ON THE MOD p LANNES ZARATI HOMOMORPHISM

ON THE MOD p LANNES-ZARATI HOMOMORPHISM
PHAN H. CHƠN AND ĐỒNG T. TRIẾT

Abstract. The mod 2 Lannes-Zarati homomorphism was constructed in [21],
which is considered as a graded associated version of the mod 2 Hurewicz map
in the E2 -term of Adams spectral sequence. The map is studied by many
authors such as Lannes-Zarati [21], Hưng [14], [15], [16], Hưng et. al. [18],
Chơn-Tri´
êt [6]. In this paper, we construct an analogue ϕs for p odd, and we
also investigate the behavior of this map for s ≤ 3.

1. Introduction and statement of results
For any pointed space X, let QX = limΩn Σn X be the infinite loop space of X.
−→

An element ξ ∈ H∗ QX = H∗ (QX; Fp ) is called a spherical class if there exists an
/ H∗ QX
element η ∈ π∗ (QX) = π∗S (X) such that h∗ (η) = ξ, where h∗ : π∗ (QX)
is the Hurewicz map. For p = 2, work of Curtis [10] shows that the Hopf invariant
one elements and the Kervaire invariant one elements in π∗ (Q0 S0 ) (if they exist)
are those whose images are nontrivial in H∗ Q0 S 0 under the mod 2 Hurewicz map
/ H∗ Q0 S 0 ; and he conjectured that there are only spherical classes
h∗ : π∗ (Q0 S 0 )
0
in H∗ Q0 S those are detected by the Hopf invariant one elements and the Kervaire
invariant one elements, where Q0 S 0 is the component of QS 0 containing the basepoint. Later, Wellington [30] generalized the Curtis’ result for p > 2 and he was
led to an analogue conjecture. These conjectures are called the classical conjecture
on spherical classes.
An algebraic approach to attack the conjecture is to study the graded associated
/ H∗ Q0 S 0 in E2 -term of the Adams
of the mod p Hurewicz map h∗ : π∗ (Q0 S 0 )

spectral sequence.
For p = 2, this homomorphism was constructed by Lannes and Zarati [21], the
so-called Lannes-Zarati homomorphism. In more detail, for each s ≥ 1, there is a
homomorphism of Singer’s type
/ Ann(Rs (F2 )# )t ∼
ϕs : Exts,s+t (F2 , F2 )
= Ann(D[s]# )t ,
A

where D[s] is the (graded) dual of the Dickson algebra D[s], Rs is the Singer’s
functor (see [21]); and we denote Ann(M ) the subspace of M consisting of all elements annihilated by all positive elements in the Steenrod algebra A. The behavior
of ϕs is actually studied by Lannes-Zarati [21] (for s ≤ 2), Hưng [14] (for s = 3),
Hưng [16] (for s = 4), Hưng-Quỳnh-Tu´ân [18] (for s = 5) and Chơn-Tri´êt [6] (for
s = 6 and stem ≤ 114).
#

Date: February 2, 2015.
2010 Mathematics Subject Classification. Primary 55P47, 55Q45; Secondary 55S10, 55T15.
Key words and phrases. Spherical classes, Hurewizc map, Lannes-Zarati homomorphism,
Adams spectral sequence.
This work is partial supported by a NAFOSTED gant.
1


2

P. H. CHƠN AND Đ. T. TRIẾT

For p > 2, from results of Zarati [31], the sth left derived functor Ds (Σ1−s Fp )
of the destabilization functor is isomorphic to ΣRs (Fp ) ∼

= ΣB[s], where B[s] is
the image of the restriction from cohomology of the symmetric group Σps to the
cohomology of the elementary p-group of rank s, Es [25]. Therefore, there exists
an analogue homomorphism, which is also called Lannes-Zarati homomorphism,
/ Ann(B[s]# )t .

ϕs : Exts,s+t
(Fp , Fp )
A

Using Goodwillie towers, Kuhn also pointed out the existence of ϕs as a graded
associated version of the Hurewicz map in the E2 -term of the Adams spectral
sequence [20]. In addition, the method of Kuhn can be apply to other generalized
cohomology theories.
In the paper, we are interested in the study of the mod p Lannes-Zarati homomorphism for p odd. We show that, up to a sign, the canonical inclusion B[s] → Γ+
s
is the chain-level representation of the dual of ϕs ,
ϕ#
s : Fp ⊗A B[s]

/ TorA
s (Fp , Fp ),

where Γ+ = ⊕s≥0 Γ+
s is the Singer-Hưng-Sum chain complex [19]. In more detail,
we obtain the following theorem, which is the first main result of the paper.
Theorem 4.6. The inclusion map ϕ˜#
s : B[s]
γ → (−1)

/ Γ+
s given by

s(s−1)
+(s+1) deg γ
2

γ

is the chain-level representation of the dual of the Lannes-Zarari homomorphism
ϕ#
s .
The theorem is an extended of Theorem 3.9 in [15] for p odd.
Let Λopp be the opposite algebra of the Lambda algebra defined by Bousfield
et. al. [3], and let R be the Dyer-Lashof algebra, which is the algebra of homology
operations acting on the homology of infinite loop spaces. The algebra R is also
isomorphic to a quotient of Λopp [10], [30]. It is well-known that B[s]# ∼
= Rs [8]
and (Γ+ )# ∼
= Λopp [19], where Rs is the subspace of R spanned by all monomials of
Rs is
length s. Therefore, in the dual, up to a sign, the canonical projection Λopp
s
the chain-level representation map of the Lannes-Zarati homomorphism ϕs , which
is given by the following corollary.
Corollary 4.7. The projection ϕ˜s : Λopp
s
ϕ˜s (λI ) = (−1)

/ Rs given by

s(s−1)
+(s+1) deg(λI )
2

QI

is the chain-level representation of the Lannes-Zarati homomorphism ϕs .
From Liulevicius [22], [23] and May [24], there exists the the power operation
P 0 acting on the cohomology of the Steenrod algebra Exts,s+t
(Fp , Fp ) whose chainA
level representation in the cobar complex is induced from the Frobenius map. Since
its representation in Λopp induces naturally an operation in R (see Lemma 5.1), and
the latter is compatible with the A-action on R (see Lemma 5.2), then there exists
an power operation acting on Ann(Rs ), which is also denoted by P 0 . Furthermore,
these power operations commute with each other through the Lannes-Zarati homomorphism ϕs (see Proposition 5.3). Using these results to study the behavior of
ϕs , we obtain the following results.


ON THE MOD p LANNES-ZARATI HOMOMORPHISM

3

Theorem 6.1. The first Lannes-Zarati homomorphism
/ Ann(B[1]# )t
ϕ1 : Ext1,1+t
(Fp , Fp )
A
is isomorphic.
This result is an analogue of the case p = 2 [21]. The behavior of ϕ2 is given by

the theorem.
Theorem 6.2. The second Lannes-Zarati homomorphism
2,2+t
/ Ann(B[2]# )t
ϕ2 : ExtA
(Fp , Fp )
is vanishing for t = 0 and t = 2(p − 1)pi+1 − 2, i ≥ 0.
From the result of Wellington [30], Ann(R2 ) is nontrivial at stem t = 0, t =
2(p−1)pi+1 −2 and t = 2(p−1)p(pi +· · ·+1). Therefore, ϕ2 is not an epimorphism.
The behavior of ϕ3 is given by the following theorem, which is the final result of
this work.
Theorem 6.4. The third Lannes-Zarati homomorphism
3,3+t
/ Ann(B[3]# )t
ϕ3 : ExtA
(Fp , Fp )
is vanishing for all t > 0.
From the above results, we observe that the map ϕs , for s ≤ 3, is only nontrivial
in positive stem corresponding with the Hopf invariant one and Kervaire invariant
one elements (if they exist). Based on these results together with the classical
conjecture on spherical classes and Hưng’s conjecture [14, Conjecture 1.2], we are
led to a conjecture, which is considered as a graded associated version of the classical
one on the spherical classes in E2 -term of the Adams spectral sequence, as follows.
Conjecture 1.1. The homomorphism ϕs vanishes in any positive stem t for s ≥ 3.
Of course, the classical conjecture on spherical class is not a consequence of Conjecture 1.1. But if Conjecture 1.1 were false on a permanent cycle in Exts,s+t
(Fp , Fp ),
A
then the classical conjecture on spherical classes could be also false.
The Singer transfer was introduced by Singer [29] (for p = 2) and Crossley [9]
(for p > 2), which is given by, for s ≥ 1,

/ Exts,s+t
T rs : [Ann(Ht BEs )]GLs
(Fp , Fp ),
A
where GLs is the general linear group. The results of Singer [29], Boardman [2], Hà
[12], Nam [27], Chơn-Hà [4], [5] (for p = 2) and Crossley [9] (for p odd) showed that
the image of Singer transfer is a big enough and worthwhile to pursue subgroup of
the Ext group. It is well-known that the Ext group is too mysterious to understand,
although it is intensively studied. In order to avoid the shortage of our knowledge
of the Ext group, we want to restrict ϕs on the image of the Singer transfer. Then
we have a weak version of Conjecture 1.1.
Conjecture 1.2. The composition
js := ϕs ◦ T rs : [Ann(Ht BEs )]GLs

/ Ann(B[s]# )t

is vanishing in any positive degree t for s ≥ 3.
From Theorem 4.6, Theorem A.2 and Proposition A.12, it is clear that, up to
a sign, the canonical inclusion B[s] → H ∗ BEs is the chain-level representation of
the dual of js . Thus, Conjecture 1.2 is equivalent to the following conjecture.


4

P. H. CHƠN AND Đ. T. TRIẾT

¯ ∗ BEs , where A¯ is the augmentation ideal
Conjecture 1.3. For s ≥ 3, B[s] ⊂ AH
of A.
For p = 2, Conjecture 1.2 and Conjecture 1.3 appeared in [14, Conjecture 1.3

and Conjecture 1.5], which were showed by Hưng-Nam in [17].
The paper is organized as follows. Section 2 is a preliminary on the SingerHưng-Sum chain complex, the Lambda algebra and the Dyer-Lashof algebra. In
Section 3 and 4, we construct the mod p Lannes-Zarati homomorphism and its
chain-level representation. Section 5 is devoted to develop the power operations.
The behavior of the Lannes-Zarati homomorphism is investigated in Section 6. The
chain-level representation of the Singer transfer in Singer-Hưng-Sum chain complex
is established in Appendix section.

2. Preliminaries
In this section, we recall some preliminaries about the Singer-Hưng-Sum chain
complex, the Lambda algebra as well as the Dyer-Lashof algebra (see [19], [3], [28]
and [8] for more detail).
2.1. The Singer-Hưng-Sum chain complex. Let Es be the s-dimensional Fp vector space, where p is an odd prime number. It is well-known that the mod p
cohomology of the classifying space BEs is given by
Ps := H ∗ BEs = E(x1 , · · · , xs ) ⊗ Fp [y1 , · · · , ys ],
where (x1 , · · · , xs ) is the basis of H 1 BEs = Hom(Es , Fp ), and yi = β(xi ) for
1 ≤ i ≤ s with β the Bockstein homomorphism.
Let GLs denote the general linear group GLs = GL(Es ). The group GLs acts
on Es and then on H ∗ BEs according to the following standard action
ais yi ,

(aij )ys =

ais xi ,

(aij )xs =

i

(aij ) ∈ GLs .

i

The algebra of all invariants of H ∗ BEs under the actions of GLs is computed by
Dickson [11] and Mùi [25]. We briefly summarize their results. For any n-tuple of
rj
non-negative integers (r1 , . . . , rs ), put [r1 , · · · , rs ] := det(yip ), and define
Ls,i := [0, · · · , ˆi, . . . , s];

Ls := Ls,s ;

qs,i := Ls,i /Ls ,

for any 1 ≤ i ≤ s.
In particular, qs,s = 1 and by convention, set qs,i = 0 for i < 0. Degree of qs,i is
2(ps − pi ). Define
Vs := Vs (y1 , · · · , ys ) :=

(λ1 y1 + · · · + λs−1 ys−1 + ys ).
λj ∈Fp

Another way to define Vs is that Vs = Ls /Ls−1 . Then qs,i can be inductively
expressed by the formula
p
qs,i = qs−1,i−1
+ qs−1,i Vsp−1 .


ON THE MOD p LANNES-ZARATI HOMOMORPHISM

5

For non-negative integers k, rk+1 , . . . , rs , set

[k; rk+1 , · · · , rs ] :=

1
k!

x1
·
x1
y1p

rk+1

·
rn
y1p

···
···
···
···
···
···

xs
·
xs

ysp

rk+1

.

·
rs
ysp

For 0 ≤ i1 < · · · < ik ≤ s − 1, we define
Ms;i1 ,...,ik := [k; 0, · · · , ˆi1 , · · · , ˆik , · · · , s − 1],
Rs;i1 ,··· ,ik := Ms;i1 ,...,ik Lsp−2 .
The degree of Ms;i1 ,··· ,ik is k + 2((1 + · · · + ps−1 ) − (pi1 + · · · + pik )) and then
the degree of Rs;i1 ,··· ,ik is k + 2(p − 1)(1 + · · · + ps−1 ) − 2(pi1 + · · · + pik ).
The subspace of all invariants of H ∗ BEs under the action of GLs is given by the
following theorem.
Theorem 2.1 (Dickson [11], Mùi [25]).
(1) The subspace of all invariants under the action of GLs of Fp [x1 , · · · , xs ] is given by
D[s] := Fp [x1 , · · · , xs ]GLs = Fp [qs,0 , · · · , qs,s−1 ].
(2) As a D[s]-module, (H ∗ BEs )GLs is free and has a basis consisting of 1 and
all elements of {Rs;i1 ,··· ,ik : 1 ≤ k ≤ s, 0 ≤ i1 < · · · < ik ≤ s − 1}. In other
words,
s

(H ∗ BEs )GLs = D[s] ⊕

Rs;i1 ,··· ,ik D[s].
k=1 0≤i1 <···

(3) The algebraic relations are given by
2
Rs;i
= 0,
k−1
Rs;i1 · · · Rs;ik = (−1)k(k−1)/2 Rs;i1 ,··· ,ik qs,0

for 0 ≤ i1 < · · · < ik < s.
Let B[s] be the subalgebra of (H ∗ BEs )GLs generated by
(1) qs,i for 0 ≤ i ≤ s − 1,
(2) Rs;i for 0 ≤ i ≤ s − 1,
(3) Rs;i,j for 0 ≤ i < j ≤ s − 1.
Mùi [25] show that the algebra B[s] is the image of the restriction from the cohomology of the symmetric group Σps to the cohomology of the elementary abelian
p-group of rank s, Es .
∗
Let Φs := H ∗ BEs [L−1
s ] be the localization of H BEs obtained by inverting Ls .
It should be noted that Ls is the product of all non-zero linear forms of y1 , . . . , ys .
So inverting Ls is equivalent to inverting all these forms. The action of GLs on
H ∗ BEs extends an action of it on Φs . Set
∆s := ΦTs s ,

s
Γs := ΦGL
,
s

where Ts is the subgroup of GLs consisting of all upper triangle matrices with 1’s
on the main diagonal.

6

P. H. CHƠN AND Đ. T. TRIẾT

Put ui := Mi;i−1 /Li−1 and vi := Vi /qi−1,0 . Then, from [19], we have
∆s = E(u1 , . . . , us ) ⊗ Fp [v1±1 , . . . , vs±1 ],
±1
Γs = E(Rs;0 , . . . , Rs;s−1 ) ⊗ Fp [qs,0
, qs,1 , . . . , qs,s−1 ].

Let ∆+
s be the subspace of ∆s spanned by all monomials of the form
(p−1)i1 −

1

u11 v1
Γ+
s

· · · uss vs(p−1)is − s ,

i

∈ {0, 1}, 1 ≤ i ≤ s, j1 ≥

1,

∆+

s .

and let
:= Γs ∩
From [19], Γ+ = ⊕s≥0 Γ+
s is a graded differential A-algebra with the differential
induced by
∂(u11 v1i1 · · · uss vsis ) =

(−1)
0,

1 +···+ s−1

i

s−1
s−1
,
vs−1
u11 v1i1 · · · us−1

s = −is = 1;
otherwise,
(2.1)

where Γ+
0 = Fp .
For any A-module M , we define the stable total power Sts (x1 , y1 , . . . , xs ys ; m),
for m ∈ M , as follows

Sts (x1 , y1 , . . . , xs ys ; m)
:=

(−1)

1 +i1 +···+ s +is

−(p−1)i1 −

uss · · · u11 v1

1

· · · uss vs−(p−1)is −

s

=0,1,ij ≥0

⊗ (β 1 P i1 · · · β s P is )(m).
For convenience, we put Sts (m) = Sts (x1 , y1 , . . . , xs ys ; m). And let Sts (M ) =
{Sts (m) : m ∈ M }.
Then Γ+ M := ⊕s≥0 (Γ+ M )s , where (Γ+ M )0 = M and (Γ+ M )s = Γ+
s Sts (M ), is
(p−1) −
∈ Γ+
a differential module with its differential given by, for γ =
γ
u
v

s
, s
s
,
and m ∈ M , where γ , ∈ Γ+
s−1 ,
∂(γSts (m)) = (−1)deg γ+1

(−1) γ , Sts−1 (β 1− P m).

(2.2)

,

∼ TorA (Fp , M ) for any A-module
In [19], Hưng and Sum showed that Hs (Γ+ M ) =
s
+
M . Therefore, Γ M is a suitable complex to compute TorA
s (Fp , Fp ).
2.2. The Lambda algebra and the Dyer-Lashof algebra. In [3], Bousfield et.
al. defined the Lambda algebra Λ, that is a differential algebra for computing the
cohomology of the Steenrod algebra. In [28], Priddy showed that the opposite of
the Lambda algebra Λopp is isomorphic to the co-Koszul complex of the Steenrod
algebra.
Recall that Λopp is a graded differential algebra generated by λi−1 of degree
2i(p − 1) − 1 and µi−1 of degree 2i(p − 1) subject to the adem relations

i+j=n

i+j=n

i+j=n

i+j
λi−1+pm λj−1+m = 0,
i
i+j
(λi−1+pm µj−1+m − µi−1+pm λj−1+m ) = 0,
i
i+j
λi+pm µj−1+m = 0,
i


ON THE MOD p LANNES-ZARATI HOMOMORPHISM

i+j=n

7

i+j
µi+pm µi−1+m = 0,
i

for all m ≥ 0 and n ≥ 0.
And the differential is given by
d(λn−1 ) =
i+j=n

d(µn−1 ) =
i+j=n

i+j
λi−1 λj−1 ,
i
i+j
(λi−1 µj−1 − µi−1 λj−1 ),
i

d(στ ) = (−1)deg σ σd(τ ) + d(σ)τ.
Let Λopp
be the subspace of Λopp spanned by all monomials of length s. By the adem
s
relations, Λopp
has an additive basis consisting of all admissible monomials (which
s
satisfying pik − k ≥ ik−1
are monomials of the form λI = λi11−1 · · · λiss−1 ∈ Λopp
s
#
be the
for 2 ≤ k ≤ s, where λi−1 is λi−1 if = 1 and µi−1 if = 0). Let (Λopp
s )
opp
∗
1
s
dual of Λs and let (λi1 −1 · · · λis −1 ) be the dual basis of the admissible basis.
By the same method of Hưng-Sum [19], it is easy to show that the map κs :

#
/ (Λopp
Γ+
s
s ) given by
(p−1)i1 −

κs (u11 v1

1

· · · uss vs(p−1)is − s ) = (−1)i1 +···+is (λi11−1 · · · λiss−1 )∗

is an isomorphism of differential modules over A.
An important quotient algebra of Λopp is the Dyer-Lashof algebra R, which is
also well-known as the algebra of homology operations acting on the homology of
infinite loop spaces.
For any admissible monomial λI = λi11−1 · · · λiss−1 ∈ Λopp
s , we define the excess
of λI or of I to be
s

s

e(λI ) = e(I) = 2i1 −

1

2(p − 1)ik +

−
k=2

s.
k=2

Then, the Dyer-Lashof algebra is the quotient of the algebra Λopp over the ideal
generated by all monomials of negative excess [10], [30].
Let β Qi be the image of λi−1 under the canonical projection. A monomial
I
Q = β 1 Qi1 · · · β s Qis is called admissible if λI is admissible. Then R has an
additive basis consisting of all admissible monomials of nonnegative excess.
Let Rs be the subspace of R spanned by all monomials of length s, then Rs
is isomorphic to B[s]# as A-coalgebras, where the A-action on R is given by the
Nishida’s relation (see May [8]).
From the above result, we observe that the restriction of κs on B[s] is isomorphism between B[s] and Rs# .
3. The Lannes-Zarati homomorphism
We want to sketch the work of Zarati [31] in this section. And we end this section
by the establishing the mod p Lannes-Zarati homomorphism.
Let M be the category of left A-modules. A module M ∈ M is called unstable
if β P i x = 0 for + 2i > deg(x) and for all x ∈ M . Let U be the full subcategory
of M consisting of all unstable modules.


8

P. H. CHƠN AND Đ. T. TRIẾT

The destabilization functor D : M → U is defined by, for M ∈ M,
D(M ) = M/EM,

i

where EM = SpanFp {β P x : + 2i > deg(x), x ∈ M }, which is a sub-A-module
of M because of the Adem relations. The functor D is right exact and admits left
derived functors Ds , s ≥ 0. Then
Ds (M ) = Hs (D(F (M ))),
for F (M ) the free resolution (or projective resolution) of M .
/ Dr−1 (P1 ⊗ M ) to be the connecting homomorDefine α1 (M ) : Dr (Σ−1 M )
phism of the functor D(−) associated to the short exact sequence
0 → P1 ⊗ M → Pˆ ⊗ M → Σ−1 M → 0,
where Pˆ is the A-module extended of P1 by formally adding a generator x−1
1 u1 of
−1 n(p−1)−1
x
u1
degree −1. The action of A on Pˆ is given by setting P n (x−1
u
)
=
1
1
1
n
−1
and β(x1 u1 ) = 1, while the summand P1 has its usual A-action. Put
αs (M ) = α1 (Ps−1 ⊗ M ) ◦ · · · ◦ α1 ( Σ−(s−1) M ),
/ Dr−s (Ps ⊗ M ).
then αs (M ) : Dr (Σ−s M )
/ D0 (Ps ⊗ M ).
When r = s, we obtain αs (M ) : Ds (Σ−s M )

Theorem 3.1 ([31, Theórème 2.5]). For any M ∈ U, the homomorphism αs (ΣM ) :
/ ΣRs M is an isomorphism of unstable A-modules, where Rs (−) is
Ds (Σ1−s M )
the Singer functor.
When M = Fp , Hải [13] showed that Rs (Fp ) ∼
= B[s]. Therefore, we have the
following corollary.
∼
=

Corollary 3.2. For s ≥ 0, αs := αs (ΣFp ) : Ds (Σ1−s Fp ) −
→ ΣB[s].
/ Fp ⊗A M factors
Because of the definition of the functor D, the projection M
through DM . Then it induces a commutative diagram
···

···

/ D(Fs M )


/ D(Fs−1 M )

/ ···

is−1

is


/ Fp ⊗A Fs−1 M

/ Fp ⊗A Fs M

/ ··· .

Here horizontal arrows are induced by the differential of F M , and i∗ is given by
is ([z]) = [1 ⊗A z].
Taking the homology, we get
/ TorA
s (Fp , M ).

is : Ds (M )

Since, for z ∈ Fs M and a > 0, is (Sq a [z]) = is ([Sq a z]) = [1 ⊗A Sq a z] = [0] ∈
F2 ⊗A Fs M , the induced map of is in homology factors through Fp ⊗A Hs (D(Fs M )).
Therefore, we have following commutative diagram
Ds (M )

is



/ TorA
s? (Fp , M )
¯is

Fp ⊗A Ds (M )

ON THE MOD p LANNES-ZARATI HOMOMORPHISM

When M = Σ1−s F2 , we obtain
¯is : Fp ⊗A Ds (Σ1−s Fp ))

9

1−s
/ TorA
Fp ).
s (Fp , Σ

For each s ≥ 1, we define
−1¯
is (1 ⊗A αs−1 )Σ : Fp ⊗A B[s]
ϕ#
s := Σ

−s
/ TorA
Fp )
s (Fp , Σ

In the dual, we have the Lannes-Zarati homomorphism for p odd
/ Ann(B[s]# )t .
ϕs : Exts,s+t
(Fp , Fp )
A
In [20], Kuhn showed that the map ϕs , for s ≥ 1, is the graded associated version

/ H∗ Q0 S 0 in the E2 -term of the Adams
of the mod p Hurewizc map h∗ : π∗ (Q0 S 0 )
spectral sequence.
4. The chain-level representation of ϕs
In this section, we construct the chain-level representation of ϕ#
s in the SingerHưng-Sum chain complex as well as the chain-level representation of ϕs in the
opposite algebra of the Lambda algebra.
For M ∈ M, recall that B∗ (M ) := ⊕s≥0 Bs (M ) is the usual bar resolution of M
with
Bs (M ) = A ⊗ A¯ ⊗ · · · ⊗ A¯ ⊗M,
s times

where A¯ is the augmentation ideal of A, which is the ideal of A generated by all
positive degree elements in A.
The element a0 ⊗ a1 ⊗ · · · ⊗ as ⊗ m ∈ Bs (M ) has homological degree s and
internal degree t = i deg(ai ) + deg(m). The total degree is s + t, i.e.
deg(a0 ⊗ a1 ⊗ · · · ⊗ as ⊗ m) = s +

deg(ai ) + deg(m).
i

The A-action on B∗ (M ) is given by
a(a0 ⊗ a1 ⊗ · · · ⊗ as ⊗ m) = aa0 ⊗ a1 ⊗ · · · ⊗ as ⊗ m,
and the differential of B∗ (M ) is given by
s−1

(−1)ei a0 ⊗ · · · ⊗ ai ai+1 ⊗ · · · ⊗ m

∂(a0 ⊗ a1 ⊗ · · · ⊗ as ⊗ m) =
i=0

− (−1)es−1 a0 ⊗ a1 ⊗ · · · ⊗ as m,
where ei = deg(a0 ⊗ · · · ⊗ ai ).
Since B∗ (M ) is the free resolution of M , by definition one has
TorA
s (N, M ) = Hs (N ⊗A B∗ (M )).
As B[s] ⊂ Γ+
s , for γ ∈ B[s], γ has an unique expansion
(p−1)i1 −

γ=

u11 v1
I=(

1

· · · uss vs(p−1)is − s .

1 ,i1 ,..., s ,is )∈I

Put
(−1)e(I) β 1− 1 P i1 ⊗ · · · ⊗ β 1− s P is ⊗Σ1−s 1 ∈ Bs−1 (Σ1−s Fp ),

γ˜ :=
I∈I

where e(I) = s +

1

+ ··· +

s

+ i1 + · · · + is .


10

P. H. CHƠN AND Đ. T. TRIẾT

Lemma 4.1. The element γ˜ ∈ EBs−1 (Σ1−s Fp ).
Proof. Fron the proofs of Lemma A.9, Lemma A.10 and Proposition A.12, one
gets that the exponents of vi ’s in the expansion of γ are nonnegative. Therefore
γ˜ ∈ Bs−1 (Σ1−s Fp ).
By the action of A,
(−1)e(I) β 1− 1 P i1 (1 ⊗ β 1− 2 P i2 ⊗ · · · ⊗ β 1− s P is ⊗Σ1−s 1).

γ˜ =
I∈I

Therefore, it is sufficient to show that
s

2i1 + (1 −

(2ik (p − 1) + (1 −

1) >

k ))

+ 1 − s,

k=2

it is equivalent to
s

2i1 −

1

s

2ik (p − 1) −

>
k=1

k.

(4.1)

k=2

Also from the proofs of Lemma A.9, Lemma A.10 and Proposition A.12, we ob(p−1)is − s
(p−1)i1 − 1
,

serve that qs,i and Rs;i,j can be written in the sum of u11 v1
· · · uss vs
where ( 1 , i1 , . . . , s , is ) satisfies (4.1), therefore so is γ.
Lemma 4.2. The element γ˜ is a cycle in EBs−1 (Σ1−s Fp ).
Proof. Let
/A

Ω : ∆+
2
(p−1)i1 −

u11 v1

1

(p−1)i2 −

u22 v2

2

→ (−1)

1 +i1 +i2

β 1− 1 P i1 β 1− 2 P i2 .

From the result of Ciampell and Lomonaco [7], one gets Γ2 ⊂ KerΩ.
/ ∆+ ⊗ ∆+ ⊗ ∆+
Consider the diagonal map ψ : ∆+

s
q−1
2
s−q−1 defined by

(p−1)i
−
k
k
k

⊗ 1 ⊗ 1,
k < q,
 uk vk
(p−1)ik − k
k (p−1)ik − k
k
ψ(uk vk
)=
1 ⊗ uk−q+1 vk−q+1
⊗ 1, i ≤ k ≤ i + 1,


(p−1)i −
k
1 ⊗ 1 ⊗ uk−q−1
vk−q−1k k , k > i + 1.
From results of Hưng and Sum [19, Corollary 3.4], ψ(Γs ) ⊂ Γi−1 ⊗ Γ2 ⊗ Γs−i−1 .
Define the homomorphism
/ A⊗(s−1) = A ⊗ · · · ⊗ A,

πs,q : ∆+
s
s−1 times

given by
(p−1)i1 −

πs,q (u11 v1

= (−1)

1

(p−1)iq+1 −

q+1
· · · uqq vq(p−1)iq − q uq+1
vq+1

q+1

· · · uss vs(p−1)is − s )

1 +···+ q +i1 +···+is

× β 1− 1 P i1 ⊗ · · · ⊗ β 1− q P iq β 1−

q+1

P iq+1 ⊗ · · · ⊗ β 1− s P is .

It is easy to see that π2,1 = Ω. Moreover, if we define ωt , ωt : ∆+
t
(p−1)i1 −

ωt (u11 v1

1

(p−1)i1 −

ωt (u11 v1

(p−1)it −

· · · ut t vt
1

t

) = (−1)i1 +···+it β 1− 1 P i1 ⊗ · · · ⊗ β 1− t P it ,

(p−1)it −

· · ·ut t vt

= (−1)

/ A⊗t given by

t

)

1 +···+ t +i1 +···+it

β 1− 1 P i1 ⊗ · · · ⊗ β 1− t P it ,


ON THE MOD p LANNES-ZARATI HOMOMORPHISM

11

then πs,q = (ωq−1 ⊗ π2,1 ⊗ ωs−q−1 )ψ. Since π2,1 (Γ2 ) = 0, then πk,q (Γk ) = 0.
By the definition of the differential in the bar resolution, one gets
s−1

∂(˜
γ ) = (−1)

deg γ
˜ +s

(πs,q ⊗ idΣ1−s Fp )(γ ⊗ Σ1−s 1).
q=1

Since γ ∈ B[s] ⊂ Γs , then πs,q (γ) = 0. Therefore, ∂(˜
γ ) = 0.
For any A-module M , from the definition of the functor D, one gets the short
exact sequence of chain complexes

/ B∗ (M )
/ D(B∗ (M ))
/ 0.
/ EB∗ (M )
0
Because B∗ (M ) is acyclic, for s ≥ 1, the connecting homomorphism
∼
=

∂∗ : Hs (D(B∗ (M )) −
→ Hs−1 (EB∗ (M ))

(4.2)

is isomorphic.
Letting M = Σ1−s Fp , one gets
∼
=

∂∗ : Ds (Σ1−s Fp ) −
→ Hs−1 (EBs−1 (Σ1−s Fp )).
Lemma 4.3. For γ ∈ B[s],
∂∗ [1 ⊗ γ˜ ] = [˜
γ ].
(p−1)i −

(p−1)i −

s
s

1
1
∈ B[s]. Then [1 ⊗
Proof. Suppose that γ = I∈I u11 v1
· · · uss vs
1−s
1−s
γ˜ ] ∈ D(Bs (Σ Fp )). Since γ˜ is a cycle in EBs−1 (Σ Fp ), then [1 ⊗ γ˜ ] is a cycle
in D(B∗ (Σ1−s Fp )). It can be pulled back by the element 1 ⊗ γ˜ ∈ Bs (Σ1−s Fp ).
In Bs (Σ1−s Fp ), we have

∂(1 ⊗ γ˜ ) = 1 ⊗ ∂(˜
γ)
(−1)e(I) β 1− 1 P i1 ⊗ · · · ⊗ β 1− s P is ⊗Σ1−s 1

+
I∈I

(−1)e(I) β 1− 1 P i1 ⊗ · · · ⊗ β 1− s P is ⊗Σ1−s 1.

=
I∈I

Thus, the proof is complete.
From the short exact sequence
0 → Σ2−s P1 → Σ2−s Pˆ → Σ1−s Fp → 0,
we have the short exact sequence of chain complexes
/ B∗ (Σ2−s P1 )

0

/ B∗ (Σ2−s Pˆ )

/ B∗ (Σ1−s Fp )

/ 0.

It induces a short exact sequence (even though E(−) is not exact)
0

/ EB∗ (Σ2−s P1 )

/ EB∗ (Σ2−s Pˆ )

/ EB∗ (Σ1−s Fp )

/ 0.

Taking homology, we have the connecting homomorphism
/ Hs−2 (EB∗ (Σ2−s P1 )).
δ(Σ2−s Fp ) : Hs−1 (EB∗ (Σ−1 Fp ))
By (4.2), one gets
αs = δ(ΣFp ⊗ Ps−1 ) ◦ · · · ◦ δ(Σ2−s Fp ) ◦ ∂∗
Put δs−1 := δ(ΣFp ⊗ Ps−1 ) ◦ · · · ◦ δ(Σ2−s Fp ). Then we have the lemma.


12

P. H. CHƠN AND Đ. T. TRIẾT

Lemma 4.4. For any γ ∈ B[s],
δs−1 ([˜
γ ]) = (−1)

s(s−1)
+(s+1) deg γ
2

[Σγ].

Proof. Assume that
(p−1)i1 −

γ=

u11 v1

1

· · · uss vs(p−1)is −

s

∈ B[s].

I∈I

By Lemma 4.2,
(−1)e(I) β 1− 1 P i1 ⊗ · · · ⊗ β 1− s P is ⊗Σ1−s 1

γ˜ =
I∈I

is a cycle in EBs−1 (Σ1−s Fp ). It can be pulled back by
(−1)e(I) β 1− 1 P i1 ⊗ · · · ⊗ β 1− s P is ⊗Σ2−s xs ys−1 ∈ EBs−1 (Σ2−s Pˆ ).

y=
I∈I

Then, in EBs−1 (Σ2−s Pˆ ),
(−1)η(I)+e(I) β 1− 1 P i1 ⊗ · · · ⊗ β 1−

∂(y) =

s−1

P is−1 ⊗Σ2−s β 1− s P is (xs ys−1 ),

I∈I

where η(I) = s + 1 + · · · + s−1 + s s .
Therefore, δ(Σ2−s Fp )([˜
γ ]) is equal to
(−1)η(I)+e(I) β 1− 1 P i1 ⊗ · · · ⊗ β 1−

s−1

P is−1 ⊗Σ2−s β 1− s P is (xs ys−1 ) .

I∈I

Repeating this process, finally we have
(−1)fI β 1− 1 P i1 (x1 y1−1 β 1− s P i2 (x2 y2−1 · · · β 1− s P is xs ys−1 ) · · · )) ,

δs−1 ([˜
γ ]) =
I∈I

where fI = e(I) + (1 + · · · + s) + s( 1 + · · · +
The lemma follows from Corollary A.13.

s ).

Combining Lemma 4.3 and Lemma 4.4, we have the following corollary.
Corollary 4.5. The map B[s]
γ → (−1)

/ DBs (Σ1−s Fp ) given by
s(s−1)
+(s+1) deg γ
2

[1 ⊗ γ˜ ]

is a chain-level representation of the homomorphism
(1 ⊗A αs )−1 Σ : Fp ⊗A B[s]

/ Fp ⊗A Ds (Σ1−s Fp ).

The chain-level representation of the dual of ϕs is given by the following theorem.

Theorem 4.6. The inclusion map ϕ˜#
s : B[s]
γ → (−1)

/ Γ+
s given by

s(s−1)
+(s+1) deg γ
2

γ

is the chain-level representation of the dual of the Lannes-Zarari homomorphism
ϕ#
s .


ON THE MOD p LANNES-ZARATI HOMOMORPHISM

13

Proof. In [28], Priddy showed that the opposite of the lambda algebra Λopp is
isomorphic to the co-Koszul complex of A, which is the quotient cocomplex of the
usual cobar resolution C ∗ (Fp ) := HomM (B∗ (Fp ), Fp ). The canonical quotient map
/ Λopp sends τ0 ξ1i to (−1) λ1− and the rest to zero. Thus, under the
ι∗ : C ∗ (Fp )
i−1
projection ιs ,
ιs (τ01− 1 ξ1j1 ⊗ · · · ⊗ τ01− s ξ1js ) = (−1)

1 +···+ s +s

λj11 −1 · · · λjss −1 .

In Section 1, we showed that the chain complex Γ+ is isomorphic to (Λopp )# ,
the dual of Λopp , via the isomorphism given by
(p−1)j1 −

κs (u11 v1

1

· · · uss vs(p−1)js − s ) = (−1)i1 +···+is (λj11 −1 · · · λjss −1 )∗ .

Thus, there exists an inclusion νs : (Γ+ Σ1−s Fp )s
(p−1)j1 −

νs (u11 v1

1

/ Bs (Σ1−s Fp ), that sends

· · · uss vs(p−1)js − s ) = (−1)e(I) 1 ⊗ β 1− 1 P j1 ⊗ · · · ⊗ β 1− s P js
= 1 ⊗ γ˜ ,

where e(I) = s + 1 + · · · + s + i1 + · · · + is .
This fact together with Lemma 4.3 and Lemma 4.4, we have the assertion of the
theorem.

∼ opp #
∼ #
Since Γ+
s = (Λs ) and B[s] = Rs via κs , we have the following corollary.
Corollary 4.7. The projection ϕ˜s : Λopp
s
ϕ˜s (λI ) = (−1)

/ Rs given by

s(s−1)
+(s+1) deg(λI )
2

QI

is the chain-level representation of the Lannes-Zarati homomorphism ϕs .
5. The power operations
This section is devoted to develop the power operations, these are useful tools
in studying the behavior of the Lannes-Zarati homomorphism in the next section.
From Liulevicius [22], [23] and May [24], there exists the power operation P 0 :
s,p(s+t)
/ ExtA
Exts,s+t
(Fp , Fp )
(Fp , Fp ). Its chain-level representation in the cobar
A
complex is given by
θ1 ⊗ · · · ⊗ θs → θ1p ⊗ · · · ⊗ θsp ,
where θi ∈ A# , the dual of the Steenrod algebra A.

/ Λopp
By the projection ιs : C s (Fp )
s , the power operation has a chain-level
opp
representation in the Λ
given by
˜ 0 (λ 1 · · · λ s ) =
P
i1 −1
is −1

λpi11 −1 · · · λpiss −1 ,
0,

1 = ··· =
otherwise.

s

= 1,

˜ 0 induces an operation θ on the Dyer-Lashof algebra
Lemma 5.1. The operation P
R given by
θ(β 1 Qi1 · · · β s Qis ) =

β 1 Qpi1 · · · β s Qpis , 1 = · · · =
0,
otherwise.

s

= 1,

Proof. It is sufficient to show that if λi1 −1 · · · λis −1 has negative excess then so does
λpi1 −1 · · · λpis −1 for s ≥ 2.


14

P. H. CHƠN AND Đ. T. TRIẾT

By inspection, one gets
s

e(λpi1 −1 · · · λpis −1 ) = 2pi1 −

2p(p − 1)ik + (s − 2)
k=2

= pe(λi1 −1 · · · λis −1 ) − (p − 1)(s − 2).
Therefore, if e(λi1 −1 · · · λis −1 ) < 0 then e(λpi1 −1 · · · λpis −1 ) < 0.
Lemma 5.2. The operation θ commutes with the action of A. In particular,
θ((β 1 Qi1 · · · β s Qis ) P k ) = (θ(β 1 Qi1 · · · β s Qis )) P pk .
Proof. It is sufficient to show the lemma in the case
We will prove the assertion by induction on s.
For s = 1, it is easy to see that

1

= ··· =

s

(5.1)

= 1.

(p − 1)(i − k) − 1
βQi−k )
k
(p − 1)(i − k) − 1
βQpi−pk ,
k

θ((βQi ) P k ) = θ((−1)k
= (−1)k
and

(θ(βQi )) P pk = βQpi P pk = (−1)pk

(p − 1)(pi − pk) − 1
βQpi−pk .
pk

Since (−1)pk (p−1)(pi−pk)−1
≡ (−1)k (p−1)(i−k)−1
pk
k
For s > 1, by the inductive hypothesis,

mod p, we have the assertion.

θ((βQi1 · · · βQis ) P k )
(−1)k+t

=θ
t

(−1)k+t

+θ
t

(−1)k+t

=

(p − 1)(i1 − k) − 1
βQi1 −k+t (βQi2 · · · βQis ) P t
k − pt

t

(p − 1)(i1 − k) − 1
Qi1 −k+t (βQi2 · · · βQis )β P t
k − pt − 1

(p − 1)(i1 − k) − 1
βQp(i1 −k+t) (βQpi2 · · · βQpis ) P pt .

k − pt

On the other hand,
(θ(βQi1 · · · βQis )) P pk = (βQpi1 · · · βQpis ) P pk
(−1)pk+j

=
j

(−1)pk+j

+
j

(−1)k+j

=
j

(p − 1)(pi1 − pk) − 1
βQpi1 −pk+j (βQpi2 · · · βQpis ) P j
pk − pj
(p − 1)(pi1 − pk) − 1
Qpi1 −pk+j (βQpi2 · · · βQpis )β P j
pk − pj − 1

(p − 1)(i1 − k) − 1
βQpi1 −pk+j (βQpi2 · · · βQpis ) P j .
k−j

ON THE MOD p LANNES-ZARATI HOMOMORPHISM

15

If j is not divisible by p then (p − 1)(pi2 − j) − 1 ≡ j − 1 mod p, while j − p ≡ j
mod p. Therefore,
(βQpi2 · · · βQpis ) P j
(−1)j+

=
j

(−1)j+

+
j

(−1)j+

=
j

(p − 1)(pi2 − ) − 1
βQpi2 −j+ (βQpi3 · · · βQpis ) P
j−p
(p − 1)(pi2 − j) − 1
Qpi2 −j+ (βQpi3 · · · βQpis )β P j
j−p −1
(p − 1)(pi2 − ) − 1

βQpi2 −j+ (βQpi3 · · · βQpis ) P = 0.
j−p

Thus,
(θ(βQi1 · · · βQis )) P pk
(−1)k+t

=
j

(p − 1)(i1 − k) − 1
βQp(i1 −k+t) (βQpi2 · · · βQpis ) P pt .
k − pt

The lemma is proved.
By Lemma 5.2, the operation θ induces an power operation on Ann(R), which
is also denoted by P 0 .
Proposition 5.3. The power operations P 0 s commute with each other through the
Lannes-Zarati homomorphism. In other words, the following diagram is commutative
P0 /
s,p(s+t)
Exts,s+t
(Fp , Fp )
ExtA
(Fp , Fp )
A
ϕs


Ann(Rs )t

ϕs
P0


/ Ann(Rs )p(s+t)−s .

Proof. It is immediate from Corollary 4.7.
6. Behavior of the Lannes-Zarati homomorphism
In this section, we use the chain-level representation map of the ϕs constructed
in the previous section to investigate its behavior.
6.1. The first Lannes-Zarati homomorphism.
Theorem 6.1. The first Lannes-Zarati homomorphism
/ Ann(B[1]# )t
ϕ1 : Ext1,1+t
(Fp , Fp )
A
is isomorphic.
Proof. As we well-known, Ext1,1+t
(Fp , Fp ) spanned by α0 of stem 0 and hi of stem
A
(p−1)i−1
0
2i(p−1)−1. These element are represented in Γ+
1 respectively be v1 and u1 v1
for i > 0.
(p−1)i−1
On the other hand, Fp ⊗ B[1] is spanned by 1 and x1 y1
for i > 0.
Applying Theorem 4.6, one gets

0
ϕ#
1 ([v1 ]) = [v0 ];

This fact follows the theorem.

(p−1)i−1

ϕ#
1 ([x1 y1

(p−1)i−1

]) = [u1 v1

].


16

P. H. CHƠN AND Đ. T. TRIẾT

6.2. The second Lannes-Zarati homomorphism.
Theorem 6.2. The second Lannes-Zarati homomorphism
/ Ann(B[2]# )t

ϕ2 : Ext2,2+t
(Fp , Fp )
A

is vanishing for t = 0 and t = 2(p − 1)pi+1 − 2, i ≥ 0.
Proof. From the results of Liulevicius [23] (see also Aikawa [1]), Ext2,2+t
(Fp , Fp )
A
spanned by the elements
i

2,2(p−1)(p
(1) hi hj = [λpi −1 λpj −1 ] ∈ ExtA

+pj )

2,2(p−1)pi +1

(2) α0 hi = [µ−1 λpi −1 ] ∈ ExtA
(3) α02 = [µ2−1 ] ∈ Ext2,2
A (Fp , Fp );

(Fp , Fp ), 0 ≤ i < j + 1;

(Fp , Fp ), i ≥ 1;
i+1

i

2,2(p−1)(2p
+p )
(4) hi;2,1 = (P 0 )i [λ2p−1 λ0 ] ∈ ExtA
(Fp , Fp ), i ≥ 0;
2,2(p−1)(pi+1 +2pi )

0 i
(5) hi;1,2 = (P ) [λp−1 λ1 ] ∈ ExtA
(Fp , Fp ), i ≥ 0;
(6) ρ = [λ1 µ−1 ] ∈ Ext2,4(p−1)+1
(Fp , Fp );
A
i+1
(p−1) (−1)j+1
0 i
˜
λ(p−j)−1 λj−1 ∈ Ext2,2(p−1)p (F2 , F2 ), i ≥ 0.
(7) λi = (P )
j=1

j
0 i

A

0

0

Here we denote (P ) = P · · · P .
i times

It is clear that monomials λpi −1 λpj −1 (i < j + 1), µ−1 λpi −1 , λp−1 λ1 are of
negative excess, therefore their images under ϕ˜2 are trivial in R2 . It implies under
ϕ2 the images of hi hj , α0 hi , and h0;1,2 are trivial. By Proposition 5.3, ϕ2 (hi;1,2 ) =
(P 0 )i ϕ2 (h0;1,2 ) = 0.

It is easy to see that ϕ2 (α02 ) = −Q0 Q0 = 0 ∈ R2 .
By inspection,
ϕ˜2 (λ2p−1 λ0 ) = −βQ2p βQ1 .
Applying adem relation, one gets
βQ2p βQ1 = −

(−1)2p+j
j

(p − 1)(j − 1) − 1
βQ2p+1−j βQj .
pj − 2p − 1

Since pj > 2p+1, then e(βQ2p+1−j βQj ) = 2(2p+1−j)−2(p−1)j = 2(2p+1−pj) <
0. Therefore, βQ2p βQ1 = 0, it implies that ϕ2 (h0;2,1 ) and then ϕ2 (hi;2,1 ) = 0.
Similarly, ϕ˜2 (λ1 µ−1 ) = βQ2 Q0 . Applying adem relation, we obtain βQ2 Q0 = 0
and therefore ϕ2 (ρ) = 0.
Finally, it is easy to verify that


(p−1)
j+1
(−1)
ϕ˜2 
λ(p−j)−1 λj−1  = −βQp−1 βQ1 = 0 ∈ R2 .
j
j=1
Therefore ϕ(λ˜0 ) = βQp−1 βQ1 . By Proposition 5.3, one gets
ϕ2 (λ˜i ) = (P 0 )i (βQp−1 βQ1 ) = −βQp

i

(p−1)

i

βQp = 0 ∈ R2 .

The proof is complete.
Remark 6.3. From the result of Wellington [30, Theorem 11.11], Ann(R2 ) is
i
i
spanned by Q0 Q0 , βQp (p−1) βQp , i ≥ 0, and Qs(p−1) Qs , s = pi + · · · + 1, i > 0.
Therefore, ϕ2 is not an epimorphism.


ON THE MOD p LANNES-ZARATI HOMOMORPHISM

17

6.3. The third Lannes-Zarati homomorphism.
Theorem 6.4. The third Lannes-Zarati homomorphism
ϕ3 : Ext3,3+t
(Fp , Fp )
A

/ Ann(B[3]# )t

is vanishing for all t > 0.
Proof. By the results of Liulevicius [23] and Aikawa [1], Ext3,3+t

(Fp , Fp ) is spanned
A
by following elements (for convenience we will write Exts,s+t
for
Ext3,3+t
(Fp , Fp ))
A
A
3,2(p−1)(p
(1) hi hj hk = [λpi −1 λpj −1 λpk −1 ] ∈ ExtA

i

+pj +pk )

3,2(p−1)(pi +pj )+1

(2) α0 hi hj = [µ−1 λpi −1 λpj −1 ] ∈ ExtA

, 0 ≤ i < j + 1 < k + 2;

, 0 ≤ i < j + 1;

i

3,2(p−1)p +2
(3) α02 hi = [µ2−1 λpi −1 ] ∈ ExtA
, i ≤ 0;
3,3
3

3
(4) α0 = [µ−1 ] ∈ ExtA ;
3,2(p−1)(pi+1 +pj )
(5) λ˜i hj = [Li λpj ] ∈ ExtA
, i, j ≥ 0, j = i + 2;
˜
(6) α0 λi = [µ−1 Li ], i ≥ 0;
i+1

i

j

3,2(p−1)(p
+2p +p )
(7) hi;1,2 hj = [λpi+1 −1 λ2pi −1 λpj −1 ] ∈ ExtA
, i, j ≥ 0, j = i +
2, i, i − 1;
3,2(p−1)(pi+1 +2pi )+1
(8) hi;1,2 α0 = [λpi+1 −1 λ2pi −1 µ−1 ] ∈ ExtA
, i ≥ 1;
i+1

i

j

2(p−1)(2p
+p +p )
(9) hi;2,1 hj = [λ2pi+1 −1 λpi −1 λpj −1 ] ∈ ExtA

; i, j ≥ 0, j = i +
2, i ± 1, i;
3,2(p−1)(2pi+1 +pi )+1
(10) hi;2,1 α0 = [λ2pi+1 −1 λpi −1 µ−1 ] ∈ ExtA
, i ≥ 1;
(11) ρα0 = [λ1 µ−1 µ−1 ] ∈ Ext3,4(p−1)+2
;
A
3,2(p−1)(3pi+2 +2pi+1 +pi
(12) hi;3,2,1 = (P 0 )i [λ3p2 −1 λ2p−1 λ0 ] ∈ ExtA
, p = 3, i ≥ 0;
3,2(p−1)(3p+2)+1
(13) h3,2,1 = [λ3p−1 λ1 µ−1 ] ∈ ExtA
, p = 3;
i+3

+2p
(14) hi;2,2,1 = (P 0 )i [λ2p3 −1 λ2p−1 λ0 ] ∈ Ext3,2(p−1)(2p
A
2
3,2(p−1)(2p +2)+1
(15) h2,2,1 = [λ2p2 −1 λ1 µ−1 ] ∈ ExtA
, p = 3;
i+2

3,2(p−1)(p
+3p
(16) hi;1,3,1 = (P 0 )i [λp2 −1 λ3p−1 λ0 ] ∈ ExtA
3,2(p−1)(p+3)+1
(17) h1,3,1 = [λp−1 λ2 µ−1 ] ∈ ExtA

, p = 3;
i+2

i+1

+pi )

i+1

+pi )

i+1

i

, p = 3, i ≥ 0;

, p = 3, i ≥ 0;

3,2(p−1)(2p
+p
+2p
(18) hi;2,1,2 = (P 0 )i [λ2p2 −1 λp−1 λ1 ] ∈ ExtA
, i ≥ 0;
i+2
i+1
3,2(p−1)(p
+2p
+3pi )
0 i

(19) hi;1,2,3 = (P ) [λp2 −1 λ2p−1 λ2 ] ∈ ExtA
, p = 3, i ≥ 0;
(20) 3 = [λ2 µ2−1 ] ∈ Ext3,6(p−1)+2
,
p
=
3;
A
(21) 3 = [λ5 µ2−1 ] Ext3,12(p−1)+2
, p = 3;
A
i+1

i

+2p )
(22) fi = (P 0 )i−1 [M1 ] ∈ Ext3,2(p−1)(p
, i ≥ 1;
A
i+1
3,2(p−1)(2p
+pi )
0 i−1
(23) gi = (P ) [N1 ] ∈ ExtA
, i ≥ 1;

where


(p−1)

Li = (P 0 )i 
j=1
p−1

M1 =
j=1


(−1)j+1
λ(p−j)−1 λj−1  , i ≥ 0;
j

(−1)j+1
(λjp−1 λ(p2 −jp)−1 λ2p−1 − 2λp2 −1 λj−1 λ2p−j−1
j
− 2λp2 −1 λp+j−1 λp−j−1 );


18

P. H. CHƠN AND Đ. T. TRIẾT
p−1

N1 =
j=1

(−1)j+1
(2λjp−1 λ(2p2 −jp)−1 λp−1 + 2λp2 +jp−1 λp2 −jp−1 λp−1
j

− λ2p2 −1 λj−1 λp−j−1 ).

By inspection, we see that hi hj hk (i < j + 1 < k + 2), α0 hi hj (i < j + 1), α02 hi ,
α0 λ˜i , hi;1,2 hj , hi;1,2 α0 , hi;1,3,1 , h1,3,1 , hi;1,2,3 , and fi are represented by cycles of
negative excess. Therefore, their images under ϕ3 are trivial.
It is easy to check that ϕ3 (α03 ) = −Q0 Q0 Q0 = 0 ∈ R3 .
i
i
j
Applying Corollary 4.7, one gets that ϕ3 (λ˜i hj ) = −βQp (p−1) βQp βQp . Applyi
i
ing adem relation, we obtain that βQp (p−1) βQp = 0, therefore ϕ3 (λ˜i hj ) = 0.
i+1
i
j
It is clear that ϕ3 (hi;2,1 hj ) = −βQ2p βQp βQp . Applying adem relation, we
i+1
i
obtain that βQ2p βQp = 0, it implies that ϕ3 (hi;2,1 hj ) = 0. Similarly, we have
ϕ3 (hi;2,1 α0 ) = 0.
By the same argument, we obtain
• ϕ3 (ρα0 ) = −βQ2 Q0 Q0 = 0;
2
• ϕ3 (h0;3,2,1 ) = −βQ3p βQ2p βQ1 = 0;
3p
• ϕ3 (h3,2,1 ) = −βQ βQ2 Q0 = 0;
3
• ϕ3 (h0;2,2,1 ) = −βQ2p βQ2p βQ1 = 0;
2
• ϕ3 (h2,2,1 ) = −βQ2p βQ2 Q0 = 0;

2
• ϕ3 (hi;2,1,2 ) = −βQ2p βQp βQ2 = 0;
• ϕ3 ( 3 ) = −βQ3 Q0 Q0 = 0;
• ϕ3 ( 3 ) = −βQ6 Q0 Q0 = 0.
Finally, by inspection, we have
p−1

ϕ3 (g1 ) = −
j=1
j

p−j

It is clear that βQ βQ

2
(−1)j
βQ2p βQj βQp−j .
j

= 0 if j < p − 1. But applying adem relation, we have
2

βQ2p βQp−1 βQ1 = 0.
Combining with Proposition 5.3, we have the assertion of the theorem.
Appendix A. The Singer transfer
The purpose of this section is to establish the chain-level representation of the
dual of the mod p Singer transfer in the Singer-Hưng-Sum chain complex. We end
this section by the computation of the image of B[s] ⊂ Γ+
s through the Singer

transfer, the result is used in Section 4.
−1
/ TorA
Let e1 (M ) : TorA
M)
r−1 (Fp , P1 ⊗ M ) be the Singer’s element,
r (Fp , Σ
which is the connecting homomorphism associated with the short exact sequence
/ P1 ⊗ M
/ Pˆ ⊗ M
/ Σ−1 M
/ 0.
0
Put es (M ) := e1 (Ps−1 ⊗ M ) ◦ · · · ◦ e1 ( Σ−(s−1) M ), then
−s
/ TorA
es (M ) : TorA
M)
r (Fp , Σ
r−s (Fp , Ps ⊗ M ).
When M = Fp and r = s, we have the dual of the mod p Singer transfer
/ Fp ⊗A Ps .
T rs# := es (Fp )Σ−s : TorA
s (Fp , Fp )


ON THE MOD p LANNES-ZARATI HOMOMORPHISM

/ E(x1 , · · · , xs )⊗Fp [y ±1 , · · · , ys±1 ]
1

Definition A.1. The homomorphism Ts : ∆+
s
is defined by
(p−1)i1 −

1

Ts (u11 v1

19

· · · uss vs(p−1)is − s )

= (−1)fs β 1− 1 P i1 (x1 y1−1 β 1− s P i2 (x2 y2−1 · · · β 1− s P is (xs ys−1 ))),

(A.1)

where fs = (1 + · · · + s) + s( 1 + · · · + s ) + i1 + · · · + is , and i1 , . . . , is are arbitrary
integers. Here we mean P i = 0 for i < 0.
Theorem A.2. The restriction of Ts on Γ+
is the chain-level representation
s , Ts |Γ+
s
of the dual of the mod p Singer transfer T rs# .
−s
1−s
/ TorA
Proof. By the definition, e1 (Σ1−s Fp ) : TorA
Fp )

P1 ) is
s (Fp , Σ
s−1 (Fp , Σ
the connecting homomorphism of the exact sequence of chain complexes
/ Γ+ Σ1−s Pˆ
/ Γ+ Σ−s Fp
/ 0.
/ Γ+ Σ1−s P1
0
(p−1)i −

(p−1)i −

s
s
1
1
∈ (Γ+ Σ−s Fp )s , it can be
Σ−s 1 · · · uss vs
For a cycle X = u11 v1
(p−1)i
(p−1)i
−
s− s
1
1
Sts (Σ1−s xs ys−1 ) ∈
· · · uss vs
pulled back to the element X = u11 v1
+ −s

+ 1−s ˆ
+ 1−s ˆ
(Γ Σ P )s . Since X is the cycle in (Γ Σ Fp )s , in Γ Σ P , one gets that ∂(X )
is equal to

(p−1)i1 −

(−1)ks +is u11 v1
where ks = s +

1

1

+ ··· +
(p−1)i1 −

[(−1)ks +is +1 u11 v1

(p−1)is−1 −

s−1
· · · us−1
vs−1

s−1
1

s−1

Sts−1 (Σ1−s β 1− s P is (xs ys−1 )),

+ s s . Therefore, e1 (Σ1−s Fp )([X]) is equal to
(p−1)is−1 −

s−1
· · · us−1
vs−1

s−1

Sts−1 (Σ1−s β 1− s P is (xs ys−1 ))].

Repeating this process, we have the assertion.
For M is unstable A-module and m ∈ M q , we define
d∗ P (x, y; m) := µ(q)

(−1)

x
y

+i

=0,1;0≤ +2i≤q

y

(q−2i)(p−1)
2

⊗ β P i (m),

where µ(q) = (h!)q (−1)hq(q−1)/2 , h = (p − 1)/2.
From Mùi’s [26] and Hưng-Sum [19], we have
Lemma A.3. For m, n ∈ H ∗ BE1 = Fp [y] ⊗ E(x)
(1) d∗ P (x1 , y1 ; Vi−1 (y2 , · · · , yi )) = Vi (y1 , · · · , yi );
h−1
(2) d∗ P (x1 , y1 ; Mi;i−1 Lh−1
;
i−1 ) = (−h!)Mi+1;i Li
∗
h deg m deg n ∗
d P (x, y; m)d∗ P (x, y; n).
(3) d P (x, y; mn) = (−1)
Lemma A.4. For m, n ∈ H ∗ BE1 = Fp [y] ⊗ E(x),
(1) Sts (mn) = Sts (m) · Sts (n);
(2) Sts (x) = (−1)s us+1 ;
(3) Sts (y) = (−1)s vs+1
Corollary A.5.
(1) d∗ P (x, y, Vip−1 ) = βP p
p−1
p−1
(2) d∗ P (x, y; Vi ) = Vi+1
;
p−1
(3) y 2 d∗ P (x, y; Ri;i−1 ) = (−h!)Ri+1;i ;
(4) y p−1 d∗ P (x, y; qi,0 ) = qi+1,0 ;
(5) St1 (ui ) = −ui+1 ;
(6) St1 (vi ) = −vi+1 .

i

(p−1)

(xy −1 ⊗ Vip−1 );


20

P. H. CHƠN AND Đ. T. TRIẾT

Lemma A.6. Let M be an A-algebra, X, Y ∈ M . For 2a ≥ deg X and 2b ≥ deg Y ,
then
β P a+b (xy −1 ⊗ XY ) = β P a (xy −1 ⊗ X)β P b (xy −1 ⊗ Y ).
Proof. Using Cartan formula, we can verify that
β P a+b (xy −1 ⊗ XY ) =

(−1)a+b−

+

x y (p−1)(a+b−

)−

⊗ β P (XY )

,

(−1)a+b−

=
,

+

(−1)

2

deg X

x y (p−1)(a+b−

)−

i+j =
+ 2 =

1

⊗ β 1 P i (X)β 2 P j (Y )




=

(−1)a−i+ 1 x 1 y (p−1)(a−i)− 1 ⊗ β 1 P i (X)

i,

1





×

(−1)b−j+ 2 x 2 y (p−1)(b−j)− 2 ⊗ β 2 P j (Y )
i,

a

= β P (xy

1

−1

⊗ X)β P b (xy −1 ⊗ Y ).

The proof is complete.
Lemma A.7. Let M be an A-algebra, X, Y ∈ M . For 2a ≥ deg X and 2b ≥ deg Y ,
then
P a+b (xy −1 ⊗ XY ) = P a (xy −1 ⊗ X)β P b (xy −1 ⊗ Y ).
Proof. Using Cartan formula, we can verify that
a

P a+b (xy −1 ⊗ XY ) =

(−1)a−i xy (p−1)(a−i)−1 ⊗ P i (X)
i=0





b

(−1)b−j y (p−1)(b−j) ⊗ P j (Y )

×
j=0

= P (xy −1 ⊗ X)β P b (xy −1 ⊗ Y ).
a

The proof is complete.
Put Ts := (−1)1+···+s Ts . Then we have the following result.
(p−1)i1 −

Lemma A.8. For elements satisfying (4.1) v I = u11 v1
(p−1)j1 −σ1
(p−1)js −σs
· · · uσs s vs
in Γ+
and v J = uσ1 1 v1
s , one gets

Ts (v I · v J ) = Ts (v I ) · Ts (v J ).

1

(p−1)is −

· · · uss vs

s


ON THE MOD p LANNES-ZARATI HOMOMORPHISM

21

Proof. We only need to prove for s = 2. The case s > 2 is proved similarly.
(p−1)(i1 +j1 )−(

T2 (v I · v J ) = T2 (u11 +σ1 v1
= (−1)

2(

× β 1−(
= (−1)2(
× β 1−(

1 +σ1 )

(p−1)(i2 +j2 )−(

2 +σ2 )

u22 +σ2 v2

)

1 +σ1 + 2 +σ2 )+i1 +j1 +i2 +j2
1 +σ1 )

P i1 +j1 (x1 y1−1 ⊗ β 1−(

2 +σ2 )

P i2 +j2 (x2 y2−1 ))

1 +σ1 + 2 +σ2 )+i1 +j1 +i2 +j2
1 +σ1 )

P i1 +j1 (x1 y1−1 ⊗ (β 1− 2 P i2 (x2 y2−1 )(β 1−σ2 P j2 (x2 y2−1 )).

Since v I and v J satisfy condition (4.1), then applying Lemma A.6 or Lemma A.7,
one gets
T2 (v I · v J ) = (−1)2(

1 +σ1 + 2 +σ2 )+i1 +j1 +i2 +j2

× [β 1− 1 P i1 (x1 y1−1 ⊗ (β 1− 2 P i2 (x2 y2−1 ))]
× β 1−σ1 P j1 (x1 y1−1 ⊗ (β 1−σ2 P j2 (x2 y2−1 ))] = T2 (v I ) · T2 (vJ ).
The lemma is proved.

(p−1)

Lemma A.9. For 1 ≤ i ≤ s, Ts (Vi

(p−1)

) = Vi

.

Proof. By inspection, we have
pi−2 (p−1)(p−1) pi−3 (p−1)(p−1)
v1

Vip−1 = v1

(p−1)(p−1) p−1
vi .

· · · vi−1

Using (A.1), one gets
pi−2 (p−1)(p−1) pi−3 (p−1)(p−1)
v1

Ts (Vip−1 ) = Ts (v1

= (−1)β P p

i−2

(p−1)

(p−1)(p−1) p−1
vi )

· · · vi−1

−1
(x1 y1−1 · · · β P p−1 (xi−1 yi−1
β P 1 (xi yi−1 )))

= d∗ P (x1 , y1 ; · · · d∗ P (xi−1 , yi−1 ; V1p−1 )) = Vip−1 .
The proof is complete.
Lemma A.10. For 1 ≤ i ≤ s, then
Ts (Ri;i−1 ) = (−1)s Ri;i−1 .
Proof. By inspection, we have
pi−2 (p−1)(p−1)

Ri;i−1 = v1

(p−1)(p−1)

· · · vi−1

(p−1)−1

ui vi

.

Therefore,
Ts (Ri;i−1 ) = (−1)s·1+(p−1)(p
× β Pp

i−2

(p−1)
i−2

= (−1)s β P p
First, we claim that β P p
A.5, it is easy to see that
β Pp

a

(p−1)

a

i−2

+···+1)+1

−1
(x1 y1−1 · · · β P p−1 (xi−1 yi−1
P 1 (xi yi−1 )))

(p−1)

(p−1)

(p−1)−1

−1
(x1 y1−1 · · · β P p−1 (xi−1 yi−1
xi yi

)).

(xy −1 Ra+1;a ) = Ra+2;a+1 . Indeed, by Corollary

p−1
1
y 2 d∗ P (x, y; Ra+1;a )
a
− 2p − 1)
1
=
(−h!)Ra+2;a+1 .
µ(2pa+1 − 2pa − 1)

(xy −1 Ra+1;a ) =

µ(2pa+1


22

P. H. CHƠN AND Đ. T. TRIẾT

By Wilson’s theorem and the Fermat’s little theorem, one gets
−h!
p−1
≡− (
)!
µ(2pa+1 − 2pa − 1)
2
≡ −(−1)
(p−1)−1

Since xi yi
the assertion.

p+1
2

2

(−1)

(−1)

p−1
2

p−1 (2pa+1 −2pa −1)(2pa+1 −2pa −2)
2
2

≡1

mod p.

= R1,0 , applying the above claim for a from 0 to i − 2, we have

Corollary A.11. For 0 ≤ k < i ≤ s,
Ts (Ri;k ) = (−1)s Ri;k .
Proof. Using Lemma A.9-A.10 together with the formula
Ri;s = Ri−1;s Vip−1 + qi−1,s Ri;i−1 ,
we have the assertion.
Proposition A.12. Let γ ∈ B[s]. Then Ts (q) = (−1)

s(s+1)
+s deg γ
2

γ.

Proof. From Lemma A.8-A.10, it is sufficient to prove Ts (Rs;i,j ) = Rs;i,j for 0 ≤
i < j ≤ s − 1.
Since
−1
−Rs;i,j = Rs;i Rs;j qs,0
−1
= Rs−1;i Rs−1;j qs−1,0
Vsp−1
−1
+ (Rs−1;i qs−1,j + Rs−1;j qs−1,i )Rs;s−1 qs−1,0

,
−1
−1
it is sufficient to show the assertion for Rk;i Rk;k−1 qk,0
and Rk−1;i Rk;k−1 qk−1,0
.
The first case, we have
p−2
−1
Rk;i Rk,k−1 qk,0
= qk−1,0
Rk−1;i uk vkp−2 .

By Lemma A.8 and Corollary A.11, one gets
p−2
−1
Ts (Rk;i Rk,k−1 qk,0
) = (−1)s Rk−1;i Ts (qk−1;0
uk vkp−2 ).

By inspection, we obtain
pk−2 (p−2)(p−1)

p−2
qk−1,0
uk vkp−2 = v1

(p−2)(p−1)

· · · vk−1

uk vkp−2 .

Therefore,
p−2
Ts (qk−1,0
uk vkp−2 ) = (−1)s+p

× β Pp

k−2

= (−1)s+p
× β Pp

k−2

(p−2)

k−2

k−2

(p−2)+···+(p−2)+1
−1
(x1 y1−1 · · · β P p−2 (xk−1 yk−1
β P 1 (xk yk−1 )))

(p−2)+···+(p−2)

(p−2)

(p−1)−1

−1
(x1 y1−1 · · · β P p−2 (xk−1 yk−1
xk yk

It is easy to see that
(p−1)−1

−1
β P p−2 (xk−1 yk−1
xk yk

(p−1)(p−2)

St1 (xk ykp−2 )

(p−1)(p−2)

u2 v2p−2 .

) = (−1)yk−1
= (−1)yk−1

)).


ON THE MOD p LANNES-ZARATI HOMOMORPHISM

23

By the same method, one gets
(p−2)(p−1)

−1
β P p(p−2)(p−1) (xk−2 yk−2
yk−1

u2 v2p−2 )

(p−1)

p−1 p−2
= (−1)[yk−1 d∗ P (xk−2 , yk−2 ; yk−1
)] St1 (u2 , v2p−2 )
p−2
u3 v3p−2 .
= (−1)q2,0

By induction, we have
p−2
p−2
Ts (qk−1,0
uk vkp−2 ) = (−1)s qk−1,0
uk vkp−2 .
−1
−1
Thus, Ts (Rk;i Rk,k−1 qk,0

) = Rk;i Rk,k−1 qk,0
.
The final case, we have
p−2
−1
Rk−1;i Rk,k−1 qk−1,0
= Rk−1;i qk−1,0
uk vkp−2 .

By the same argument, we have the assertion.
From this proposition, we have the following corollary.
Corollary A.13. For any γ =

(p−1)i1 −

I∈I

u11 v1

1

(p−1)is −

· · · uss vs

s

∈ B[s], then

(−1)i1 +···+is β 1− 1 P i1 (x1 y1−1 β 1− s P i2 (x2 y2−1 · · · β 1− s P is (xs ys−1 ))) = γ.

I∈I

Acknowledgement. The paper was completed while the frist author was visiting
the Vietnam Institute for Advanced Study in Mathematics (VIASM). He thanks
the VIASM for support and hospitality.
References
1. T. Aikawa, 3-dimensional cohomology of the mod p Steenrod algebra, Math. Scand. 47 (1980),
no. 1, 91–115. MR 600080 (82g:55023)
2. J. M. Boardman, Modular representations on the homology of powers of real projective space,
Algebraic topology (Oaxtepec, 1991), Contemp. Math., vol. 146, Amer. Math. Soc., Providence, RI, 1993, pp. 49–70. MR 1224907 (95a:55041)
3. A. K. Bousfield, E. B. Curtis, D. M. Kan, D. G. Quillen, D. L. Rector, and J. W. Schlesinger,
The mod-p lower central series and the Adams spectral sequence, Topology 5 (1966), 331–342.
4. P. H. Chơn and L. M. Hà, On may spectral sequence and the algebraic transfer, Manuscripta
Math. 138 (2012), no. 1-2, 141–160.
, On the May spectral sequence and the algebraic transfer II, Topology Appl. 178
5.
(2014), 372–383. MR 3276753
6. P. H. Chơn and Đ. T. Triết, On the behavior of the Lannes-Zarati homomorphism, preprint
(2014).
7. A. Ciampella and L. A. Lomonaco, The universal Steenrod algebra at odd primes, Comm.
Algebra 32 (2004), no. 7, 2589–2607. MR 2099919 (2005f:55014)
8. F. R. Cohen, T. J. Lada, and J. P. May, The homology of iterated loop spaces, Lecture Notes
in Mathematics, Vol. 533, Springer-Verlag, Berlin, 1976. MR 0436146 (55 #9096)
9. M. D. Crossley, A(p) generators for H ∗ V and Singer’s homological transfer, Math. Z. 230
(1999), no. 3, 401–411. MR 1679985 (2000g:55025)
10. E. B. Curtis, The Dyer-Lashof algebra and the Λ-algebra, Illinois J. Math. 19 (1975), 231–246.
MR 0377885 (51 #14054)
11. L. E. Dickson, A fundamental system of invariants of the general modular linear group with a
solution of the form problem, Trans. Amer. Math. Soc. 12 (1911), no. 1, pp. 75–98 (English).
12. L. M. Hà, Sub-Hopf algebras of the Steenrod algebra and the Singer transfer, Proceedings

of the School and Conference in Algebraic Topology, Geom. Topol. Monogr., vol. 11, Geom.
Topol. Publ., Coventry, 2007, pp. 81–105. MR 2402802 (2009d:55027)
13. N. D. H. Hải, Résolution injective instable de la cohomologie modulo p d’un spectre de Thom
et application, Ph.D. thesis, Université Paris 13, 2010.


24

P. H. CHƠN AND Đ. T. TRIẾT

14. N. H. V. Hưng, Spherical classes and the algebraic transfer, Trans. Amer. Math. Soc. 349
(1997), no. 10, 3893–3910. MR 1433119 (98e:55020)
15.
, Spherical classes and the lambda algebra, Trans. Amer. Math. Soc. 353 (2001), no. 11,
4447–4460 (electronic).
16.
, On triviality of Dickson invariants in the homology of the Steenrod algebra, Math.
Proc. Cambridge Philos. Soc. 134 (2003), no. 1, 103–113.
17. N. H. V. Hưng and T. N. Nam, The hit problem for the Dickson algebra, Trans. Amer. Math.
Soc. 353 (2001), no. 12, 5029–5040.
18. N. H. V. Hưng, V. T. N. Quỳnh, and N. A. Tu´
ân, On the vanishing of the Lannes-Zarati
homomorphism, C. R. Math. Acad. Sci. Paris 352 (2014), no. 3, 251–254. MR 3167575
19. N. H. V. Hưng and N. Sum, On Singer’s invariant-theoretic description of the lambda algebra:
a mod p analogue, J. Pure Appl. Algebra 99 (1995), no. 3, 297–329. MR 1332903 (96c:55024)
20. N. Kuhn, Adams filtration and generalized Hurewicz maps for infinite loopspaces, preprint
(2014), />21. J. Lannes and S. Zarati, Sur les foncteurs dérivés de la déstabilisation, Math. Z. 194 (1987),
no. 1, 25–59. MR 871217 (88j:55014)
22. A. Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations,
Proceedings of the National Academy of Sciences of the United States of America 46 (1960),

no. 7, pp. 978–981.
23.
, The factorization of cyclic reduced powers by secondary cohomology operations, Mem.
Amer. Math. Soc. No. 42 (1962), 112. MR 0182001 (31 #6226)
24. J. P. May, A general algebraic approach to Steenrod operations, The Steenrod Algebra and its
Applications (Proc. Conf. to Celebrate N. E. Steenrod’s Sixtieth Birthday, Battelle Memorial
Inst., Columbus, Ohio, 1970), Lecture Notes in Mathematics, Vol. 168, Springer, Berlin, 1970,
pp. 153–231. MR 0281196 (43 #6915)
25. H. Mùi, Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci.
Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 319–369. MR 0422451 (54 #10440)
, Cohomology operations derived from modular invariants, Math. Z. 193 (1986), no. 1,
26.
151–163.
27. T. N. Nam, Transfert algébrique et action du groupe linéaire sur les puissances divisées modulo
2, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 5, 1785–1837. MR 2445834 (2009g:55025)
28. S. B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), no. 1, pp. 39–60.
29. W. M. Singer, The transfer in homological algebra, Math. Z. 202 (1989), no. 4, 493–523.
MR 1022818 (90i:55035)
30. R. J. Wellington, The unstable Adams spectral sequence for free iterated loop spaces, Mem.
Amer. Math. Soc. 36 (1982), no. 258, viii+225. MR 646741 (83c:55028)
31. S. Zarati, Dérivés du foncteur de déstabilisation en caractéristiques impaire et application,
Ph.D. thesis, Université Paris-Sud (Orsay), 1984.
Department of Mathematics and Applications, Saigon University, 273 An Duong
Vuong, District 5, Ho Chi Minh city, Vietnam
E-mail address:
Department of Mathematics and Applications, Saigon University, 273 An Duong
Vuong, District 5, Ho Chi Minh city, Vietnam
E-mail address: