The affirmative answer to Singer's conjecture on the algebraic transfer of rank four

During the last decades, the structure of mod-2 cohomology of the Steenrod ring $\mathscr {A}$ became a major subject in Algebraic topology. One of the most direct attempt in studying this cohomology by means of modular representations of the general linear groups was the surprising work [Math. Z. 202 (1989), 493–523] by William Singer, which introduced a homomorphism, the so-called algebraic transfer, mapping from the coinvariants of certain representation of the general linear group to mod-2 cohomology group of the ring $\mathscr A.$ He conjectured that this transfer is a monomorphism. In this work, we prove Singer's conjecture for homological degree $4.$


Introduction
Everywhere in the text of this article, we will be working over the field F 2 ∼ = Z/2Z of characteristic 2 and taking (co)homology with coefficients in F 2 . It is well-known that the calculation of the stable homotopy groups of spheres π S * (S 0 ) is one of the most central and intractable problems in Algebraic topology. Historically, in the 1950s, Serre [24] used his spectral sequence to study this problem. In the late 1950s, Adams [1] constructed his celebrated spectral sequence that converges to π S * (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 A over F 2 . This cohomology has been explicitly computed by Adem [3] for q = 1, by Adams [2] and Wall [31] for q = 2, by Adams [2] and Wang [32] for q = 3, by Lin [12] for q = 4, by Lin [12] and Chen [6] for q = 5. However, it is still largely mysterious for all q > 5. With an idea that we can study the structure of Ext q A (F 2 , F 2 ) through the modular invariant theory, in 1989, W. Singer Đ. V. Phúc [25] introduced a 'transfer' homomorphism of rank q, which passes from coinvariants of a certain representation of the general linear group GL(q) over F 2 to mod-2 cohomology of A . It has been shown that this transfer is highly nontrivial (see the works by Boardman [4], Minami [13], Bruner, Hà and Hu'ng [5], Hu'ng [9], Hà [8], Nam [15], Hu'ng and Quỳnh [10], Cho'n and Hà [7], Sum [29], the author [17][18][19][20][21][22][23], and others). In order to better understand it, we offer some related issues. Let us denote by V ⊕q ∼ = 1 i q (Z/2Z) a rank q elementary abelian 2-group, which is considered as q-dimensional vector F 2 -space. It is known, H * (V ⊕q ) ∼ = S(V ⊕q * ), the symmetric algebra over the dual space V ⊕q * ≡ H 1 (V ⊕q ) of V ⊕q . Pick u 1 , . . . , u q to be a basis of H 1 (V ⊕q ). Then, it has been shown that P q := H * (V ⊕q ) ∼ = F 2 [u 1 , . . . , u q ], the connected Z-graded polynomial algebra on generators of degree 1, equipped with the canonical unstable algebra structure over A . By dualizing, H * (V ⊕q ) ∼ = Γ(a 1 , . . . , a q ), the divided power algebra generated by a 1 , . . . , a q , each of degree one, where a i ≡ a (1) i is dual to u i . It is to be noted that this algebra and the polynomial algebra P q are not in general isomorphic as F 2 GL(q)-modules. Now, let us recall that the algebra A consists of the Steenrod squaring operations Sq i for i 0. The operations Sq 0 and Sq 2 i , i 0, constitute a system of multiplicative generators for A (see also Walker and Wood [30]). 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.
consisting of all elements that are annihilated by all Sq i for i > 0. The group GL(q) acts regularly on V ⊕q and therefore on P q and H * (V ⊕q ). This action commutes with that of the algebra A and so acts F 2 ⊗ A P q and P A H * (V ⊕q ). Singer [25] constructed a homomorphism from P A H n (V ⊕q ) to Ext q,q+n A (F 2 , F 2 ), which commutes with two Sq 0 's on P A H n (V ⊕q ) and Ext q,q+n A (F 2 , F 2 ) (see also [4,14]). 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 Σ ∞ (B(V ⊕q ) + ) −→ Σ ∞ (S 0 ) between the suspension spectrum in stable homotopy category. Singer [25] demonstrated that the 'total' transfer {T r q (F 2 )} q 0 is an algebra homomorphism and that T r q (F 2 ) is an isomorphism for q = 1, 2. Afterwards, Boardman [4] stated that T r 3 (F 2 ) is also an isomorphism. Remarkably, in mostly all the decade 1980s, Singer believed that T r q (F 2 ) is an isomorphism for all q. However, in the rank 5 case, he himself claimed in [25]  As shown above, Singer's transfer is an isomorphism in ranks 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 [5,8,9,15] and [10]. More precisely, in [5] This result refuted a prediction by Minami in [14] that the localization of the algebraic transfer given by inverting Sq 0 is an isomorphism. More explicitly, when q = 4 and n = 12 · 2 s − 4, following [5, Cor. 1.2], the localization of the fourth transfer given by inverting Sq 0 is not an epimorphism. As a continuation of the work [5], Hu'ng proved in [9] that   Alternatively, since the total transfer {T r q (F 2 )} q 0 is a homomorphism of algebras and T r q (F 2 ) is an isomorphism for q = 1, 2, 3, all decomposable elements in Ext 4 A (F 2 , F 2 ) belong to the image of T r 4 (F 2 ). Now, based on the results of [5,[8][9][10]15] on the image of T r 4 (F 2 ), Singer's conjecture 1.1 for T r 4 (F 2 ) turns out to be equivalent to the following.
if n is not bad.
Here n is called bad if it equals to the stem of one element in the three families {g s | s 1}, {D 3 (s)| s 0} and {p s | s 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.

Đ. V. Phúc
The algebraic transfer we are describing is closely related to the hit problem in literature [16] of determination of a minimal generating set for the F 2 -algebra P q , considered as an unstable A -module. The reader is familiar with an event that if F 2 is an 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 Z-graded vector space over the field F 2 : where the homogeneous components (QP q ) n := (F 2 ⊗ A P q ) n of degrees n are F 2 GL(q)-submodules of QP q . Usually, one would investigate this tensor product. Its structure was systematically depicted by Peterson [16] for q = 1, 2, by Kameko's thesis [11] for q = 3, and by Sum [27,28] for q = 4. So far it has been thoroughly studied for more than three decades by many topologist (see also [4,5,11, 17-19, 21, 23, 26-28, 30, 33]), but it remains unanswered for q 5. We also emphasize that in general, it is not easy to compute or even estimate the dimension of QP q in each positive degree. Most notably, in his thesis [11], Kameko conjectured that an upper bound on the dimension of (QP q ) n is the order of the factor group of GL(q) by the Borel subgroup B q , i.e., for any n 0. However, in 2010, the famous work [26] of Sum refuted the above prediction. In order to reduce the process of the calculation of QP q in each certain degree, one considers the arithmetic function μ : N → N, which is defined by Theorem 1.6. For each non-negative integer n, the following assertions are true: (i) (QP q ) n = 0 if and only if μ(n) > q (see Peterson's conjecture [16], Wood [33]); The statement (i) is given by Peterson's conjecture. Peterson himself confirmed it for q 2. Afterwards, Wood proved the general case under a stronger form.
To close this section, we recall the already known results on Ext q, * is generated by h i h j for j i 0 and j = i + 1 (see Adams [2] and Wall [31]); Adams [2] and Wang [32]); Lin [12]).

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 T r 4 as vector spaces for all n. By this and theorem 1.6, we shall compute the dimension of the domain of T r 4 (F 2 ) in the internal degrees n satisfying μ(n) < q = 4. In these cases, due to Sum [27], 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, T r 4 (F 2 ) is an isomorphism in these internal degrees, The theorem has been proved by Bruner, Hà and Hu'ng [5,Prop. 4 [29,Thm. 4.1] for item (i) and by the present author for items (ii), (iii) where t = 3 (see [20,21]) and item (iv) (see [22]). It is remarkable that in [20,21], we have proved the theorem for the cases (s, t) ∈ {(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 [9]. 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 [29].
Case n := n s, t = 2 s+1 − t, 1 t 3, s 1. According to Sum [29], the dimension of the domain of T r 4 (F 2 ) in degree n s t is determined by , and s 2, 0 if t = 2, and 1 s 2, 1 if t = 2, and s 3, 0 if t = 3, and s 1.
On the other side, by theorem 1.7, one gets   F 2 ), and so they are in the image of T r 4 (F 2 ). It is well-known that by Hu'ng [9], the indecomposable elements D 3 (0) and D 3 (1) are not in the image of T r 4 (F 2 ) (see also theorem 1.3) and that by Hà [8], the indecomposable element d 0 is in the image of T r 4 (F 2 ) (see also theorem 1.4(i)). These results together with the equalities (2.2) and (2.3) show that T r 4 (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 = 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 [5] and Hu'ng [9], respectively. The remaining cases have been proved by the present author in [20][21][22]. More precisely, with item (ii), we have the following case.
(2.5) 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 1, t = 3, and s 1.
Next, for item (iii) with t = 3, we have the following case.
Case n := n s, t = 2 s+t + 2 s − 2, t 1, t = 3, s 1. By theorem 1.7, we obtain if t = 4, and s 5,  [20,21], the dimension of the domain of T r 4 (F 2 ) in degree n s, t is determined by (2.7) 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 1, t = 3 and s 1.
Based on theorems 1.3, 1.4, the equalities (2.6), (2.7) and the facts that {h s | s 0} ⊂ Im(T r 1 (F 2 )) and the total algebraic transfer {T r q (F 2 )} q 0 is a homomorphism of algebras (see Singer [25]), we may claim that T r 4 (F 2 ) is an isomorphism in degrees n s, t for all t 1, t ∈ {3, 7}, and any s 1. When s = 1 and t = 7, it has been shown in the proof of theorem 7.3 in Hu'ng [9] that the fourth transfer is a monomorphism, but it is not an epimorphism in the internal degree n 1,7 .
We should note that the remaining cases of s, u, t where dim F 2 ⊗ GL(4) P A H ns, u, t (V ⊕4 ) = 0 = dim Ext 4,4+ns, u, t A (F 2 , F 2 ) are described as follows: s = 1, u = 1 and t 1; s = 2, u = 1 and t 2; s 3, u = 1 and t 1; s = 1, u = 2 and t = 1; s = 1, u 3 and t 1; s 3, u 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 1, u = 1 and t 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 [20][21][22].
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 1. Let us recall that due to Sum [27], the dimension of (QP 4 ) ns, 3 is given by the following table: 3 136 180 195 A monomial basis of (QP 4 ) ns, 3 is also given in the same paper [27]. Taking this basis, together with a computational technique similar to that of our works in [21,22], we obtain that the GL(4)-invariant space (QP 4 ) is trivial if s = 2 and is 1-dimensional if s = 2. As it is known, (4) ns, 3 , the dimensions of the domain of T r 4 (F 2 ) in degrees n s, 3 are determined by On the other hand, from the result by Lin [12], it follows that (2.11) 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 1.
It is well-known that by Hu'ng [9], the indecomposable element p 1 is not in the image of T r 4 (F 2 ) (see also theorem 1.3) and that by Hu'ng and Quỳnh [10], the indecomposable element p 1 is in the image of T r 4 (F 2 ) (see also theorem 1.4(iii)). At the same time, as shown above, the decomposable elements h 3 1 h 5 , h 2 0 h s h s+3 , for s 3, and h 1 h 2 s−1 h s+3 for s 5 are in the image of T r 4 (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 = 4, and that T r 4 (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 [5], Hu'ng [9], Hà [8], Nam [15], Hu'ng and Quỳnh [10], we obtain the following corollaries. It is to be noted that in each degree n (j) s , 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 [8], Nam [15], Hu'ng and Quỳnh [10], Singer's transfer is an epimorphism in the bidegree (4, 4 + n (j) s ) for 1 j 4 and all s > 0. Indeed, following theorem 1.7, for each positive integer s, we have (2.16) By Singer [25], the decomposable elements h 2 s−1 h s+2 h s+4 , h 2 s−1 h s+1 h s+3 and h 2 s−1 h s+2 h s+5 are detected by the fourth transfer, T r 4 (F 2 ) for any s > 0. On the other hand, by theorem 1.4, {d s | s 0} ⊂ Im(T r 4 (F 2 )) (see Hà [8]), {e s | s 0} ⊂ Im(T r 4 (F 2 )) (see Hà [8]), {f s | s 0} ⊂ Im(T r 4 (F 2 )) (see Nam [15]) and {p s | s 0} ⊂ Im(T r 4 (F 2 )) (see Hu'ng and Quỳnh [10]). So, the degrees n (j) s are not bad and T r 4 (F 2 ) is an epimorphism in those degrees for 1 j 4 and any s > 0. Thus, to prove that T r 4 (F 2 ) is an isomorphism in degrees n (j) s , we need only to show that conjecture 1.5 holds true in these degrees n (j) s . The proof is presented as follows.
Proof. It is straightforward to check that μ(n , for all s 2 and j = 3. (2.17) 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 [9, Cor. 6.2]:  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 (j) s ) for 5 j 7 and arbitrary s 1.
We remark that by the previous works in [5] and [9], Singer's transfer is not an epimorphism in the bidegree (4, 4 + n (j) s ) for 5 j 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 2 s−1 h s+3 h s+6 and h 2 s−1 h 2 s+5 are detected by the fourth transfer, T r 4 (F 2 ). However, by theorem 1.2, g s ∈ Im(T r 4 (F 2 )) (see [5]), and by theorem 1.3, p s ∈ Im(T r 4 (F 2 )) (see [9]) and D 3 (s) ∈ Im(T r 4 (F 2 )) (see [9]). So, the degrees n (j) s are bad and T r 4 (F 2 ) is not an epimorphism in those degrees for 5 j 7 and any s > 0. Thus, to prove that T r 4 (F 2 ) is a monomorphism in degrees n (j) s , we shall show that conjecture 1.5 holds true in these degrees n (j) s . Note that the case j = 5 was proved by Bruner, Hà and Hu'ng [5]. More explicitly, in [5], the authors show that the coinvariant space F 2 ⊗ GL(4) P A H n (5) s (V ⊕4 ) is trivial for any s > 0. This together with (2.19) imply that conjecture 1.5 is true in the degree n (5) s for every positive integer s. We now prove the corollary for the cases j = 6 and 7.
By the main theorem and corollaries 2.1, 2.2, we see that Corollary 2.3. Conjecture 1.5 is true for all n and so is, conjecture 1.1 for T r 4 (F 2 ).