On behavior of the Lannes Zarati homomorphism

In this paper, we construct the chainlevel of the LannesZarati homomorphism on the lambda algebra. Using this chainlevel, we investigate the vanishing of the LannesZarati homomorphism of rank five and six. Keywords: Adams spectral sequence, Lambda algebra, DyerLashof algebra, LannesZarati homomorphism, Hurewicz homomorphism. 2000 MSC: Primary 55P47, 55Q45, 55S10, 55T15

On behavior of the Lannes-Zarati homomorphismI

Phan Hoàng Chơn∗, Đồng Thanh Triết

Faculty of Mathematics - Application, Saigon University, 273 An Duong Vuong, District 5,

Ho Chi Minh city, Vietnam.

Abstract

In this paper, we construct the chain-level of the Lannes-Zarati homomorphism

on the lambda algebra Using this chain-level, we investigate the vanishing of the Lannes-Zarati homomorphism of rank five and six

Keywords: Adams spectral sequence, Lambda algebra, Dyer-Lashof algebra, Lannes-Zarati homomorphism, Hurewicz homomorphism

2000 MSC: Primary 55P47, 55Q45, 55S10, 55T15

1 Introduction

Let Ps:= H∗(B(Z/2)s) ∼= F2[x1, · · · , xs] be the polynomial algebra on s gen-erators x1, · · · , xs, each of degree 1 The general linear group GLs:= GL(s, F2) and the mod 2 Steenrod algebra A both act on Ps in the usual way Let Dsbe the Dickson algebra (see Dickson [8]), which is the algebra of invariants

Ds:= F2[x1, · · · , xs]GLs Since the action of A and of GLsupon Ps commute with each other, Ds is an A-module

In [16], Lannes and Zarati constructed, for each s > 0, a homomorphism (so-called the Lannes-Zarati homomorphism)

ϕs: ExtsA(F2, F2) // PD∗

s mapping from the cohomology of the Steenrod algebra to the subspace of the dual of the Dickson algebra spanned by all elements annihilated by all positive degree Steenrod operations, P Ds∗ They also showed that ϕs, for s ≤ 2, are

I This work is partial supported by the grant of NAFOSTED.

∗ Corresponding author

Email addresses: chonkh@gmail.com (Phan Hoàng Chơn),

dongthanhtriet.dhsg@gmail.com (Đồng Thanh Triết)

non-trivial (i.e ϕ1 is an isomorphim and ϕ2 is an epimorphism) Moreover, these homomorphism correspond to an associated graded of the Hurewicz map

H : π∗S(S0) ∼= π∗(Q0S0) // H∗(Q0S0)

of the base-point component Q0S0of the infinite loop space QS0= lim

−→ΩnΣnS0 Beside, the results of Adams [1] and Browder [3] show that the Hopf invariant one and the Kervaire invariant one spherical classes (if exist) are respectively detected by certain permanent cycles in Ext1,∗A (F2, F2) and in Ext2,∗A (F2, F2) Therefore, the results of Lannes and Zarati closely corresponds to the classical conjecture on the spherical classes, which states that only the classes of Hopf invariant one and Kervaire invariant one (if exist) are detected by the Hurewicz homomorphism (see Curtis [7] and Wellington [28])

As above discussion, Hưng set up a conjecture [10] that ϕs= 0 in any positive stems for s > 2 This conjecture corresponds to an associated graded version of the classical conjecture on the spherical classes (see [11]) The conjecture has been proved for s = 3 [13] and s = 4 [12] Furthermore, Hưng and Peterson [13] also proved that ϕs is zero on decomposable elements for s > 2 The fact that

ϕ3 and ϕ4are trivial in any positive stems [13], [12] bases on the computations

of Wang [27] (for Ext3,∗A (F2, F2)), Lin-Mahowald [17] (for Ext4,∗A (F2, F2)) and the defining the basis of D3and D4as modules over the Steenrod algebra (so-called the “hit” problem for Ds) More detail, since stems where the domain and the rank of ϕs, s = 3, 4, are nontrivial are not compatible with each other though

ϕs, s = 3, 4, then these homomorphisms are vanishing Recently, using the ac-tion of the squaring operaac-tion on the Dickson algebra, Hưng-Quỳnh [14] showed that Hưng’s conjecture is true on almost known elements of the cohomology of the Steenrod algebra

In this paper, we construct the chain-level map of the Lannes-Zarati ho-momorphism on the lambda algebra We show that the Lannes-Zarati homo-morphism is induced by the canonical projection from the lambda algebra onto the Dyer-Lashof algebra Therefore, the squaring operation acting on lambda algebra (see Lin-Mahowald [17]) induces naturally the squaring on the Dyer-Lahof These squaring commute with each other through the Lannes-Zarati homomorphism Using these facts, we show that ϕ5= 0 in any positive stems and that ϕ6= 0 in all indecomposable elements of stems not greater than 114 The advantage of our method is that we need not use the “hit” problem for

Ds In this work, we also show that the action of the “big” Steenrod algebra

on H∗(B(Z/2)s)Σ s (see Singer [26]) induces an action on D∗s We expect that this action becomes an useful tool to study the “hit” problem of Ds and the Lannes-Zarati homomorphism

The paper is divided into five sections The first two sections are preliminar-ies In section 2, we review some main points of the lambda algebra, the Dyer-Lashof algebra as well as the Lannes-Zarati homomorphism The chain-level

of the Lannes-Zarati homomorphism over the lambda algebra is constructed in Section 3 Section 4 contains some discussion about the squaring operations and the action of the “big” Steenrod algebra on Ds The main results are present in

the last section.

2 Preliminaries

In this section, we review some main points of the lambda algebra, the Dyer-Lashof algebra as well as the Lannes-Zarati homomorphism

2.1 The lambda algebra and the Dyer-Lashof algebra

The lambda algebra, Λ, is the differential bigraded algebra with unit over

F2generated by symbols λi, i ≥ 0, of bidegree (1, i) subject to adem relations

λaλb=X

t

t − b − 1 2t − a



for all a, b ≥ 0 Here nk
is interpreted as the coefficient of xk in expansion of (x + 1)nso that it is defined for all integer n and all non-nagative integer k (see Chơn-Hà [6])

The differential of the lambda algebra is given by

δ(λn) =X

j

n − j − 1

j + 1



Let Λs be the subspace of the lambda algebra spanned by all monomials in the λiof the length s It is well-known that Λshas an additive basis consisting of all the admissible monomials, which are monomials in the form λI = λi1 λis where i1≤ 2i2, , is−1≤ 2is

It should be note that our definition of the lambda algebra follows that of Singer [24], which is opposite (by the canonical reversing-order map) to the original version in Bousfield et al [2]

Let M be a graded right A-module, then Λ ⊗ M is a differential module It

is the direct sum of Λs⊗ M of homological degree s Its differential is given by

δ(λ ⊗ x) = δ(λ) ⊗ x +X

i≥0

λλi⊗ (xSqi+1)

From Bousfield et al [2] and Priddy [22], there is the canonical isomorphism

Exts,s+tA (F2, M ) ∼= Hs,t(Λ ⊗ M )

When M = F2, we have the isomorphism Exts,s+tA (F2, F2) ∼= Hs,t(Λ)

An important subalgebra of the Λ is the Dyer-Lashof algebra R, which is the algebra of homology operations acting on the homology of infinite loop spaces For any monomial λI = λi1 λis, we define the excess of λI to be e(I) = e(λI) = i1−i2−· · ·−is Then the Dyer-Lashof algebra is defined as the quotient

of lambda algebra by the two-sided ideal generated by all monomials of negative excess

Denote by Qi is the image of λi through the quotient map Λ // R A monomial QI is called admissible if it is the image of an admissible monomial

λI Then the set of all admissible monomials QI of non-negative excess provides

an additive basis of R Let Rsbe the subspace of R spanned by all monomials

of length s As a coalgebra (see Madsen [19], and Mùi [21]) Rsis isomorphic to the dual of the Dickson algebra, D∗

s 2.2 The Lannes-Zarati homomorphism

For any left A-module M , the destabilization of M is defined by

D(M ) = M/EM, where EM = SpanF

2{Sqkx : k > deg(x), x ∈ M } Because D(−) is the right exact functor, HomF2(D(−), F2) is the left exact functor Let Ds be the s-th right derived functor of the left exact functor HomF2(D(−), F2) Then

Ds(M ) = Hs(HomF2(DF (M ), F2)), where F (M ) is the free resolution (or projective resolution) of M

Define e∗1(M ), which is the dual of the Singer’s element e1(M ), to be the connecting homomorphism of the functor HomF2(D(−), F2) associated with the short exact sequence

0 // M ⊗ P1 // M ⊗ ˆP // Σ−1M // 0, where ˆP = Span{xi : i ≥ −1}, which is an A-module with the A-action given

by Sqn(x−1) = xn−1 Put

e∗s(M ) = (e∗1(M )) ◦ · · · ◦ ((e∗1(M ) ⊗ Ps−1)), where Ps:= H∗(B(Z/2)s) ∼= F2[x1, , xs], each xi of degree 1

Then e∗s(M ) : D0(M ⊗ Ps) // Ds(Σ−sM ) As Singer’s result [25], e∗s(M ) factors through the coinvariant of HomF2(M ⊗ Ps, F2) = M∗⊗ H∗(B(Z/2)s) under the action of the general linear group GLs= GL(s, F2)

In [16], Lannes and Zarati show that

Theorem 2.1 (Lannes-Zarati [16]) Let Ds ⊂ Ps be the Dickson algebra of

s variables Then αs = e∗s(ΣF2) : ΣD∗s // Ds(Σ1−s

F2) is an isomorphic of internal degree 0

Because of the definition of the functor D, the projection M //F2⊗AM factors through DM Then it induces a commutative diagram

· · · // D(FnM )

in

// D(Fn−1M )

i n−1

// · · ·

· · · //F2⊗AFnM //F2⊗AFn−1M // · · ·

Here the horizontal arrows are induced by the differential of F M , and i∗is given by

in([z]) = [1 ⊗Az]

Passing the functor HomF2(−, F2), we have a commutative diagram

· · · // HomF2(F2⊗AFn−1M, F2)

j n−1

// HomF2(F2⊗AFnM, F2)

jn

// · · ·

· · · // HomF2(DFn−1M, F2) // HomF2(DFnM, F2) // · · · Taking the cohomology, we have

js: ExtsA(F2, M ) // Ds(M )

[x] 7→ [x]

Since the positive degree Steenrod operations act trivially on ExtA(F2, M ), the image of js is contained in P Ds(M ), where P Ds(M ) is the subspace of

Ds(M ) spanned by all elements annihilated by all the positive-degree Steenrod operations Therefore, we have the homomorphism

js: ExtsA(F2, M ) // PDs(M )

When M = Σ1−s

F2, we obtain

js: ExtsA(F2, Σ1−sF2) // PDs(Σ1−sF2)

In [16], Lannes and Zarati defined, for each s > 0, a homomorphism

ϕs= Σ(e∗s)−1(Σ1−sF2)jsΣ−1 : ExtsA(F2, Σ−sF2) // PD∗

s, and they showed that ϕsis the algebraic version of the Hurewicz map

H : π∗S(S0) ∼= π∗(Q0S0) // H∗(Q0S0)

However, the proof of this assertion is unpublished, but it is sketched in Goerss [9] and Lannes [15]

3 The chain-level of ϕs on the lambda algebra

In this section, we construct an representation of ϕs in the lambda algebra Let M be a left A-module Recall that B∗(M ) = ⊕s≥0Bs(M ) is the usual bar resolution of M with

Bs(M ) = A ⊗ I ⊗ · · · ⊗ I

s times

⊗M,

where I is the argumentation ideal of A, which is the ideal of A generated by all the positive degree Steenrod operations

The element a0⊗ a1 ⊗ · · · ⊗ as⊗ m ∈ Bs(M ) has homological degree s and internal degree P

ideg(ai) + deg(m) The bar resolution B∗(M ) is the differential A-module with the A-action given by

a(a0⊗ a1⊗ · · · ⊗ as⊗ m) = aa0⊗ a1⊗ · · · ⊗ as⊗ m,

and the differential given by

∂(a0⊗ a1⊗ · · · ⊗ as⊗ m) =

s−1 X

i=0

a0⊗ · · · ⊗ aiai+1⊗ · · · ⊗ m + a0⊗ a1⊗ · · · ⊗ asm

Let C∗(M ) = HomA(B∗(M ), F2) ∼= HomF2(F2⊗AB∗(M ), F2) be the cobar resolution of M with

Cs(M ) = A∗⊗ · · · ⊗ A∗

s times

⊗M∗,

where A∗and M∗are respective the dual of A and M Because Exts,∗A (F2, M ) is isomorphic to Hs(C∗(M )), the cobar resolution C∗(M ) is a suitable cocomplex

to compute the cohomology of the Steenrod algebra

As we well-known, (see Madsen [19] and Mùi [21]), Ds∗is isomorphic to Rs

In this section, we show that the canonical projection ˜ϕs: Λs→ Rsinduces the Lannes-Zarati homomorphism ϕs

For any q = Qi1 Qis ∈ Rs, we put ˜q = ξi1 +1

1 ⊗ · · · ⊗ ξi s +1

1 ⊗ Σ1−s1 ∈

Cs(Σ1−s

F2)

Lemma 3.1 The isomorphism αs: ΣD∗s // Ds(Σ1−s

F2) is given by

αs(Σq) = [˜q] + orther terms

Proof We have αs= e∗s(ΣF2) = e∗1(Σ2−sF2) ◦ · · · ◦ (e∗1(ΣF2) ⊗ Ps−1), where

e∗1(Σ−iF2) ⊗ Ps−i−2: Di+1(Σ−iPs−i−1) // Di+2(Σ−i−1Ps−i−2)

is the connecting homomorphism of the functor HomF2(D(−), F2) associated with the short exact sequence

0 // Σ−iPs−i−1 // Σ−iP ⊗ Pˆ s−i−2 // Σ−i−1Ps−i−2 // 0

Because of the short exact sequence

0 // EB∗(M ) // B∗(M ) // DB∗(M ) // 0, and B∗(M ) acyclic, we obtain the isomorphism

∂∗: Hs−1(HomF2(EB∗(M ), F2)) ∼= Ds(M )

Moreover, the short exact sequence

0 // ΣPs // ˆP ⊗ ΣPs−1 // Ps−1 // 0 induces the short exact sequence

0 // EB∗(ΣPs) // EB∗( ˆP ⊗ ΣPs−1) // EB∗(Ps−1) // 0

So we have the homomorphism

σ(Ps−1) : Hs−1(HomF2(EB(ΣPs), F2)) // Hs(HomF2(EB(Ps−1), F2)) Therefore, αs= ∂∗◦ σ(Σ1−s

F2) ◦ · · · ◦ σ(ΣPs−1)

For any admissible monomial q = Qi1 Qis ∈ D∗

s such that Σq is a cycle in HomF2(EΣPs, F2), it can be pulled back to the same element in HomF2(EΣ ˆP ⊗

Ps−1, F2) In HomF2(EΣPs, F2), we have

δ(Σq) =

i 1 +1 X

a=0

ξa1⊗ (ΣQi1· · · Qis)Sqa +Xξr

1

1 · · · ξr1`

` ⊗ (ΣQi 1· · · Qi s)Sq(r1, , r`1) = 0, where the second sum is taken over all the string (r1, , r1

`) 6= (1, 0, , 0) Therefore, in HomF2(EΣ ˆP ⊗ Ps−1),

δ(Σq) =

i1+1 X

a=0

 i1− a 2i1+ 2 − a



ξa1⊗ ΣQ−1(Qi2 Qis)Sqa−i1 −1 +Xξr11· · · ξr1`

` ⊗ ΣQ−1Qi02· · · Qi0s, where the second sum is taken over all the string (r11, , r1`) 6= (1, 0, , 0) and (ΣQi1· · · Qis)Sq(r1, , r1`) = ΣQ−1Qi02· · · Qi0s

Observation that if i1− a ≥ 0, then 2i1− a + 2 > i1− a So i1 −a

2i 1 +2−a
= 1

if and only if i1− a = −1 It implies

δ(Σq) = ξi1 +1

1 ⊗ ΣQ−1(Qi2 Qis) +Xξr

1

1 · · · ξr1`

` ⊗ ΣQ−1Qi02· · · Qi0s Hence,

σ(Ps−1)(Σq) = ξi1+1

1 ⊗ (Qi2 Qis) +hXξr11· · · ξr1`

` ⊗ Qi02· · · Qi0s

i Repeating this process, we obtain

σ(Σ1−sF2) ◦ · · · ◦σ(Ps−1)(Σq)

= [ξi1 +1

1 ⊗ · · · ⊗ ξis−1 +1

1 ⊗ Σ1−sQis] +hXξ1r1· · · ξr`1

` ⊗ · · · ⊗ ξrs−11

1 · · · ξrs−1`

` ⊗ Σ1−sQi(s−1)s

i

By the same argument, passing the homomorphism ∂ , we get

∂∗◦ σ(Σ1−s

F2) ◦ · · · ◦σ(Ps−1)(Σq)

= [ξi1 +1

1 ⊗ · · · ⊗ ξis +1

1 ⊗ Σ1−s1]

+hXξr11· · · ξr1`

` ⊗ · · · ⊗ ξ1rs· · · ξr`s

` ⊗ Σ1−s1i

= [˜q] + [z]

The proof is complete

Theorem 3.2 The canonical projection ˜ϕs : Λs // Rs ∼= D∗

s induces the Lannes-Zarati homomorphism ϕs

Proof In [23], Priddy showed that the lambda algebra is isomorphic to the Koszul cocomplex of F2, which is the quotient of the cobar resolution C∗(F2) = HomA(B∗(F2), F2) Therefore, there is a projection ι : C∗(F2) // Λ sending

ξa 7→ λa−1 and ξr

i 7→ 0 for all i ≥ 2 This map induces the isomorphism Exts,s+tA (F2, F2) ∼= Hs,t(Λ)

Under ι, the image of [z] is trivial in Λ So

ι([˜q + z]) = Σλi1· · · λis The theorem is proved

Proposition 3.3 The homomorphism ϕ = ⊕s≥1ϕs is an algebra homomor-phism

Proof Since the canonical projection ˜ϕ = ⊕s≥1ϕ˜s : Λ −→ R is the algebra homomorphism, the assertion is immediate from Theorem 3.2

4 The action of the “big” Steenrod algebra

From Liulevicius [18], May [20], there exists the Steenrod operations acting

on the Ext-groups

Sqi: Exts,tA (F2, F2) // Exts+i,2t

A (F2, F2), i ≥ 0

These Sqi

s satisfy the usual adem relations, but Sq0is no longer identity The definition of the operations uses “cup-i products” on a projective resolution of

F2as a A-module

Beside, in [26], Singer defines the operations

Sqi: (Γs,t)Σs // (Γs+i,2t)Σs+i, i ≥ 0, where (Γs,t)Σs is the subspace of Γs,t = Ht−s(B(Z/2)s) spanned by all coin-variants under the action of the symmetric group These Sqis satisfy the usual relations without the relation Sq0= 1

Let Γ = {Γs,t : s ≥ 0, t ≥ 0} be the bigraded algebra Let a1 · · · as be the dual of xi1

1 · · · xis

s ∈ H∗(B(Z/2)s), then the product on Γ is given by

Γs,t⊗ Γs0 ,t 0 // Γs+s 0 ,t+t 0

a(i1 )

1 · · · a(i s )

s ◦ a(is+1 )

1 · · · a(is+s0 )

s 0 7→ a(i1 )

1 · · · a(i s )

s a(is+1 ) s+1 · · · a(is+s0 )

s+s 0 The product induces a product on ΓΣ, which is the bigraded algebra with (ΓΣ)s,t= (Γs,t)Σs Then the action of Sqi is given by

Sqi(a(n)k ) =







a(2n+1)k if i = 0, (a(n)k )2 if i = 1,

0 if i > 1, and the Cartan formula

In this section, we show that Sqis induce the action on the dual of the Dickson algebra Let I = {Is,t : s ≥ 0, t ≥ 0} and J = {Js,t: s ≥ 0, t ≥ 0}, where Is,t and Js,tare ideals of Γs,tgenerated respectively by {σ(x) + x|x ∈ Γs,t, σ ∈ Σs} and {g(x) + x|x ∈ Γs,t, g ∈ GLs} It is sufficient to prove that actions of Sqis commute with the action of τ = (1 0) modulo an element in J

Let Sq(u) =P

i≥0Sqiui be the formal power series, and let

ai(x) =X

k≥0

a(k)i xk; ¯ai(x) =X

k≥0

a(2k+1)i x2k

Modulo an element in I, we have

Sq(u)a1(x2) = ¯a1(x) + ua1(x)a2(x) (4.1) Lemma 4.1 Let a, b ∈ {a1, a2· · · } ∈ Γ, then

τ (¯a(x)¯b(y)) = ¯a(x)¯b(x + y) (4.3) Proof The first is immediate as follows

τ (a(x)b(y)) = (a + b)(x)b(y) = a(x)b(x)b(y)

= a(x)



 X

i,j

b(i)b(j)xiyj



= a(x)X

i+j

b(i+j) X

i

i + j i



xiyj

!

= a(x)X

i+j

b(i+j)(x + y)i+j = a(x)b(x + y)

For the second formula, we observe that 2k+1+ii
= 0 (mod 2) if i odd and

that 2i
= 2i
(mod 2) Therefore,

τ (¯a(x)¯b(y)) = X

i (a + b)(2i+1)x2i

!

 X

j

b(2j+1)y2j





i−k

a(2i−2k+1)x2i−2k

! X

k

b(2k)x2k

!

 X

j

b(2j+1)y2j





= ¯a(x)X k+j

b(2j+2k+1) X

k

2k + 2j + 1 2k



x2ky2y

!

= ¯a(x)X k+j

b(2j+2k+1) X

k

2k + 2j 2k



x2ky2y

!

= ¯a(x)¯b(x + y), since 2ni
= 0 (mod 2) for i odd

Using the lemma, we have the following proposition

Proposition 4.2 For a, b ∈ {a1, a2, · · · } ∈ Γ1,2 and for any formal variables

u, x, y, we have

Sq(u)(τ (a(x2)b(y2))) = τ (Sq(u)(a(x2)b(y2))) (mod J )

Proof Since coproduct ψ(Sq(u)) = Sq(u) ⊗ Sq(u), using Cartan formula and (4.1), we have

Sq(u)(a(x2)b(y2)) = (Sq(u)a(x2))(Sq(u)b(y2))

= [¯a(x) + ua(x)a0(x)][¯b(y) + ub(y)b0(y)]

= ¯a(x)¯b(y) + u2a(x)b(y)a0(x)b0(y) (mod I), where a0, b0∈ {a1, a2, · · · } \ {a, b} Therefore, using Lemma 4.1,

τ (Sq(u)(a(x2)b(y2))) = ¯a(x)¯b(x + y) + u2a(x)b(x + y)a0(x)b0(y) (mod I)

On the other hand,

Sq(u)(τ (a(x2)b(y2))) = Sq(u)(a(x2)b((x + y)2))

= ¯a(x)¯b(x + y) + u2a(x)b(x + y)a0(x)b0(x + y) (mod I) Hence,

τ (Sq(u)(a(x2)b(y2))) + Sq(u)(τ (a(x2)b(y2)))

= u2a(x)b(x + y)[a0(x)b0(y) + a0(x)b0(x + y)] (mod I)

= u2a(x)b(x + y)[a0(x)b0(y) + τ0(a0(x)b0(y)]

= 0 (mod J ), where τ0∈ G sends a0 to a0+ b0 and fixes other variables