On generators of the unstable \mathscr A-module H^{*}((K(\mathbb F_2, 1))^{\times 5}) in a generic degree and applications
aa r X i v : . [ m a t h . A T ] F e b On the hit problem for the unstable A -module P = H ∗ (( K ( F , × , F ) and application D- ă. ng Võ Phúc
Faculty of Education Studies, University of Khánh Hòa,01 Nguyễn Chánh, Nha Trang, Khánh Hòa, Viet Nam
Abstract
Let us consider the prime field of two elements, F . One of the open problems in Algebraictopology is the hit problem for a module over the mod 2 Steenrod algebra A . The problem asksa minimal set of generators for the polynomial algebra P m := F [ x , x , . . . , x m ] regarded as aconnected unstable A -module on m variables x , . . . , x m , each of degree one. The algebra P m isthe cohomology with F -coefficients of the product of m copies of the Eilenberg-MacLan complex K ( F , . The hit problem has been thoroughly studied for 35 years in a variety of contexts bymany authors and completely solved for m ≤ . Furthermore, it has been closely related to someclassical problems in the homotopy theory and applied in investigating the m -th Singer algebraictransfer T r A m [34]. This transfer is one of the useful tools for describing the Adams E -term,Ext m,m + ∗ A ( F , F ) = H m,m + ∗ ( A , F ) . The aim of this work is to continue our study of the hit problem of five variables. At the sametime, this result will be applied to the investigation of the fifth transfer of Singer and the modularrepresentation of the general linear group of rank 5 over F . More precisely, we extend a result in[28] on the hit problem for A -module P in the generic degree 5(2 t −
1) + 18 . t with t ≥ . Theresult confirms Sum’s conjecture [38] on the relation between the minimal set of A -generators forthe polynomial algebras P m − and P m in the case m = 5 and the above generic degree. Moreover,by using our result [28] and a presentation in the λ -algebra of T r A , we show that the non-trivialelement h e = h f ∈ Ext , − . ) A ( F , F ) is in the image of the fifth transfer and that T r A is an isomorphism in the bidegree (5 , −
1) + 18 . )) . In addition, the behavior of
T r A in the bidegree (5 , t −
1) + 18 . t )) when t ≥ Keywords:
Adams spectral sequences; Steenrod algebra; Lambda algebra; Peterson hit problem; Algebraic transfer
Let O S ( i, F , F ) denote the set of all stable cohomology operations of degree i, with coefficientin the prime field F . Then, the F -algebra A := L i ≥ O S ( i, F , F ) is called the mod 2 Steenrodalgebra . In other words, the algebra A is the algebra of stable operations on the mod 2 cohomology.In [17], Milnor observed that this algebra is also a graded connected cocommutative Hopf algebraover F . Therefore, its dual A ∗ is a commutative algebra and in the same paper, Milnor showed that A ∗ is a polynomial algebra. In many cases, the resulting A -module structure on H ∗ ( X, F ) providesadditional information about the topological space X ; for instance, the CW-complexes C P / C P and S ∨ S have cohomology rings that agree as a graded commutative F -algebras, but are differentas a module over A . Later, the Steenrod algebra is widely studied by mathematicians whoseinterests range from algebraic topology and homotopy theory to manifold theory, combinatorics,representation theory, and more.
Email address: [email protected] (D- ă. ng Võ Phúc)
Preprint submitted to Elsevier s is well-known that the cohomology with F -coefficients of the Eilenberg-MacLan complex K ( F ,
1) is isomorphic to F [ x ] , the polynomial ring of degree 1 in one variable. Hence, based uponthe cohomology K¨unneth formula, we have an isomorphism of F -algebras P m := H ∗ (( K ( F , × m , F ) (cid:27) F [ x ] ⊗ F F [ x ] ⊗ F · · · ⊗ F F [ x m ] | {z } m times (cid:27) F [ x , . . . , x m ] , in which x i ∈ H (( K ( F , × m , F ) for every i. Since P m is the cohomology of a CW-complex, itis equipped with a structure of unstable module over A . It has been known (see also [36]) that A is spanned by the Steenrod squares Sq i of degree i for i ≥ P m is given by Sq i ( x t ) = x t if i = 0 ,x t if i = 1 , ( the unstable condition )0 if i > , and Sq i ( uv ) = X α i Sq α ( u ) Sq i − α ( v ) , for all u, v ∈ P m ( the Cartan formula ) . The study of the hit problems was initiated in a variety of contexts by Peterson [23], Priddy [32],Singer [35], and Wood [42]. The problem seeks a minimal set of A -generators for P m . Nowadays,it is a central problem of Algebraic topology and applied to homotopy theory. The hit problemhas been proved surprisingly difficult by many authors. (See [8], [9], [10], [18, 19], [20], [22],[24, 25, 26, 27, 28], [37, 38], [40], and others.)Let us recall that if consider F as a trivial A -module, then solving the hit problem is todetermine a basis of F -graded vector space { ( P m ) n / (( P m ) n ∩ A P m ) } n ≥ , where A is generated bythe Steenrod squares of positive degrees, and ( P m ) n denotes the F -subspace of P m consisting of allthe homogeneous polynomials of degree n in P m . (Note that A P m is the set of the "hit" elementsin A -module P m . ) The above vector space is isomorphic to Q ⊗ m := F ⊗ A P m , a representationof the general linear group GL m := GL ( m, F ) over F . The structure of Q ⊗ m has been treated for m ≤
4: see [23, 10, 37]. Furthermore, it plays an important role in the study of the E -term of theAdams spectral sequence (Adams SS), Ext m,m + ∗ A ( F , F ) via the m -th Singer algebraic transfer [34] T r A m : ( F ⊗ GL m P A (( P m ) ∗ )) n → Ext m,m + n A ( F , F ) = H m,m + n ( A , F ) . Here P A (( P m ) ∗ ) denotes the subspace of ( P m ) ∗ consisting of all A -annihilated elements. Note that( P m ) ∗ = H ∗ (( K ( F , × m , F ) (cid:27) Γ( a , . . . , a m ) , the divided power algebra generated by a , . . . , a m , where a j = a (1) j is dual to x j with respect tothe basis of P m consisting of all monomials in x , . . . , x m . It is emphasized that the algebra ( P m ) ∗ has a right A -module structure. The right action of A on this algebra is given by( a ( j ) t ) Sq k = j − kk ! a ( j − k ) t = Sq k ∗ ( a ( j ) t )and Cartan’s formula, where Sq k ∗ denotes the dual Steenrod squares. A natural question arises:Why do we need to calculate the E -term of the Adams SS? Because this is involved in determiningthe stable homotopy groups of spheres. These groups are pretty fundamental and interesting.However, they are also not fully-understood subjects yet. Therefore, computing explicitly theseproblems is an important task of Algebraic topology.It has been shown (see [3], [34]) that the algebraic transfer is highly nontrivial, more precisely,that T r A m is an isomorphism for 0 < m < L m ≥ T r A m is a homomorphism of bigradedalgebras with respect to the product by concatenation in the domain and the usual Yoneda productfor the Ext group. Moreover, Singer also showed in [34] that T r A is an isomorphism in someinternal degrees and that T r A is not an epimorphism in the internal degree nine by using invarianttheory tools. In the same paper, he then made the following prediction. onjecture 1. The m -th transfer map is a monomorphism, for all m > . Singer’s conjecture has been studied for over 30 years by many mathematicians (e.g., [3], [5], [7],[9], [11], [16], [21], [26, 27, 28, 29, 30], [38]). Unfortunately, this conjecture is only confirmed for m ≤ A -module P m is still open for m ≥ . The subject of thiswork is the case of five variables. More explicitly, we extend our result in [28] on the hit problemfor P in the generic degree 5(2 t −
1) + 18 . t with t ≥ . Using this result for t = 0 , we examineConjecture 1 for T r A in respective degree.From now on we denote by Q ⊗ mn the F -subspace of Q ⊗ m consisting of all the classes representedby the elements in ( P m ) n . Our primary tool for invesgating the hit problem is the F GL m -modulesepimorphism of Kameko [10]:( g Sq ∗ ) ( m,m +2 n ) : Q ⊗ mm +2 n −→ Q ⊗ mn . [ Q ≤ j ≤ m x a j j ] [ Q ≤ j ≤ m x aj − j ] if a j odd, j = 1 , , . . . , m, . This map induces the dual homomorphism g Sq ∗ : ( Q ⊗ mm +2 n ) GL m → ( Q ⊗ mn ) GL m of the so-called Kameko g Sq , g Sq : ( F ⊗ GL m P A (( P m ) ∗ )) n → ( F ⊗ GL m P A (( P m ) ∗ )) n + m . Note that ( F ⊗ GL m P A (( P m ) ∗ )) n is isomorphic as an F -vector space to ( Q ⊗ mn ) GL m , the subspaceof invariants under the usual action of the group GL m . Since A is a cocommutative Hopf algebra,there is the squaring operation Sq : Ext m,m + n A ( F , F ) = H m,m + n ( A , F ) → Ext m, m +2 n A ( F , F ) = H m, m +2 n ( A , F ) , which share most of the properties with Sq i on the cohomology of spaces (see [15]). However,it is not the identity in general. (We note that Sq is a special case of the squaring operations Sq i : Ext s,t A ( U , V ) → Ext s + i, t A ( U , V ) , where U is an A -coalgebra and V is an A -algebra.) Moreover,the squaring operation Sq commutes with the Kameko g Sq via the m -th Singer transfer (see [3],[16]). In [10], Kameko showed that if m = ξ ( n ) = min n γ ∈ N : α ( n + γ ) ≤ γ o , then the map ( g Sq ∗ ) ( m,m +2 n ) is an isomorphism of F GL m -modules. Here, α ( n ) denotes the numberof 1’s in the dyadic expansion of the positive integer n. Based upon this event and a result in Wood[42], we need only to study the structure of Q ⊗ mn in the "generic" degree n of the "special" form (seealso [27]): r (2 t −
1) + d. t , where 0 < ξ ( d ) < r < m, and t ≥ . We now survey our main results. Let us consider the above generic degree with r = m = 5, d = 18 and t ≥ . Then, we have the following remarks.
Remark 2.
Since α ( d +1) = 3 > , α ( d +2) = 2 , α ( d +3) = 3 , α ( d +4) = 3 < , etc, ξ ( d ) = 2 < . Moreover, it is easy to see that ξ (5(2 t −
1) + 18 . t ) = 5 for any t > . So, the iterated Kameko map(( g Sq ∗ ) (5 , t − . t ) ) t − : Q ⊗ t − . t → Q ⊗ − . is an isomorphism of F GL -modules for all t ≥ . This follows thatdim( Q ⊗ t − . t ) = dim( Q ⊗ − . ) , for t ≥ . Therefore, we need only to study Q ⊗ t − . t for t = 0 and t = 1 . The case t = 0 has beenexplicitly computed by us in [28]. For t = 1 , we see that the Kameko map( g Sq ∗ ) (5 , − . ) : Q ⊗ − . → Q ⊗ − . s an epimorphism of F GL -modules, hence Q ⊗ − . (cid:27) Ker(( g Sq ∗ ) (5 , − . ) ) M Q ⊗ − . . Since Q ⊗ is 730-dimensional in degree 5(2 −
1) + 18 . (see [28]), we need to determine the kernelof the homomorphism ( g Sq ∗ ) (5 , − . ) . Our method can be summarized as follows:(i) A mononial in P is assigned a weight vector ω of degree 5(2 − . , which stems from thebinary expansion of the exponents of the monomial. The space of indecomposable elementsKer(( g Sq ∗ ) (5 , − . ) ) is then decomposed into a direct sum of ( Q ⊗ − . ) and thesubspaces ( Q ⊗ ) ω > indexed by the weight vectors ω. Here [ F ] ω = [ G ] ω in ( Q ⊗ − . ) ω if the polynomial ( F − G ) of degree 5(2 −
1) + 18 . is hit, modulo a sum of monomials ofweight vectors less than ω. From the previous results of Peterson [23], Kameko [10], Sum [37],and of us [30], we can easily determine ( Q ⊗ − . ) . (ii) The monomials in a given degree are lexicographically ordered first by weight vectors and thenby exponent vectors. This leads to the concept of admissible monomial below: a monomial isadmissible if, modulo hit elements, it is not equal to a sum of monomials of smaller orders.The space ( Q ⊗ − . ) ω > above is easily seen to be isomorphic to the space generated byadmissible monomials of the weight vector ω. (iii) In a given (small) degree, we first lists all possible weight vectors of an admissible monomial.This is done by first using a criterion of Singer [34] on the hit monomials, and then combiningwith a result of Kameko [10] and Sum [37] of the form " X Z r (or ZY t ) admissible implying Z admissible, under some mild conditions".(iv) Finally, we then claims the (strict) inadmissibility of some explicit monomials. The proof isgiven for a typical monomial in each case by explicit computations. Then, based on some ho-momorphisms of Sum [37] and recent results in [18, 24], we obtain a basis of ( Q ⊗ − . ) ω > . This approach is important and different from one of Kameko [10]. The MAGMA computeralgebra system [14] has been used for experimentation leading up to some of the results below.In order to state our main result precisely, we provide some notations and related notions.
Weight vector and exponent vector . In what follows: For a natural number n, denote by α t ( n ) the t -th coefficients in dyadic expansion of n. This means α ( n ) = P t ≥ α t ( n ) . Further, n can be represented as follows: n = P t ≥ α t ( n )2 t , where α t ( n ) ∈ { , } , t = 0 , , . . . . Consider themonomial X = x u x u . . . x u m m ∈ P m , we define two sequences associated with X by ω ( X ) := ( ω ( X ) = X ≤ j ≤ m α ( u j ) , ω ( X ) = X ≤ j ≤ m α ( u j ) , . . . , ω j ( X ) = X ≤ j ≤ m α t − ( u j ) , . . . )and ( u , u , . . . , u m ) . They are called the weight vector and the exponent vector of X, respectively.Let ω = ( ω , ω , . . . , ω t , . . . ) be a sequence of non-negative integers. Then, the sequence ω arecalled the weight vector if ω t = 0 for t ≫ . We define deg( ω ) = P t ≥ t − ω t . The sets of all theweight vectors and the exponent vectors are given the left lexicographical order.For a weight vector ω, we denote two subspaces associated with ω : P ≤ ωm = h{ X ∈ P m | deg( X ) = deg( ω ) , ω ( X ) ≤ ω }i , P <ωm = h{ X ∈ P m | deg( X ) = deg( ω ) , ω ( X ) < ω }i . Let us now consider the homogeneous polynomials F and G in P m with deg( F ) = deg( G ) . Wedefine the following binary relations " ≡ " and " ≡ ω " on P m (see [27]):(i) F ≡ G if and only if ( F − G ) ∈ A P m . If F ≡ F is "hit" (i.e., F can be written as afinite sum F = P i> Sq i ( F i ) for some polynomials F i ). ii) F ≡ ω G if and only if F, G ∈ P ≤ ωm and ( F − G ) ∈ (( A P m ∩ P ≤ ωm ) + P <ωm ) . By a simple computation, we conclude that the above binary relations are equivalence ones. Then,from the equivalence relation " ≡ ω ", we have the quotient space (see [27])( Q ⊗ m ) ω = P ≤ ωm / (( A P m ∩ P ≤ ωm ) + P <ωm ) , and dim Q ⊗ mn = P deg( ω )= n dim( Q ⊗ m ) ω , dim( Q ⊗ mn ) GL m ≤ P deg( ω )= n dim(( Q ⊗ m ) ω ) GL m . Assumethat X = x u x u . . . x u m m and Y = x v x v . . . x v m m are the monomials of the same degree in P m . Wewrite u, v for the exponent vectors of X and Y, respectively. We say that u < v if there is a positiveinteger d such that u j = v j for all j < d and u d < v d , and that X < Y if and only if one of thefollowing holds:(i) ω ( X ) < ω ( Y );(ii) ω ( X ) = ω ( Y ) and u < v. Admissible monomial and inadmissible monomial . A monomial X ∈ P m is said tobe inadmissible if there exist monomials Y , Y , . . . , Y k such that Y j < X for 1 ≤ j ≤ k and X ≡ P ≤ j ≤ k Y j . Then, X is said to be admissible if it is not inadmissible.Obviously, the set of all the admissible monomials of degree n in P m is a minimal set of A -generators for P m in degree n. So, Q ⊗ mn is an F -vector space with a basis consisting of all theclasses represent by the admissible monomials of degree n in P m . Let P m and P > m denote the A -submodules of P m spanned all the monomials x t x t . . . x t m m suchthat Q ≤ j ≤ m t j = 0 , and Q ≤ j ≤ m t j > , respectively. We put ( Q ⊗ m ) := F ⊗ A P m , and ( Q ⊗ m ) > := F ⊗ A P > m . Then, as well known, we have Q ⊗ m = ( Q ⊗ m ) L ( Q ⊗ m ) > . For a polynomial F ∈ P m , we denote by [ F ] the classes in Q ⊗ m represented by F. If ω is a weightvector and F ∈ P ≤ ω , then denote by [ F ] ω the classes in ( Q ⊗ m ) ω represented by F. For a subset B ⊂ P m , denote by [ B ] = { [ F ] : F ∈ B } . If B ⊂ P ≤ ωm , then denote by [ B ] ω = { [ F ] ω : F ∈ B } . Let us denote C ⊗ mn the set of all admissible monomials of degree n in P m . Assume that ω is aweight vector of degree n. We denote by( C ⊗ mn ) ω := C ⊗ mn ∩ P ≤ ωm , ( C ⊗ mn ) ω := C ωm ∩ P m , ( C ⊗ mn ) ω > := C ωm ∩ P > m , ( Q ⊗ mn ) ω := ( Q ⊗ m ) ω ∩ ( Q ⊗ mn ) , and ( Q ⊗ mn ) ω > := ( Q ⊗ m ) ω ∩ ( Q ⊗ mn ) > . Note that [( C ⊗ mn ) ω ] ω , [( C ⊗ mn ) ω ] ω and [( C ⊗ mn ) ω > ] ω are respectively the bases of the F -vectorspaces ( Q ⊗ mn ) ω , ( Q ⊗ mn ) ω and ( Q ⊗ mn ) ω > . We now return to the kernel of the Kameko map ( g Sq ∗ ) (5 , − . ) . Based on the previousresults in [10, 34, 37, 39], we obtain the following, which is one of main results of this note.
Theorem 3.
Consider the weight vector ω = (3 , , , , of degree −
1) + 18 . . Then, wehave an isomorphism
Ker( g Sq ∗ ) (5 , − . ) (cid:27) ( Q ⊗ − . ) M ( Q ⊗ − . ) ω > . Remark 4.
We pointed out in [30] that( Q ⊗ n ) (cid:27) M ≤ s ≤ M ℓ ( J )= s ( Q ⊗ J n ) > , where Q ⊗J = h [ x t j x t j . . . x t s j s ] | t i ∈ N , i = 1 , , . . . , s }i ⊂ Q ⊗ with J = ( j , j , . . . , j s ) , ≤ j <. . . < j s ≤ m , 1 ≤ s ≤ , and ℓ ( J ) := s denotes the length of J . This implies thatdim(( Q ⊗ n ) )) = X ≤ s ≤ s ! dim(( Q ⊗ sn ) > ) , or all n ≥ . On the other hand, since ξ (5(2 −
1) + 18 . ) = 3 , by Peterson [23] and Wood[42], the spaces Q ⊗ − . and Q ⊗ − . are trivial. By Kameko [10], ( Q ⊗ − . ) > is15-dimensional. (We note that Q ⊗ − . is isomorphic to Q ⊗ . ) By Sum [37], ( Q ⊗ − . ) > has dimension 165 . From these data, we claimdim(( Q ⊗ − . ) = 15 . ! + 165 . ! = 975 . We compute ( Q ⊗ − . ) ω > by explicitly determining the set ( C ⊗ − . ) ω > . Then, weget the following theorem.
Theorem 5.
There exist exactly admissible monomials of degree −
1) + 18 . in P > suchthat their weight vectors are ω. This implies that ( Q ⊗ − . ) ω > has dimension . Combining Remarks 2 and 4, Theorems 3 and 5 with the fact that Q ⊗ = ( Q ⊗ ) L ( Q ⊗ ) > , wededuce the following. Corollary 6.
For the generic degree t −
1) + 18 . t , we find that Q ⊗ t − . t is -dimensionalif t = 0 and that Q ⊗ t − . t is -dimensional if t ≥ . In [38], Sum gave a conjecture on the relation between the minimal set of A -generators for thepolynomial algebras P m and P m − . This conjecture helps us to reduce remarkably in solving thehit problem in some generic degrees. From the results of Peterson [23], Kameko [10] and Sum [37],the conjecture satisfies for m ≤ . Sum proved in [38] that the conjecture is true in the 5-variablecase and the degree 5(2 t −
1) + 10 . t for t a non-negative integer. By previous results of the presentauthor and Sum (see [24, 25, 26, 27, 28]), the conjecture is also true for m = 5 and in the degrees4 . (2 t − , t −
1) + 6 . t , t −
1) + 8 . t , and 4 . t − t ≥ , t ∈ Z . Based on the proofof Theorem 5, we claim that Sum’s conjecture holds also in generic degree 5(2 t −
1) + 18 . t . As an application of Corollary 6, we use this result for the case t = 0 , together with theresults of Lin [12], and Chen [6] on the Adams E -term, Ext , ∗ A ( F , F ) , to obtain informationabout the behavior of the fifth Singer algebraic transfer in bidegree (5 , −
1) + 18 . )).More precisely, by Lin [12], and Chen [6], we have Ext , t − . t ) A ( F , F ) = h h t +1 e t i and0 , h t +1 e t = h t f t for all t ≥ . where h t = ( Sq ) t ( h ) is the Adams element in Ext , t A ( F , F )(see [1]) and 0 , f t = ( Sq ) t ( f ) ∈ Ext , . t +1 A ( F , F ) , for any t ≥ . So, to invesgative
T r A in the above bidegree, we compute the dimension of ( F ⊗ GL P A (( P ) ∗ )) − . by using abasis of the F -vector space Q ⊗ − . . (We note that the computation of the GL m -coinvariants F ⊗ GL m P A (( P m ) ∗ ) 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 thegeneral problem.) Moreover, this method can be used to determine the dimensions of the spaces( F ⊗ GL m P A (( P m ) ∗ )) n in small degrees n for m ≥ . In a previous paper [28], we have proven that
Proposition 7.
The following statements are true: i) If Y ∈ C ⊗ − . , then ω := ω ( Y ) is one of the following sequences: ω [1] := (2 , , , , ω [2] := (2 , , , ω [3] := (2 , , ,ω [4] := (4 , , , , ω [5] := (4 , , , ω [6] := (4 , , . ii) | ( C ⊗ − . ) ω [ k ] | =
300 if k = 1 ,
15 if k = 2 , ,
10 if k = 3 ,
110 if k = 4 ,
280 if k = 6 . ote that | ( C ⊗ − . ) ω [ k ] | = | ( C ⊗ − . ) ω > k ] | for k = 2 ,
3, and | ( C ⊗ − . ) ω | =0 = | ( C ⊗ − . ) ω | . Moreover, dim( Q ⊗ − . ) = P ≤ k ≤ | ( C ⊗ − . ) ω [ k ] | = 730 . Usingthese results, we explicitly compute the action of the group GL on the subspaces ( Q ⊗ − . ) ω [ k ] , for 1 ≤ k ≤ . Theorem 8.
We have the following: i) (( Q ⊗ − . ) ω [ k ] ) GL = 0 , k = 1 , , , , . ii) (( Q ⊗ − . ) ω [4] ) GL = h [ µ ] ω [4] i , where µ = x x x x x + x x x x x + x x x x x + x x x x x + x x x x x + x x x x x + x x x x x + x x x x x + x x x x x + x x x x x + x x x x x + x x x x x . Since ( F ⊗ GL P A (( P ) ∗ )) . (2 − . F − vectorspace (cid:27) ( Q ⊗ . (2 − . ) GL , by Theorem 8, we claimdim( F ⊗ GL P A (( P ) ∗ )) . (2 − . = dim( Q ⊗ . (2 − . ) GL ≤ X ≤ k ≤ dim(( Q ⊗ − . ) ω [ k ] ) GL ≤ . (0.1)Now, as well known, the Lambda algebra Λ is one of the tools to study the Adams E -term,Ext m,m + ∗ A ( F , F ) . This algebra was constructed in the "six author paper" [4], and numerous authorshave studied it over the past half-century. Moreover, it is viewed as the E -term of the classicalAdams SS converging to the 2-component of the stable homotopy groups of spheres. Let us recallthat Λ is the bigraded differential F -algebra generated by λ d ∈ Λ ,d of degree d for d ≥ λ i λ i + d +1 = X j ≥ d − j − j ! λ i + d − j λ i +1+ j ( i ≥ , d ≥ . The differential is given by ∂ ( λ d − ) = X j ≥ d − j − j ! λ d − j − λ j − ( d ≥ . Denote by Λ m, ∗ the F -vector subspace of Λ spanned all monomials in λ d of the length m. It iswell-known that this subspace has a basis consisting of all admissible monomials (i.e., those of theform Q ≤ k ≤ m λ d k , where d k ≤ d k +1 for all 1 ≤ k ≤ m − δ = { δ m | m ≥ } : { ( P m ) ∗ | m ≥ } → { Λ m, ∗ | m ≥ } = Λ , which induces the Singer algebraic transfer. Here the map δ m : ( P m ) ∗ → Λ m, ∗ is considered as apresentation of T r A m over the algebra Λ and given in terms of generating function as follows: δ m : a [ x , x , . . . , x m ] → λ [ v , v , . . . , v m ] , where a [ x , x , . . . , x m ] denotes the formal sum P j ,...,j m ≥ ( Q ≤ k ≤ m a ( j k ) k )( Q ≤ k ≤ m x j k k ) . This map isnot an A -homomorphism and determined by the following inductive formula: δ m ( a ( j ,...,j m ) ) = ( λ j if m = 1 , P i ≥ j m δ m − ( Q ≤ k ≤ m − Sq i − j m ∗ ( a ( j k ) k )) λ i if m > , for any a ( j ,...,j m ) := Q ≤ k ≤ m a ( j k ) k ∈ ( P m ) ∗ . Here Sq i − j m ∗ ( a ( j k ) k ) = a ( j k ) k Sq i − j m = (cid:0) j k − i + j m i − j m (cid:1) a ( j k − i + j m ) k with the binomial coefficient reduced mod 2 . They pointed out in the same paper [7] that if a ( j ,...,j m ) ∈ P A (( P m ) ∗ ) , then δ m ( a ( j ,...,j m ) ) is a cycle in Λ m, ∗ and T r A m ([ a ( j ,...,j m ) ]) = [ δ m ( a ( j ,...,j m ) )] . By a direct computation using this data for m = 5 and Theorem 8, we conclude that roposition 9. The non-zero element h e ∈ Ext , − . ) A ( F , F ) is in the image of T r A . The above proposition is also proved by the following facts: The non-zero elements h and e are in the image of T r A and T r A , respectively (see [34], [11]). In addition, L m ≥ T r A m is a homo-morphism of algebras. However, we proved Proposition 9 by using the representation of T r A overthe Lambda algebra. This method can be applied to homological degrees higher under certain con-ditions. For instance (see [7]), the elements h P h ∈ Ext , A ( F , F ) , and h P h ∈ Ext , A ( F , F )are not in the image of the Singer transfer map. Thus, since Ext , − . ) A ( F , F ) is one-dimensional, by Proposition 9, we get(0.2) dim( F ⊗ GL P A (( P ) ∗ )) − . ≥ . Then, combining the inequalities (0.1) and (0.2), we find that
Corollary 10.
The fifth transfer is an isomorphism when acting on ( F ⊗ GL P A (( P ) ∗ )) − . . This result confirms that Conjecture 1 is true in the case of rank 5 and the internal degree5(2 t −
1) + 18 . t for t = 0 . Comments and open issues.
From the above results, we can see a quite interesting eventthat the F -vector space Q ⊗ is 730-dimensional in degree 5(2 −
1) + 18 . but the space of its GL -coinvariants is only one-dimensional. In general, it is quite efficient in using the results of the hitproblem of five variables to study F ⊗ GL P A (( P ) ∗ ) . This provides a valuable method for verifyingSinger’s open conjecture on the fifth algebraic transfer. Now, we will discuss on Conjecture 1 in thecase m = 5 and the degree 5(2 t − . t = 23 . t − , for all t ≥ . Recall that the iterated Kamekohomomorphism (( g Sq ∗ ) (5 , . t − ) t − : Q ⊗ . t − → Q ⊗ . − is an F GL -module isomorphism for all t ≥ . So, by the results of Lin [12] and Chen [6] on Ext , . − A ( F , F ), to check Conjecture 1in the above degree, we need only to determine GL -coinvariants of Q ⊗ . t − for t = 1 . It is well-known (see [34]) that the family { h t | t ≥ } ⊂ Im(
T r A ) . On the other hand, in [21], Nam showed thefamily { f t | t ≥ } ⊂ Im(
T r A ) . Combining these with the fact that L m ≥ T r A m is a homomorphismof algebras, we deduce that the non-zero element h e = h f ∈ Ext , . − A ( F , F ) is in theimage of T r A . In addition, since the Kameko map ( g Sq ∗ ) (5 , . − : Q ⊗ . − → Q ⊗ . − is anepimorphism of GL -modules, by Theorem 8, we conclude that1 ≤ dim(( F ⊗ GL P A (( P ) ∗ )) . − ) ≤ dim(Ker( g Sq ∗ ) (5 , . − ) GL + 1 . Furthermore, all elements of ( F ⊗ GL P A (( P ) ∗ )) . − are of the form ( ζ [ ψ ( X )] + [ Y ]) ∗ , where ζ ∈ F , ψ : ( P ) . − → ( P ) . − is determined by ψ ( X ) = Q ≤ j ≤ x j X for all X ∈ ( P ) . − , and Y ∈ ( P ) . − such that [ Y ] belongs to Ker( g Sq ∗ ) (5 , . − . Based on Theorem 8, [ X ] is theonly non-zero element in ( Q ⊗ . − ) GL . Calculating the elements ( ζ [ ψ ( X )]+[ Y ]) ∗ is hard. However,based upon our results in [27, 30] about the GL -invariants of Q ⊗ in some generic degrees, andthe above computations, we give the following prediction. Conjecture 11.
For each t ≥ , the kernel of the Kameko map ( g Sq ∗ ) (5 , . t − is trivial under theaction of the group GL . Consequently, ( F ⊗ GL P A (( P ) ∗ )) . t − is 1-dimensional. By Theorem 3, this conjecture is an immediate consequence of the following conjecture.
Conjecture 12.
Under the above notations, the spaces ( Q ⊗ − . ) and ( Q ⊗ − . ) ω > aretrivial under the action of the group GL . From the results of Sum [37] on the structure of Q ⊗ − . , the dimension of (( Q ⊗ − . ) ) GL can be verified. However, the computations of (( Q ⊗ − . ) ω > ) GL are not easy. Since h t f t = h t +1 e t ∈ Ext , . t − A ( F , F ) is in the image of T r A , if Conjecture 12 is true, then T r A is also somorphism when acting on ( F ⊗ GL P A (( P ) ∗ )) . t − for t ≥ , i.e., Conjecture 1 holds for m = 5 and the internal degree 23 . t − . We hope that our predictions are correct. If not, theSinger conjecture will be disproved. We leave these issues as future research and we appreciatereaders interested in solving them.Our basic tools in the article are the epimorphism of the Kameko map, the Steenrod squares,some homomorphisms of Sum [37], and the representation of the fifth Singer transfer over theLambda algebra.The contains of this note will be published in detail elsewhere.
Acknowledgments.
The author would like to give my deepest sincere thanks to Professor WilliamSinger for enlightening conversations that led to this paper.
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