
for $1\leq j\leq 4$
and all $s >0.$
Indeed, following theorem 1.7, for each positive integer $s,$
we have
By Singer [Reference Singer25], the decomposable elements $h_{s-1}^{2}h_{s+2}h_{s+4},$
$h_{s-1}^{2}h_{s+1}h_{s+3}$
and $h_{s-1}^{2}h_{s+2}h_{s+5}$
are detected by the fourth transfer, $Tr_4(\mathbb {F}_2)$
for any $s >0.$
On the other hand, by theorem 1.4, $\{d_s|\, s\geq 0\}\subset {\rm Im}(Tr_4(\mathbb {F}_2))$
(see Hà [Reference Hà8]), $\{e_s|\, s\geq 0\}\subset {\rm Im}(Tr_4(\mathbb {F}_2))$
(see Hà [Reference Hà8]), $\{f_s|\, s\geq 0\}\subset {\rm Im}(Tr_4(\mathbb {F}_2))$
(see Nam [Reference Nam15]) and $\{p_s|\, s\geq 0\}\subset {\rm Im}(Tr_4(\mathbb {F}_2))$
(see Hu’ng and Quỳnh [Reference Hu’ng and Quỳnh10]). So, the degrees $n_s^{(j)}$
are not bad and $Tr_4(\mathbb {F}_2)$
is an epimorphism in those degrees for $1\leq j\leq 4$
and any $s >0.$
Thus, to prove that $Tr_4(\mathbb {F}_2)$
is an isomorphism in degrees $n_s^{(j)}$
, we need only to show that conjecture 1.5 holds true in these degrees $n_s^{(j)}.$
The proof is presented as follows.
Proof. It is straightforward to check that $\mu (n_s^{(3)}) = 4$
for every $s > 0$
and $\mu (n_s^{(j)}) = 4$
for every $s > 1$
and $j\neq 3.$
Then, one has the following isomorphisms, which are special cases of a result in Hu’ng [Reference Hu’ng9, Cor. 6.2]:
Due to the equality (2.7) and the inequalities (2.13), (2.15) in the proof of the main theorem, we get the equalities of dimensions in (2.18) below (except the concrete values 1 or 2), which is also a special case of Hu’ng [Reference Hu’ng9, Cor. 6.2]:
Thus, the equalities (2.16), (2.17) and (2.18) indicated that conjecture 1.5 is true in degrees $n_s^{(j)}$
for $1\leq j\leq 4$
and all $s >0.$
The corollary follows.
Corollary 2.2 Let us consider the following generic degrees:
where $s$
is an arbitrary positive integer. Then, conjecture 1.5 also holds true in these degrees and Singer's transfer is a monomorphism, but it is not an epimorphism in the bidegree $(4,\,4+n_s^{(j)})$
for $5\leq j\leq 7$
and arbitrary $s\geq 1.$
We remark that by the previous works in [Reference Bruner, Hà and Hu’ng5] and [Reference Hu’ng9], Singer's transfer is not an epimorphism in the bidegree $(4,\,4+n_s^{(j)})$
for $5\leq j\leq 7$
and all $s >0.$
Indeed, due to theorem 1.7, for each positive integer $s,$
one gets
We see that for each integer $s > 0,$
the decomposable elements $h_{s-1}^{2}h_{s+3}h_{s+6}$
and $h_{s-1}^{2}h^{2}_{s+5}$
are detected by the fourth transfer, $Tr_4(\mathbb {F}_2).$
However, by theorem 1.2, $g_s\not \in {\rm Im}(Tr_4(\mathbb {F}_2))$
(see [Reference Bruner, Hà and Hu’ng5]), and by theorem 1.3, $p'_s\not \in {\rm Im}(Tr_4(\mathbb {F}_2))$
(see [Reference Hu’ng9]) and $D_3(s)\not \in {\rm Im}(Tr_4(\mathbb {F}_2))$
(see [Reference Hu’ng9]). So, the degrees $n_s^{(j)}$
are bad and $Tr_4(\mathbb {F}_2)$
is not an epimorphism in those degrees for $5\leq j\leq 7$
and any $s >0.$
Thus, to prove that $Tr_4(\mathbb {F}_2)$
is a monomorphism in degrees $n_s^{(j)}$
, we shall show that conjecture 1.5 holds true in these degrees $n_s^{(j)}.$
Note that the case $j = 5$
was proved by Bruner, Hà and Hu’ng [Reference Bruner, Hà and Hu’ng5]. More explicitly, in [Reference Bruner, Hà and Hu’ng5], the authors show that the coinvariant space $\mathbb {F}_2 \otimes _{GL(4)}P_{\mathscr A}H_{n_s^{(5)}}(V^{\oplus 4})$
is trivial for any $s > 0.$
This together with (2.19) imply that conjecture 1.5 is true in the degree $n_s^{(5)}$
for every positive integer $s.$
We now prove the corollary for the cases $j = 6$
and $7.$
Proof. It is easy to check that $\mu (n_s^{(6)}) = 4,$
for every $s > 1,$
and $\mu (n_s^{(7)}) = 4,$
for every $s > 2.$
So, we get the following isomorphisms, which are special cases of corollary 6.2 in Hu’ng [Reference Hu’ng9]:
On the other hand, from the proof of theorem 7.3 in Hu’ng [Reference Hu’ng9], we have
Then, by the equalities (2.19), (2.20) and (2.21), it may be concluded that conjecture 1.5 is also true in the degree $n_s^{(6)}$
and $n_s^{(7)}$
for any $s > 0.$
The corollary is proved.
Acknowledgments
The results of this work were reported at the Mathematical Conference on Algebra – Number theory – Geometry and Topology (DAHITO), Thai Nguyen Pedagogical University, in October 2021, Vietnam, and at the 27th International Conference of International Academy of Physical Sciences (CONIAPS XXVII) on Recent Advances in Pure and Applied Algebra (RAPAA), Department of Mathematics, National Institute of Technology Jamshedpur, India $\& $
International Academy of Physical Sciences (IAPS), Prayagraj, in October 2021, India. The author is grateful to the organizers and participants of these conferences for fruitful discussions. The author would also like to express his deep gratitude to Professor Nguyen Sum, his thesis advisor, for having proposed this research topic, and having devoted a great deal of time to discuss details of the research with the author. The author appreciates the anonymous referee for his careful corrections to and valuable comments on the original version of this work. Especially, he proposed conjecture 1.5 that led to the improvement of the exposition quality of the paper. I would like to express my sincere gratitude to him for this fun suggestion. (This paper is dedicated to the people of Ukraine and the memory of Professor Reginald Wood.)
