SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER

Volume 349, Number 10, October 1997, Pages 3893–3910 S 0002-9947(97)01991-0

SPHERICAL CLASSES AND THE ALGEBRAIC TRANSFER

NGUY ˆE˜ N H. V. HU’NG

Abstract. _{We study a weak form of the classical conjecture which predicts}

that there are no spherical classes inQ0S0except the elements of Hopf invari-ant one and those of Kervaire invariinvari-ant one. The weak conjecture is obtained by restricting the Hurewicz homomorphism to the homotopy classes which are detected by the algebraic transfer.

LetP_k =F2[x1, . . . , x_k] with|xi|= 1. The general linear groupGL_k=

GL(k,F2) and the (mod 2) Steenrod algebraAact onP_kin the usual manner. We prove that the weak conjecture is equivalent to the following one: The canonical homomorphismj_k :F2⊗

A

(P_kGLk)→(F2⊗ APk

)GLk induced by the identity map onP_k is zero in positive dimensions fork >2. In other words, every Dickson invariant (i.e. element ofP_kGLk) of positive dimension belongs toA+·P_kfork >2, where A+ denotes the augmentation ideal ofA. This conjecture is proved fork= 3 in two different ways. One of these two ways is to study the squaring operationSq0onP(F2 ⊗

GLk P∗

k), the range ofj∗k, and

to show it commuting throughj_k∗ with Kameko’sSq0 on F2⊗

GLk

P(P_k∗), the domain ofj_k∗. We compute explicitly the action of Sq0 onP(F2 ⊗

GLk P∗

k) for k≤4.

1. Introduction

The paper deals with the spherical classes in_Q0S0_{, i.e. the elements belonging}

to the image of the Hurewicz homomorphism

H :_πs_∗(_S0)∼=_π∗(_Q0S0)→_H∗(_Q0S0)_.

Here and throughout the paper, the coefficient ring for homology and cohomology is alwaysF₂, the field of 2 elements.

We are interested in the following classical conjecture.

Conjecture 1.1. (conjecture on spherical classes). There are no spherical classes in_Q0S0_{, except the elements of Hopf invariant one and those of Kervaire invariant}

one.

(See Curtis [9] and Wellington [21] for a discussion.)

Let _Vk be an elementary abelian 2-group of rank _k. It is also viewed as a _k -dimensional vector space overF₂. So, the general linear group _GLk =_GL(_k,F₂)

Received by the editors April 7, 1995.

1991Mathematics Subject Classification. Primary 55P47, 55Q45, 55S10, 55T15.

Key words and phrases. Spherical classes, loop spaces, Adams spectral sequences, Steenrod algebra, invariant theory, Dickson algebra, algebraic transfer.

The research was supported in part by the DGU through the CRM (Barcelona). c

1997 American Mathematical Society

acts on _Vk and therefore on _H∗(_BVk) in the usual way. Let _Dk be the Dickson algebra of_k variables, i.e. the algebra of invariants

Dk:=H∗(BVk)GLk=∼F2[x1, . . . , xk]GLk,

where _Pk = F₂[_{x1, . . . , xk}] is the polynomial algebra on _k generators _{x1, . . . , xk}, each of dimension 1. As the action of the (mod 2) Steenrod algebraAand that of GLk onPk commute with each other,Dk is an algebra overA.

One way to attack Conjecture 1.1 is to study the Lannes–Zarati homomorphism ϕk :Extk,k_A +i(F2,F2)→(F2⊗

ADk) ∗

i ,

which is compatible with the Hurewicz homomorphism (see [12], [13, p.46]). The domain of_ϕkis the_E2-term of the Adams spectral sequence converging to_πs

∗(S0)∼=

π∗(Q0S0). Furthermore, according to Madsen’s theorem [15] which asserts thatDk

is dual to the coalgebra of Dyer–Lashof operations of length_k, the range of_ϕk is a submodule of_H∗(_Q0S0_{). By compatibility ofϕ}

kand the Hurewicz homomorphism

we mean_ϕk is a “lifting” of the latter from the “_E∞-level” to the “_E2-level”. Let_hr denote the Adams element in_ExtA1,2r(F_2,F₂). Lannes and Zarati proved in [13] that_ϕ1 is an isomorphism with{_ϕ1(_hr)|_r≥0} forming a basis of the dual ofF₂⊗

AD1andϕ2 is surjective with{ϕ2(h

2

r)|r≥0}forming a basis of the dual of

F2⊗

AD2

. Recall that, from Adams [1], the only elements of Hopf invariant one are represented by_{h1, h2, h3} of the stems _i= 2r_{−1 = 1,3,7, respectively. Moreover,}

by Browder [5], the only dimensions where an element of Kervaire invariant one would occur are 2(2r_{−1), forr >0, and it really occurs at this dimension if and}

only if _h2_r is a permanent cycle in the Adams spectral sequence for the spheres. Therefore, Conjecture 1.1 is a consequence of the following:

Conjecture 1.2. ϕk = 0 in any positive stemifork >2.

It is well known that the Ext group has intensively been studied, but remains very mysterious. In order to avoid the shortage of our knowledge of the Ext group, we want to restrict_ϕk to a certain subgroup of Ext which (1) is large enough and worthwhile to pursue and (2) could be handled more easily than the Ext itself. To this end, we combine the above data with Singer’s algebraic transfer.

Singer defined in [20] the algebraic transfer T rk:F2⊗

GLk

P Hi(BVk)→ExtAk,k+i(F2,F2),

where _{P H∗}(_BVk) denotes the submodule consisting of allA-annihilated elements in _H∗(_BVk). It is shown to be an isomorphism for _k ≤2 by Singer [20] and for k= 3 by Boardman [4]. Singer also proved that it is an isomorphism for_k= 4 in a range of internal degrees. But he showed it is not an isomorphism for _k = 5. However, he conjectures that_{T rk} is a monomorphism for any_k.

Our main idea is to study the restriction of_ϕk to the image of_{T rk}.

Conjecture 1.3. (weak conjecture on spherical classes). ϕk·T rk :F2 ⊗ GLk P H∗(BVk)→P(F2 ⊗ GLk H∗(BVk)) := (F2⊗ ADk) ∗

In other words, there are no spherical classes in _Q0S0, except the elements of Hopf invariant one and those of Kervaire invariant one, which can be detected by the algebraic transfer.

A natural question is: How can one express_ϕk·_{T rk}in the framework of invariant theory alone, and without using the mysterious Ext group?

Let_jk:F₂⊗

A(P

GLk

k )→(F2⊗

APk)

GLk _{be the natural homomorphism induced by}

the identity map on_Pk. We have

Theorem 2.1. ϕk·T rk is dual to jk, or equivalently,

jk=T r∗k·ϕ∗k.

By this theorem, Conjecture 1.3 is equivalent to

Conjecture 1.4. jk= 0 in positive dimensions fork >2.

This seems to be a surprise, because by an elementary argument involving taking averages, one can see that if _H⊂_GLk is a subgroup of odd order then the similar homomorphism jH :F2⊗ A(P H k )→(F2⊗ APk) H

is an isomorphism. Furthermore,_j1 is iso and_j2 is mono. Obviously,_jk = 0 if and only if the composite

Dk =P_kGLkproj→ F2⊗ A(P GLk k ) jk →(F₂⊗ APk) GLk→⊂ F 2⊗ APk

is zero. So, Conjecture 1.4 can equivalently be stated in the following form.

Conjecture 1.5. Let _D+_k, A+ _{denote the augmentation ideals in D}

k and A,

re-spectively. Then_D+_k ⊂ A+_·P

k for anyk >2.

The domain and range of _jk both are still mysterious. Anyhow, they seem easier to handle than the Ext group. They both are well-known for _k = 1_,2. Furthermore, on the one hand, (F₂⊗

APk)

GLk_{is computed fork= 3 by Kameko [11],}

Alghamdi–Crabb–Hubbuck [3] and Boardman [4]. On the other hand,F₂⊗

A(P

GLk

k )

is determined by Hu’ng–Peterson [18] for_k= 3 and 4. Let F₂⊗ A(P GL_{) :=} L k≥0 F2⊗ A(P GLk k ) and (F2⊗ AP) GL _:= L k≥0 (F₂⊗ APk) GLk_{. They}

are equipped with canonical coalgebra structures. We get

Proposition 3.1. j = L_jk : F₂⊗ A(P GL_{) → (F} 2⊗ AP) GL _{is a homomorphism of} coalgebras. Let_Sq0_{:P H}

∗(BVk)→P H∗(BVk) be Kameko’s squaring operation that

com-mutes with the Steenrod operation_Sq0_:Extk,t

A (F2,F2)→Ext_Ak,2t(F2,F2) through

the algebraic transfer _{T rk} (see [11], [3], [4], [17]). Note that _Sq0 _{is completely}

different from the identity map. We prove

Proposition 4.2. There exists a homomorphism Sq0:_P(F₂ ⊗

GLk

H∗(BVk))→P(F2 ⊗ GLk

H∗(BVk)),

which commutes with Kameko’s _Sq0 through the homomorphism _j∗_k.

Theorem 3.2. jk∗ = 0 in positive dimensions for k = 3. In other words, there is

no spherical class in _Q0S0 which is detected by the triple algebraic transfer. We compute explicitly the action of_Sq0 on_P(F₂ ⊗

GLk

H∗(_BVk)) for_k= 3 and 4 in Propositions 5.2 and 5.4.

The paper contains six sections and is organized as follows.

Section 2 is to prove Theorem 2.1. In Section 3, we assemble the_jk for_k ≥0 to get a homomorphism of coalgebras_j =L_jk. By means of this property of_j we give there a proof of Theorem 3.2. Section 4 deals with the existence of the squaring operation_Sq0on_P(F₂ ⊗

GLk

H∗(_BVk)) that leads us to an alternative proof for Theorem 3.2. This proof helps to explain the problem. In Section 5, we compute explicitly the action of_Sq0 on_P(F₂ ⊗

GLk

H∗(_BVk)) for_k≤4. Finally, in Section 6 we state a conjecture on the Dickson algebra that concerns spherical classes.

Acknowledgments

I express my warmest thanks to Manuel Castellet and all my colleagues at the CRM (Barcelona) for their hospitality and for providing me with a wonderful work-ing atmosphere and conditions. I am grateful to Jean Lannes and Frank Peterson for helpful discussions on the subject. Especially, I am indebted to Frank Peter-son for his constant encouragement and for carefully reading my entire manuscript, making several comments that have led to many improvements.

2. Expressing ϕ_k·T r_k in the framework of invariant theory First, let us recall how to define the homomorphism_jk.

We have the commutative diagram

F2⊗ A(P GLk k ) - F2⊗_APk , f jk ? ? HH HH HHp _HHj PGLk k - Pk ⊂

where the vertical arrows are the canonical projections, and _je_k is induced by the inclusion _PGLk k ⊂Pk. Obviously,p(PkGLk)⊂(F2⊗ APk) GLk_{. So,j}e k factors through (F₂⊗ APk) GLk _{to give rise to} jk : F2⊗ A(P GLk k ) → (F2⊗ APk) GLk_, jk(1⊗ AY) = 1⊗AY,

for any polynomial_Y ∈_Dk=_PGLk

k .

The goal of this section is to prove the following theorem.

Theorem 2.1. jk =T rk∗·ϕ∗k.

Now we prepare some data in order to prove the theorem at the end of this section.

First we sketch Lannes–Zarati’s work [13] on the derived functors of the desta-bilization. Let D be the destabilization functor, which sends an A-module _M to the unstable A-moduleD(_M) = _M/B(_M), where _B(_M) is the submodule of _M generated by all_Sqi_{uwithu∈M, i >|u|.}

Dis a right exact functor. LetD_k be its _k-th derived functor for_k≥0.

Suppose_{M1, M2} areA-modules. Lannes and Zarati defined in [13,§2] a homo-morphism

∩: _Extr

A(M1, M2)⊗ Ds(M1) → Ds−r(M2),

(_{f, z}) 7→ _f∩_{z ,} as follows.

Let_F∗(_Mi) be a free resolution of_Mi,_i= 1_,2. A class _f ∈_Extr

A(M1, M2) can

be represented by a chain map_F :_F∗(_M1)→_F∗−r(_M2) of homological degree−_r. We write _f = [_F]. If _z = [_Z] is represented by _Z ∈ _F∗(_M1), then by definition f∩z= [_F(_Z)].

Let_M be anA-module. We set_r=_s=_k,_M1= Σ−k_M,M

2=Pk⊗M, where

as before_Pk=F₂[_{x1, . . . , xk}], and get the homomorphism

∩:_Extk_A(Σ−k_{M, Pk}⊗_M)⊗ D_k(Σ−k_M)→_Pk⊗_{M .} Now we need to define the Singer element_ek(_M)∈_Extk

A(Σ−kM, Pk⊗M) (see

Singer [20, p. 498]). Let_Pb₁be the submodule ofF₂[_{x, x}−1_{] spanned by all powers}

xi_{withi≥ −1, where|x|= 1. TheA-module structure onF}

2[x, x−1] extends that

of_P1=F₂[_x] (see Adams [2], Wilkerson [22]). The inclusion_P1 ⊂_Pb₁ gives rise to a short exact sequence ofA-modules:

0→_P1→_Pb₁→Σ−1F₂→0_. Denote by_e1 the corresponding element in_Ext1

A(Σ−1F2, P1). Definition 2.2. (Singer [20]). (i) _ek =_e1⊗ · · · ⊗_e1 | {z } ktimes ∈Extk A(Σ−kF2, Pk).

(ii) _ek(_M) =_ek⊗_M ∈_Extk_A(Σ−k_{M, Pk}⊗_M)_,for_M anA-module. Here we also denote by_M the identity map of _M.

The cap-product with_ek(_M) gives rise to the homomorphism ek(M) : Dk(Σ−kM) → D0(Pk⊗M)≡Pk⊗M,

ek(M)(z) = ek(M)∩z .

As F₂ is an unstable A-module, the following theorem is a special case (but would be the most important case) of the main result in [13].

Theorem 2.3 (Lannes–Zarati [13]). Let_Dk ⊂_Pk be the Dickson algebra of_k vari-ables. Then _ek(ΣF₂) :D_k(Σ1−kF₂)→ Σ_Dk is an isomorphism of internal degree 0.

Next, we explain in detail the definition of the Lannes–Zarati homomorphism ϕk:ExtkA(Σ−kF2,F2)→(F2⊗

ADk) ∗_,

which is compatible with the Hurewicz map (see [12], [13]).

Let _N be an A-module. By definition of the functor D, we have a natural homomorphism: D(_N)→F₂⊗

AN. SupposeF∗(N) is a free resolution ofN. Then

· · · - F2⊗

AFk(N) - F2A⊗Fk−1(N) - · · · .

? ?

ik ik−1

· · · - DFk(N) - DFk−1(N) - · · ·

Here the horizontal arrows are induced from the differential in_F∗(_N), and ik[Z] = [1⊗

AZ]

for_Z∈_Fk(_N). Passing to homology, we get a homomorphism ik: F2⊗ ADk(N) → T or A k(F2, N), 1⊗ A[Z] 7→ [1⊗AZ]. Taking_N = Σ1−k_F 2, we obtain a homomorphism ik:F2⊗ ADk(Σ 1−k_F 2)→T orAk(F2,Σ1−kF2).

Note that the suspension Σ :F₂⊗

ADk→F2⊗AΣDk and the desuspension

Σ−1:_{T ork}A(F_2,Σ1−kF₂)−→∼= _{T or}A_k(F_2,Σ−kF₂)

are isomorphisms of internal degree 1 and (−1), respectively. This leads us to

Definition 2.4. (Lannes–Zarati [13]). The homomorphism _ϕk of internal degree 0 is the dual of ϕ∗k= Σ−1ik 1⊗ Ae −1 k (ΣF2) Σ :F₂⊗ ADk →T or A k(F2,Σ−kF2).

Now we recall the definition of the algebraic transfer. Consider the cap-product ExtrA(Σ−kF2, Pk)⊗T orAs(F2,Σ−kF2) → T orAs−r(F2, Pk),

(_{e, z}) 7→ _e∩_{z .}

Taking_r=_s=_kand_e=_ek as in Definition 2.2, we obtain the homomorphism T r∗k: T orkA(F2,Σ−kF2) → T orA0(F2, Pk) ≡ F2⊗

APk,

T r∗_k[1⊗

AZ] = ek∩[Z] = [1⊗AE(Z)] ≡ 1⊗AE(Z),

for_Z ∈_Fk(Σ−k_F

2), whereek= [E] is represented by a chain mapE:F∗(Σ−kF2)→

F∗−k(Pk).

Singer proved in [20] that_ek is_GLk-invariant, hence _Im(_{T r}∗_k)⊂(F₂⊗

APk)

GLk _.

This gives rise to a homomorphism, which is also denoted by_{T rk}∗, T rk∗:T orAk(F2,Σ−kF2)→(F2⊗

APk)

GLk_.

Definition 2.5. (Singer [20]). The_k-th algebraic transfer_{T rk} :F₂⊗

GLk

P H∗(_BVk)

→Extk,k_A +∗(F_2,F₂) is the homomorphism dual to_{T r}∗_k. We have finished the preparation of the needed data.

Proof of Theorem 2.1. Note that the usual isomorphism ExtkA(Σ−kF2, Pk)

∼

=

−→ExtkA(Σ1−kF2,ΣPk)

sends_ek(F₂) to _ek(ΣF₂) =_ek(F₂)⊗ΣF₂. Moreover, if_ek(F₂) = [_E] is represented by a chain map _E : _F∗(Σ−kF₂)→ _F∗−k(_Pk) then _ek(ΣF₂) = [_EΣ] is represented by the induced chain map _EΣ : _F∗(Σ1−kF₂) → _F∗−k(Σ_Pk), which is defined by EΣ= ΣEΣ−1.

By Theorem 2.3,_ek(ΣF₂) is an isomorphism. So, for any_Y ∈_Dk, there exists a representative of_e−_k1(ΣF₂)Σ_Y, which is denoted by_EΣ−1Σ_Y ∈_Fk(Σ1−k_F

2), such

that_{EΣ EΣ}−1Σ_Y= Σ_Y.

The cap-product with_ek(ΣF₂) = [_EΣ] induces the homomorphism

f T r∗k: T orAk(F2,Σ1−kF2) → T or0A(F2,ΣPk) ≡ F2⊗ A Σ_Pk, f T r∗k[1⊗ AZ] = ek(ΣF2)∩[Z] = [1A⊗EΣ(Z)] ≡ 1⊗AEΣ(Z), for_Z∈_Fk(Σ1−k_F

2). It is easy to check thatT r∗k= Σ−1T rf

∗ kΣ. Moreover, set e ϕ∗k= Σϕ∗kΣ−1=ik 1⊗ Ae −1 k (ΣF2) :F₂⊗ AΣDk→T or A k(F2,Σ1−kF2).

Obviously,_{T r}∗_k·_ϕ∗_k= Σ−1_{T r}f∗_k·_ϕe∗_kΣ. Now, for any_Y ∈_Dk, we have T r∗k·ϕ∗k(1⊗ AY) = Σ −1_{T r}f∗ k·ϕe∗kΣ(1⊗ AY) = Σ−1_{T r}f∗_k·_ϕe∗_k(1⊗ AΣY) = Σ−1_{T r}f∗_k 1⊗ AE −1 Σ ΣY = Σ−1 1⊗ AEΣ(E −1 Σ ΣY) = 1⊗ AΣ −1_(ΣY) = 1⊗ AY .

By definition of_jk, we also have_jk(1⊗

AY

) = 1⊗

AY .

The theorem is proved.

3. The homomorphism of coalgebras _j=L_jk

The canonical isomorphism _Vk ∼= _{V`</sub>×<sub>Vm</sub>, for <sub>k</sub> = <sub>`}+_m, induces the usual inclusion_GLk⊃_{GL`</sub>×<sub>GLm</sub>and the usual diagonal ∆ :<sub>Pk</sub>→<sub>P`}⊗_Pm. Therefore, it induces two homomorphisms

∆_D:F₂⊗ A(P GLk k )→ F2⊗ A(P GL` ` ) ⊗ F2⊗ A(P GLm m ) , ∆_P: (F₂⊗ APk) GLk→₍F 2⊗ AP`) GL` ⊗₍F 2⊗ APm) GLm_.

Here and in what follows,⊗means the tensor product overF₂, except when other-wise specified.

Set F2⊗ AD=F2⊗A(P GL_{) :=}M k≥0 F2⊗ A(P GLk k ), (F₂⊗ AP) GL_:=M k≥0 (F₂⊗ APk) GLk_.

It is easy to see thatF₂⊗

A(P

GL_{) and (F} 2⊗

AP)

GL_{are endowed with the structure of}

a cocommutative coalgebra by ∆_D and ∆_P, respectively. The coalgebra structure of (F₂⊗

AP)

GL_{was first given by Singer [20].}

Proposition 3.1. j = L_jk : F₂⊗ A(P GL_{) → (F} 2⊗ AP) GL _{is a homomorphism of} coalgebras.

Proof. This follows immediately from the commutative diagram

(F₂⊗ AD` )⊗(F<sub>2</sub>⊗ ADm ) j`⊗jm - (F₂⊗ AP`) GL` ⊗₍F₂⊗ APm) GLm_. ? ? ∆D ∆P F2⊗ ADk -jk (F₂⊗ APk )GLk

Remark. According to Singer [20],_{T r}∗=L_{T r}∗_k is a homomorphism of coalgebras. One can see that _ϕ∗ =L_ϕ∗_k is also a homomorphism of coalgebras. Then, so is j=_{T r}∗·_ϕ∗. This is an alternative proof for Proposition 3.1.

Now let F2 ⊗ GLP H∗ (_BV) :=M k≥0 F2⊗ GLk P H∗(BVk) ∼ =M k≥0 (F₂⊗ APk) GLk ∗ , P(F₂ ⊗ GLH∗ (_BV)) :=M k≥0 P(F₂ ⊗ GLk H∗(BVk))∼= M k≥0 F2⊗ A(P GLk k ) _∗ . Passing to the dual, we obtain the homomorphism of algebras

j∗:F₂ ⊗

GLP H∗

(_BV)→_P(F₂ ⊗

GLH∗

(_BV))_.

As an application of_j∗, we give here a proof for Conjecture 1.4 with_k= 3.

Theorem 3.2. j3:F2⊗

A(P

GL3

3 )→(F2⊗

AP3)

GL3 _{is zero in positive dimensions.}

Proof. We equivalently show that j3∗:F2 ⊗

GL3P H∗

(_BV3)→_P(F₂⊗

GL3H∗

(_BV3)) is a trivial homomorphism in positive dimensions.

F2 ⊗ GL3P H∗

(_BV3) is described by Kameko [11], Alghamdi–Crabb–Hubbuck [3] and Boardman [4] as follows. F₂⊗

GL1P H∗

(_BV1) has a basis consisting of _hr, _r ≥ 0, where _hr is of dimension 2r_{−1 and is sent by the isomorphism T r}

1 to the

[4], F₂ ⊗

GL3P H∗

(_BV3) has a basis consisting of some products of the form _hrhsht, where_{r, s, t}are non-negative integers (but not all such appear), and some elements ci(i≥0) with dim(ci) = 2i+3+ 2i+1+ 2i−3.

We will show in Lemma 3.3 that any decomposable element in_P(F₂⊗

GL3H∗

(_BV3)) is zero. Then, since_j∗is a homomorphism of algebras,_j3∗sends any element of the form_hrhsht to zero.

On the other hand, by Hu’ng–Peterson [18],F₂⊗

AD3 is concentrated in the

di-mensions 2s+2−4 (_s ≥ 0) and 2r+2+ 2s+1 −3 (_{r > s >} 0). Obviously, these dimensions are different from dim(_ci) for any_i. Then_j3∗ also sends_ci to zero.

To complete the proof of the theorem, we need to show the following lemma.

Lemma 3.3. Let _Dk=F₂⊗

ADk. Then the diagonal

∆_D:_D3→_D1⊗_D2⊕_D2⊗_D1 is zero in positive dimensions.

Proof. Let us recall some informations on the Dickson algebra_Dk. Dickson proved in [10] that_Dk ∼=F2[Qk−1, Qk−2, . . . , Q0], a polynomial algebra on k generators,

with |_Qs| = 2k −2s. Note that _Qs depends on _k, and when necessary, will be denoted_Qk,s. An inductive definition of_Qk,s is given by

Definition 3.4.

Qk,s=Q2k−1,s−1+vk·Qk−1,s,

where, by convention,_Qk,k= 1,_Qk,s= 0 for _{s <}0 and vk =

Y λi∈F2

(_λ1x1+· · ·+_{λk−1xk−1}+_xk)_. Dickson showed in [10] that

vk= k−1 X s=0 Qk−1,sx2 s k .

Now we turn back to the lemma.

Since ∆_D is symmetric, we need only to show that the diagonal ∆ :_D3→_D2⊗_D1

is zero in positive dimensions.

For abbreviation, we denote_{x1, x2, x3} by_{x, y, z}, respectively, _Qi=_Q3,i(_{x, y, z}) for_i= 0_,1_,2,_qi=_Q2,i(_{x, y}) for_i= 0_,1. As is well known, F₂⊗

AD1 has the basis

{z2s−1_{|s≥0}, andF} 2⊗

AD2has the basis{q

2s−1

1 |s≥0}. By Hu’ng–Peterson [18],

F2⊗

AD3has the basis

{Q22s−1(s≥0), Q2

r₋₂s₋₁

2 Q2

s₋₁

1 Q0(r > s >0)}.

For _k ≤ 3, every monomial in _{Q0, . . . , Qk−1} which does not belong to the given basis is zero in F₂⊗

ADk

. Note that the analogous statement is not true for_k ≥4 (see [18]).

Using the above inductive definitions of_Qk,sand _vk, we get Q0=q02z+q0q1z2+q0z4,

or

∆(_Q0) =_q2₀⊗_z+_q0q1⊗_z2+_q0⊗_z4_.

This implies easily that every term in ∆(_Q2₂r−2s−1_Q12s−1_Q0) is divisible by _q0, so it equals zero inF₂⊗

AD2 as shown above. In other words,

∆(_Q2₂r−2s−1_Q2₁s−1_Q0) = 0_. Similarly, Q2=q21+v3=q21+q0z+q1z2+z4, or ∆(_Q2) =_q2₁⊗1 +_q0⊗_z+_q1⊗_z2+ 1⊗_z4_, ∆(_Q2₂s−1) = (_q12⊗1 +_q0⊗_z+_q1⊗_z2+ 1⊗_z4)2s−1_.

By the same argument as above, we need only to consider terms in ∆(_Q2₂s−1) which are not divisible by_q0. Such a term is some product of powers of _q2

1⊗1,q1⊗z2,

1⊗_z4_{. If it contains a positive power ofzthen this power is even and it equals zero}

inF₂⊗ AD1. Otherwise, it should beq 2(2s−1) 1 ⊗1. Obviously,q2(2 s₋₁₎ 1 equals zero in F2⊗ AD2 . So, ∆(_Q2₂s−1) = 0 for_{s >}0.

In summary, ∆ = 0 in positive dimensions. The lemma is proved. Then, so is Theorem 3.2.

As _{T r3} is an isomorphism (see Boardman [4]), we have an immediate conse-quence.

Corollary 3.5. ϕ3: Ext3A,3+i(F2,F2)→(F2⊗

AD3) ∗

i is zero in every positive stem

i.

4. The squaring operation: the existence

Liulevicius was perhaps the first person who noted in [14] that there are squar-ing operations _Sqi _{: Ext}k,t

A(F2,F2) → Ext_Ak+i,2t(F2,F2), which share most of the

properties with _Sqi _{on cohomology of spaces. In particular, Sq}i_{(α) = 0 if i > k,}

Sqk_{(α) = α}2 _{for α∈ Ext}k,t

A (F2,F2), and the Cartan formula holds for the Sqi’s.

However,_Sq0_{is not the identity. In fact,}

Sq0_{: Ext}k,t

A(F2,F2) → Ext_Ak,2t(F2,F2),

[_b1|_{. . .}|_bk] 7→ [_b2₁|_{. . .}|_b2_k]_, in terms of the cobar resolution (see May [16]).

Recall that_H∗(_BVk) is a divided power algebra H∗(_BVk) = Γ(_{a1, . . . , ak})

generated by_{a1, . . . , ak}, each of degree 1, where_aiis dual to _xi ∈_H1_(BV

k). Here

and in what follows, the duality is taken with respect to the basis of _H∗(_BVk) consisting of all monomials in_{x1, . . . , xk}.

Let _γt be the _t-th divided power in _H∗(_BVk) and for any _a ∈ _H∗(_BVk) let a(t)_=γ

t(a). Soa(it)is the element dual toxti. One has

a(2i r)a (2r) i = 0, and a(it)=a(2 r1₎ i · · ·a(2 rm₎ i if_t= 2r1_{+· · ·+ 2}rm_{, 0}≤_r 1<· · ·< rm.

In [11] Kameko defined a_GLk-homomorphism

Sq0: _{P H∗}(_BVk) → _{P H∗}(_BVk)_, a1(i1)· · ·a (ik) k 7→ a (2i1+1) 1 · · ·a (2ik+1) k , where_a(₁i1)· · ·_ak(ik)is dual to _xi1₁ · · ·_xik k . (See also [3].)

Crabb and Hubbuck gave in [8] a definition of_Sq0that does not depend on the chosen basis of_H∗(_BVk) as follows. The element _a(_Vk) =_a1· · ·_ak is nothing but the image of the generator of Λk(_Vk) under the (skew) symmetrization map

Λk(_Vk)→_Hk(_BVk) = Γ_k(_Vk) = (_Vk⊗ · · · ⊗_Vk

| {z }

ktimes

)_Σk.

Let_F:_H∗(_BVk)→_H∗(_BVk) be the Frobenius homomorphism defined by_F(_x) = x2_{for anyx, and letc:H}

∗(BVk)→H∗(BVk) be the degree-halving dual

homomor-phism. It is obviously a surjective ring homomorhomomor-phism. Then _Sq0 _{can be defined}

by

Sq0(_c(_y)) =_a(_Vk)_{y .}

Since_y∈ker_cif and only if_a(_Vk)_y= 0,_Sq0is a monomorphism of_GLk-modules. Further, it is easy to see that_cSq2i+1

∗ = 0,cSq2∗i =Sq∗ic. SoSq0 mapsP H∗(BVk)

to itself.

Using a result of Carlisle and Wood [6] on the boundedness conjecture, Crabb and Hubbuck also noted in [8] that for any_d, there exists_t0such that

Sq0:_{P H2}t_d+(2t_−1)k(BV_k)→P H₂t+1_d+(2t+1_−1)k(BV_k)

is an isomorphism for every_t≥_t0.

Kameko’s _Sq0 _{is shown to commute with Sq}0 _onExtk

A(F2,F2) through the

al-gebraic transfer _{T rk} by Boardman [4] for _k = 3 and by Minami [17] for general k.

One denotes also by_Sq0_{the operation}

Sq0:F₂ ⊗

GLk

P H∗(_BVk)→F₂ ⊗

GLk

P H∗(_BVk)

induced by Kameko’s_Sq0_{. It preserves the product. Further, fork= 3, it satisfies}

Sq0(_hrhsht) =_{hr+1hs+1ht+1, Sq}0(_ci) =_ci+1 (see Boardman [4]).

Lemma 4.1. Sq2∗r+1Sq0= 0,Sq∗2rSq0=Sq0Sqr∗.

Proof. We need a formal notation. Namely, for_a∈_H1(_BVk), set (_a(t)₎[2]_=a(2t)_.

We start with a simple remark.

Let_x∈_H1(_BVk); then_Sqr(_xs) = s_rxs+r. Let_adenote the dual element of_x. Then, by dualizing, Sq∗r(a(t)) = t−r r a(t−r). As a consequence,_Sq2r+1 ∗ (a(2t+1)) = 0 and Sq2∗r(a(2t)) = 2_t−2_r 2_r a(2t−2r)= t−r r a (t−r)[2]_=Sqr ∗a(t) [2] . Let_α=_a(₁i1)· · ·_a(_kik). By the Cartan formula, we have

Sqr∗Sq0(α) =Sqr∗(a(21i1+1)· · ·a (2ik+1) k ) = X r1+···+rk=r Sqr1∗ (a1(2i1+1))· · ·Sq∗rk(a (2ik+1) k ).

The term corresponding to (_{r1, . . . , rk}) equals 0 if at least one of_{r1, . . . , rk} is odd. Hence_Sq∗2r+1_Sq0(_α) = 0. Furthermore, Sq2_∗rSq0(_α) = X r1+···+rk=r Sq2_∗r1(_a(2₁i1+1))· · ·_Sq2rk ∗ (a(2kik+1)) = X r1+···+rk=r n Sq∗2r1(a1(2i1))· · ·Sq2∗rk(a (2ik) k ) o a1· · ·ak = X r1+···+rk=r n Sq∗r1(a(1i1)) o[2] · · ·nSqrk ∗ (a(kik)) o[2] a1· · ·ak =_Sq0_Sq∗r(_α)_. The lemma is proved.

Proposition 4.2. For every positive integer _k, there exists a homomorphism Sq0:_P(F₂ ⊗

GLk

H∗(BVk))→P(F2⊗ GLk

H∗(BVk))

that sends an element of degree _n to an element of degree 2_n+_k and makes the following diagram commutative:

F2 ⊗ GLk P H∗(BVk) -j∗_k P(F₂⊗ GLk H∗(BVk)). ? ? Sq0 Sq0 F2 ⊗ GLk P H∗(_BVk) j -∗ k P(F₂⊗ GLk H∗(_BVk))

Proof. Since _Sq0 : _H∗(_BVk)→_H∗(_BVk) is a_GLk-homomorphism, we can define Sq0

D= 1⊗ GLk

H∗(_BVk) - F2 ⊗ GLk H∗(_BVk) ? ? Sq0 Sq0D H∗(_BVk) - F2⊗ GLk H∗(_BVk)_, where the horizontal arrows are the canonical projections.

Next, we show that _Sq0

D sends the primitive part to itself. In other words,

suppose_α∈_H∗(_BVk) satisfies Sqr∗(1⊗ GLk α) = 1 ⊗ GLk Sq∗rα= 0

for any_{r >}0; we want to show that Sqr∗(Sq0(1 ⊗

GLk

α)) = 0

for any_{r >}0. By definition of_Sq0and Lemma 4.1, we have for every_{r >}0 Sq∗r(Sq0(1⊗ GLk α)) = 1⊗ GLk Sq∗rSq0(α) = ( 1⊗ GLk Sq0_(Sqr/2 ∗ (α)), r even, 0_{, r}odd_, = ( Sq0_Sq∗r/2_{(1 ⊗} GLk α)_{, r}even_, 0_{, r}odd_, = 0_.

Therefore, the above commutative diagram gives rise to a commutative diagram

P H∗(BVk) -f j∗_k P(F₂ ⊗ GLk H∗(BVk)). ? ? Sq0 Sq0_D P H∗(_BVk) -f j∗k P(F₂ ⊗ GLk H∗(_BVk))

By definition of_jk, the homomorphism_je_k∗factors throughF₂ ⊗

GLk

P H∗(_BVk) and the previous diagram induces the commutative diagram stated in the proposition, in which_Sq0_D is re-denoted by_Sq0for short. The proposition is proved.

As an application of Proposition 4.2, we give an alternative proof of Theorem 3.2. By Kameko [11], Alghamdi et al. [3] and Boardman [4],F₂⊗

GL3P H∗

(_BV3) has a basis consisting of some products of the form_hrhshtand certain elements_ci(_i≥0) with_Sq0(_ci) =_ci+1 for any_i≥0.

By Lemma 3.3, _j3∗ vanishes on any product _hrhsht. Making use of Proposi-tion 4.2, one has

One needs only to show that_j3∗(_c0) = 0. Recall that dim(_c0) = 8. The only element of dimension 8 in _D3 is _Q2₂. Obviously, _Q2₂ =_Sq4_Q2. So _P(F₂ ⊗ GL3H∗ (_BV3))₈ = (F₂⊗ AD3

)∗₈= 0. Therefore,_j3∗(_c0) = 0. Theorem 3.2 is proved.

5. The squaring operation: an explicit formula for k≤4 Let_{d(ik−1,... ,i0)}be the dual element of_Qkik₋−₁1· · ·_Qi0₀ ∈_Dk, where the duality is taken with respect to the basis of _Dk consisting of all monomials in the Dickson invariants_{Qk−1, . . . , Q0}. It is well-known that P(F₂ ⊗ GL1H∗ (_BV1)) = Span{_d(2s₋₁₎|s≥0}, P(F₂ ⊗ GL2H∗ (_BV2)) = Span{_d(2s_−1,0)|s≥0}.

By means of the definition of_Sq0_{one can easily show that}

Sq0(_d(2s₋₁₎) =d₍₂s+1₋₁₎,

Sq0(_d(2s_−1,0)) =d₍₂s+1_−1,0).

In this section we compute_Sq0_{explicitly onP(F} 2⊗

GLk

H∗(BVk)) fork= 3 and 4.

Theorem 5.1 (Hu’ng–Peterson [18]). _{P D3}∗:=_P(F₂ ⊗

GL3H∗

(_BV3))has a basis con-sisting of

d(2s−1,0,0), s≥0,

d(2r−2s−1,2s−1,1), r > s >0.

They are of dimensions 2s+2_{−4 and2}r+2_{+ 2}s+1_{−3, respectively.}

Remark. It is easy to check that _{P D}∗₃ has at most one non-zero element of any dimension.

Proposition 5.2. Sq0_{:P D}∗

3→P D3∗ is given by

Sq0(_d(2s_−1,0,0)) = 0,

Sq0(_d(2r₋₂s_−1,2s_−1,1)) =d₍₂r+1₋₂s+1_−1,2s+1_−1,1).

Proof. For brevity, we denote _{x1, x2, x3} by _{x, y, z} and _{a1, a2, a3} by _{a, b, c}, respec-tively.

The first part of the proposition is an immediate consequence of dimensional in-formation. To prove the second part we start by recalling that, from Definition 3.4, we have

Q3,0=Q0=x4y2z1+ (symmetrized).

Suppose _{m, n} are non-negative integers. Let _xα_yβ_zγ be the biggest monomial in _Qm_2Qn₁ with respect to the lexicographic order on (_{α, β, γ}). We claim that xα+4yβ+2zγ+1 appears exactly one time in_Qm_{2 Q}n_1Q0, or equivalently

Qm2Qn1Q0=xα+4yβ+2zγ+1+ (other terms).

Indeed, suppose to the contrary that it appears more than once in_Qm_{2 Q}n_1Q0. That means there exists a monomial_xα0_yβ0_zγ0 in_Qm_{2 Q}n₁, which is different from_xα_yβ_zγ, and a permutation_σon the set{4_,2_,1}such that

xα+4yβ+2zγ+1=_xα0+σ(4)_yβ0+σ(2)_zγ0+σ(1)_.

Since_α+ 4 =_α0+_σ(4) and 4≥_σ(4), this implies_α≤_α0. Combining this with the fact that (_{α, β, γ}) is the biggest monomial in_Qm_2Qn₁with respect to the lexicographic order on (_{α, β, γ}), one gets_α=_α0 and_σ(4) = 4. Similarly,_β=_β0,_γ=_γ0and_σis the identity permutation. This contradiction proves (a), or equivalently

d(m,n,1)= 1⊗a(α+4)b(β+2)c(γ+1)+ (other terms).

(b)

Here and throughout the proof,⊗means the tensor product over_GL3. By definition of the squaring operation

Sq0(1⊗_a(α+4)_b(β+2)_c(γ+1)) = 1⊗_a(2α+9)_b(2β+5)_c(2γ+3)_. (c)

Now a direct computation using Definition 3.4 shows that Q2Q1Q0=x12y4z+x10y6z+x10y5z2+x10y4z3

+_x9_y6_z2+_x9_y5_z3+_x8_y6_z3+_x8_y5_z4+ (symmetrized)_.

Note that _x9_y5_z3 and its symmetrized terms are the only terms of the form xoddyoddzoddin_Q2Q1Q0. On the other hand,

Q22mQ21n =x2αy2βz2γ+ (other terms),

where_x2α_y2β_z2γ_{is the biggest monomial in this polynomial with respect to the}

lex-icographic order on (2_α,2_β,2_γ). Focusing on monomials of the form_xodd_yodd_zodd

and using the same argument as in the proof of (a), we have Q22m+1Q21n+1Q0=x2α+9y2β+5z2γ+3+ (other terms).

This is equivalent to

1⊗_a(2α+9)_b(2β+5)_c(2γ+3)=_{d(2m+1,2n+1,1)}+ (other terms)_. (d)

Combining (b), (c) and (d), we get

Sq0(_d(m,n,1)) =_{d(2m+1,2n+1,1)}+ (other terms)_. Applying this for (_{m, n,}1) = (2r₋₂s_−1,2s_{−1,1), we obtain}

Sq0(_d(2r₋₂s_−1,2s_−1,1)) =d₍₂r+1₋₂s+1_−1,2s+1_−1,1)+ (other terms).

In addition, _Sq0 _{maps P D}∗

3 to itself (by Proposition 4.2) andP D3∗ consists of at

most one non-zero element of any dimension. So the proposition is proved.

Theorem 5.3 (Hu’ng–Peterson [18]). _{P D4}∗:=_P(F₂ ⊗

GL4H∗

(_BV4))has a basis con-sisting of

d(2s−1,0,0,0), s≥0,

d(2r−2s−1,2s−1,1,0), r > s >0,

d(2t₋₂r_−1,2r₋₂s_−1,2s_−1,2), t > r > s >1,

d(2r−2s+1−2s−1,2s−1,2s−1,2), r > s+ 1>2.

They are of dimensions 2s+3_{−8, 2}r+3_{+ 2}s+2_{−6, 2}t+3 _{+ 2}r+2_{+ 2}s+1_{−4 and}

2r+3_{+ 2}s+1_{−4, respectively.}

Proposition 5.4. Sq0:_{P D}∗₄→_{P D4}∗ is given by Sq0(_d(2s_−1,0,0,0)) =Sq0(d₍₂r₋₂s_−1,2s_−1,1,0)) = 0,

Sq0(_d(2t₋₂r_−1,2r₋₂s_−1,2s_−1,2)) =d₍₂t+1₋₂r+1_−1,2r+1₋₂s+1_−1,2s+1_−1,2),

Sq0(_d(2r₋₂s+1₋₂s_−1,2s_−1,2s_−1,2)) =d₍₂r+1₋₂s+2₋₂s+1_−1,2s+1_−1,2s+1_−1,2).

Proof. We denote_{x1, x2, x3, x4}by_{x, y, z, t}and_{a1, a2, a3, a4}by_{a, b, c, d}, respectively, for brevity.

The first part of the proposition is an immediate consequence of dimensional information.

We claim that _Q0 =_x8_y4_z2_{t+ (symmetrized). It can be checked by a routine}

computation using Definition 3.4. Here we give an alternative argument. Indeed, the Dickson algebra_D4 ∼=F2[Q3, Q2, Q1, Q0] has exactly one non-zero element of

dimension 15. To check the equality we need only to show that the right hand side is_GL4-invariant. Recall that_GL4is generated by the symmetric group Σ₄and the transformation_x7→_x+_y, _y 7→_y, _z7→ _z, _t 7→_t. So, it suffices to check that the right hand side is invariant under this transformation. We leave it to the reader.

Suppose _{m, n, p, q} are non-negative integers with _{q >} 0. Let _xα_yβ_zγ_tδ _{be the}

biggest monomial in _Qm

3Qn2Qp1Qq0−1 with respect to the lexicographic order on

(_{α, β, γ, δ}). By the same argument as in the proof of Proposition 5.2 we have Qm3Qn2Qp1Q q 0=xα+8yβ+4zγ+2tδ+1+ (other terms). In other words, d(m,n,p,q)= 1⊗a(α+8)b(β+4)c(γ+2)d(δ+1)+ (other terms). (a)

Here and throughout this proof,⊗denotes the tensor product over_GL4. By definition of the squaring operation

Sq0(1⊗_a(α+8)_b(β+4)_c(γ+2)_t(δ+1)) = 1⊗_a(2α+17)_b(2β+9)_c(2γ+5)_d(2δ+3)_. (b)

Using the same method that we used to compute_Q0above, we can show that Q3Q2= X s1+s2+s3+s4=20 si=0 or a power of 2 xs1ys2zs3ts4, Q1= X s1+s2+s3+s4=14 si=0 or a power of 2 xs1ys2zs3ts4 . In particular, we have

Q3Q2= (x16y2zt+ symmetrized) + (other terms),

Q1= (x8y4zt+ symmetrized) + (other terms).

Here, in both cases, any other term is of the form_xeven_yeven_zeven_teven. So Q3Q2Q1= (x17y9z5t3 + symmetrized) + (other terms),

where _x17_y9_z5_t3 _{and its symmetrized terms are the only terms of the form}

xodd_yodd_zodd_todd_{in Q}

3Q2Q1. On the other hand,

Q32mQ22nQ21pQ02q−2=x2αy2βz2γt2δ+ (other terms),

where_x2α_y2β_z2γ_t2δ _{is the biggest monomial in the polynomial with respect to the}

xoddyoddzoddtoddand use the same argument as in the proof of Proposition 5.2 to get Q23m+1Q22n+1Q21p+1Q02q−2=x2α+17y2β+9z2γ+5t2δ+3+ (other terms), or equivalently 1⊗_a(2α+17)_b(2β+9)_c(2γ+5)_d(2δ+3)=_{d(2m+1,2n+1,2p+1,2q−2)}+ (other terms)_. (c)

Combining (a), (b) and (c), we get

Sq0(_d(m,n,p,q)) =_{d(2m+1,2n+1,2p+1,2q−2)}+ (other terms)_.

Apply this for (_{m, n, p, q}) = (2t−2r−1_,2r−2s−1_,2s−1_,2) and (2r−2s+1−2s− 1_,2s−1_,2s−1_,2). Combining the resulting formulas and the facts that_Sq0 maps P D4∗to itself (by Proposition 4.2) and thatP D4∗has at most one non-zero element

of any dimension, we obtain the last two formulas of the proposition. 6. Final remark Recall that_Dk :=F₂⊗ ADk. Let ∆_D:_Dk→ M `+m=k `,m>0 D`⊗Dm

be the diagonal defined at the beginning of Section 3.

Conjecture 6.1. (Hu’ng–Peterson [19]). The diagonal ∆_D is zero in positive di-mensions for any_{k >}2.

This conjecture is proved in Lemma 3.3 for _k = 3 and has been proved for 2_{< k <}10 in [19]. It implies that_jk∗ (respectively,_ϕk) vanishes on the decompos-able elements inF₂⊗

GLk

P H∗(BVk) with respect to the product given by Singer [20]

and discussed in Section 3 (respectively, in _Extk

A(F2,F2) with respect to the cup

product) for 2_{< k <}10.

Note added in proof. Conjecture 6.1 has been established by F. Peterson and the author in the final version of [19].

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