On May spectral sequence and the algebraic transfer II

Submitted exclusively to the London Mathematical Society
doi:10.1112/0000/000000

On May spectral sequence and the algebraic transfer II
Phan Hoàng Chơn and Lê Minh Hà
Abstract
We study the algebraic transfer constructed by Singer [26] using technique of the May spectral
sequence. We show that the two squaring operators, defined by Kameko [12] and Nakamura
[21], on the domain and range respectively, of our E2 version of the algebraic transfer are
i
compatible. We also prove that the two Sq 0 -family ni ∈ Ext5,36·2
(Z/2, Z/2), i ≥ 0, and ki ∈
A
i
7,36·2
ExtA
(Z/2, Z/2), i ≥ 1, are in the image of the algebraic transfer.

1. Introduction and statement results
This paper is a continuation of our previous paper [7], which we will refer to as Part I. In
Part I, we use the May spectral sequence (MSS for short), to compute the kernel and image of
the algebraic transfer, introduced by Singer [26], which is an algebra homomorphism
⊕s ϕs : TorA
∗,∗ (Z/2, Z/2)

/ ⊕s [Z/2 ⊗A H ∗ (BVs )]GLs

(1.1)

from the homology of the mod 2 Steenrod algebra A (Steenrod [27], Milnor [18]) to the space
of A-generators of the (mod 2) cohomology of elementary abelian 2-group Vs of rank s, for

s ≥ 0. The cohomology ring H ∗ (BVs ), which is a polynomial algebra in s generators, all in
degree 1, is both a module over the mod 2 Steenrod algebra as well as the general linear
group GLs = GL(Vs ). Moreover, these two module structures are compatible, so that one has
an induced action of GLs on the space Z/2 ⊗A H ∗ (BVs ). It is sometime more convenient to
consider the dual of (1.1). At rank s, it has the form
ϕ∗s : [PA H∗ (BVs )]GLs → Exts,s+∗
(Z/2, Z/2),
A

(1.2)

where PA H∗ (BVs ) denote the subspace of the divided power algebra H∗ (BVs ) consisting of all
elements that are annihilated by all positive degree Steenrod squares, and MG is the standard
notation for the module of G-coinvariants.
Our interests in the map (1.1) (or its dual (1.2)) lies in the fact that on the one hand, the dual
of its domain is the cohomology of the Steenrod algebra, Ext∗,∗
A (Z/2, Z/2), which is the initial
page of the Adams spectral sequence converging to stable homotopy groups of the spheres [1],
therefore, it is an object of fundamental importance in algebraic topology. On the other hand,
the target of (1.1) is the subject of the so-called “the hit problem”, proposed by F. Peterson
[23] (see Wood [31]). The hit problem, which is originated from cobordism theory, has deep
connection with modular representation theory of the general linear group, and it is believed
that tools from modular representation theory can be used to understand the structure of the
Ext group.
We refer to the introduction of Part I for a detailed survey of known facts about the algebraic
transfer. Briefly, it is known that ϕs is an isomorphism for s ≤ 3 (see Peterson [23], Singer [26],
Kameko [12], Boardman [2]), and together with Bruner-Hà-Hưng [4], Hưng [10], Nam [22],

2000 Mathematics Subject Classification 55P47, 55Q45, 55S10, 55T15 (primary).
This work is partially supported by a NAFOSTED grant No. 101.11-2011.33

Page 2 of 13

PHAN HOÀNG CHƠN AND LÊ MINH HÀ

Hà [9], Chơn-Hà [7], complete information about the behaviour of ϕ4 was obtained in Part I
where we showed that p0 is detected by the cohomological algebraic transfer.
In [7], we initiated the use of the (homology) May spectral sequence to compute the
algebraic transfer. This method allows us to not only recover previous known results with little
computation involved, but also obtain new detection and nondetection results in degrees where
computation of the hit problem seems out of reach at the moment. However, the computation
remains difficult, partly because while the target of the algebraic transfer (1.1) is essentially a
polynomial ring which is relatively easy to work with, the domain is the Tor group, whose rich
structure, such as the action of the Steenrod algebra, is hard to exploit.
To overcome this difficulty, in this paper, we first dualize the construction in [7] to
construct a representation of the algebraic transfer in the cohomological E2 -term of the May
spectral sequence. An application of this construction is given in Section 3. Recall that in the
∗,∗
ExtA
(Z/2, Z/2) groups, there is an action of the (big) Steenrod algebra (see Liulevicius [14]
or May [17]), where the operation Sq 0 is no longer the identity map. In his thesis [12], Kameko
constructed an operation
Sq 0 : [PA Hd (BVs )]GLs

/ [PA H2d+s (BVs )]GLs ,

that corresponds to the operation Sq 0 on Ext groups. Kameko’s operation has been extremely
useful in the study of the hit problem and for computation of the algebraic transfer. An
observation of Vakil [30] indicates that Kameko’s squaring operation is compatible with the

May filtration, and thus induces a similar operation when passing to the associated graded.
On the other hand, Nakamura [21] also constructed a family of squaring operations which
are all compatible with higher differentials in the May spectral sequence. It should be pointed
out that his method of construction is quite different from the usual one such as described
in May [17], since it is known that the general framework provided in May [17] yields trivial
map in the cohomology of the associated graded algebra E 0 A. In section 4, we showed that
under the representation of the algebraic transfer in the E2 terms of the May spectral sequence
described in Section 3, the induced Kameko squaring operation corresponds to Nakamura’s
squaring operation.
Using the construction above, we have the following, which is our main result.
i

Theorem 1.1 (see also Corollary 5.3). The family {ni ∈ Ext5,36·2
(Z/2, Z/2) : i ≥ 0} is
A
detected by the algebraic transfer (1.2).
7,∗
Bruner [3] has shown that the relation k1 = h2 h5 n0 holds in ExtA
(Z/2, Z/2). Since it is well∗
∗
known that the total transfer ϕ = ⊕s≥1 ϕs is an algebra homomorphism (see Singer [26]), we
obtain an immediate corollary.

i

Corollary 1.2. The family {ki ∈ Ext7,36·2
(Z/2, Z/2) : i ≥ 1} is in the image of the
A
seventh algebraic transfer.
We do not know whether k0 ∈ Ext5,36

(Z/2, Z/2) also belongs to the image of ϕ∗5 or not.
A
The paper is divided into five sections. Sections 2 and 3 are preliminaries. In section 2,
we recall basic facts about May spectral sequence and in section 3, we present the algebraic
transfer and its representation in the E2 -term of the cohomological May spectral sequence.
We apply the above construction to show in Section 4 that a version of Kameko’s squaring
operation which has been extremely useful in the study of the hit problem is compatible with
Nakamura’s squaring operation the May spectral sequence. The final section contains the proof


ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER II

Page 3 of 13

of the main results of this paper that the two families ni , i ≥ 0 and kj , j ≥ 1 in Ext∗,∗
A (Z/2, Z/2)
are in the image of the algebraic transfer (1.2).

2. The May Spectral Sequence
In this section, we review the construction of the May spectral sequence. Our main references
are May [15, 16] and Tangora [29]. May’s chain complex for the cohomology of the associated
graded algebra E 0 A was subsumed in Priddy’s theory of Koszul resolution [24]. Let A denotes
the mod 2 Steenrod algebra and A∗ be its linear dual. All A-modules are assumed to have
finite type and non-negatively graded.
2.1. The associated graded algebra E 0 A
The Steenrod algebra is filtered by powers of its augmentation ideal A¯ by setting Fp A = A if
0
0
¯ ⊗−p if p < 0. Let E 0 A = ⊕p,q Ep,q
p ≥ 0 and Fp A = (A)

A, where Ep,q
A = (Fp A/Fp−1 A)p+q , be
the associated graded algebra. According to a well-known theorem of Milnor and Moore [19],
E 0 A is a primitively generated Hopf algebra which is isomorphic to the universal enveloping
algebra of its restricted Lie algebra of its primitive elements. In this case, the primitives are
the Milnor generators Pji (see Milnor [18]). The following result from May’s thesis remains
unpublished, but is known and used widely.
Theorem 2.1 (May [15]). The algebra E 0 A is a primitively generated Hopf algebra. It
is isomorphic to the universal enveloping algebra of the restricted Lie algebra of its primitive
elements {Pkj |j ≥ 0, k ≥ 1}. Moreover,
k
(i) Pji , P k = δi,k+ Pj+
for i ≥ k;
j
(ii) ξ(Pk ) = 0, where ξ is the restriction map (of its restricted Lie algebra structure).
Here, δi,k+ is the usual Kronecker delta. An element θ ∈ Fp A but θ ∈ Fp−1 A is said to have
weight −p. The following result determines the weight of any given Milnor generator Sq(R).
Theorem 2.2 (May [15]). The weight w(R) of a Milnor generator Sq(R), where R =
(r1 , r2 , . . .), is w(R) = i iα(ri ) where α(m) is the function that counts the number of 1 in the
binary expansion of m.
In particular, the weight of Pji is just the subscript j. In fact, May’s argument identifies
Sq(R) with the monomial (Pji )aij in the associated graded, where ri =
aij 2j is the binary
expansion of ri . In the language of Priddy’s theory of Koszul resolution [24], Theorem 2.1
states that E 0 A is a Koszul algebra with Koszul generators {Pkj |j ≥ 0, k ≥ 1} and quadratic
relations:
Pji P k = P k Pji if i = k + ,

i−
Pji P i− + P i− Pji + Pj+

= 0,

Pji Pji = 0.

The following theorem is first proved in May’s thesis, but see also Priddy [24] for a modern
treatment.
Theorem 2.3 (May [15], Priddy [24]). The cohomology of E 0 A, H ∗ (E 0 A), is isomorphic
to the homology of the complex R, where R is the polynomial algebra over Z/2 generated by


Page 4 of 13

PHAN HOÀNG CHƠN AND LÊ MINH HÀ

{Rji |i ≥ 0, j ≥ 1} each of degree 2i (2j − 1) and with the differential is given by
j−1
i+k
Rki Rj−k
.

δ(Rji ) =
k=1

Moreover, cup products in H (E A) correspond to products of representative cycles in R.
∗

0

We will need a more general version of the above theorem for the cohomology of E 0 A with
non-trivial coefficients, which can be derived easily from the discussion in Section 4 of Priddy’s

seminal paper [24]. Let Rs denote the subspace of R consisting of all monomials in Rji of
length s. If M is a right E 0 A-module, we can form a cocomplex R ⊗ M where in degree s is
Rs ⊗ M . The differential, which is again denoted as δ, is given by
RRts ⊗ mPts ,

δ(R ⊗ m) = δ(R) ⊗ m +

(2.1)

s,t

for all R ∈ Rs and all m ∈ M . According to Priddy [24, Section 4], there exists a natural
isomorphism:
/ Exts,t0 (Z/2, M ).
(2.2)
Θ : H s,t (R ⊗ M )
E A
2.2. The May spectral sequence
We will be working with the cohomology version of the May spectral sequence. Let A∗ be
¯ ∗ . Then A∗ admits a filtration where F p A∗ = 0 if p ≥ 0 and
the dual of A and let A¯∗ = (A)
p ∗
∗
¯
F A = (A/Fp−1 A) if p < 0. If M is an A-module, let M ∗ be the Z/2-graded dual of M . The
comodule M ∗ is filtered by setting
F p M ∗ = {m ∈ M ∗ |α∗ (m) ∈ F p A∗ ⊗ M ∗ },
where α∗ is the structure map of the A∗ -comodule M ∗ . Clearly F p M ∗ = 0 for p ≥ 0 and when
0
0

M∗ =
M ∗ , where Ep,q
p < 0, we have F p M ∗ ⊆ F p−1 M ∗ . Thus (E 0 M )∗ ∼
= E 0 M ∗ = ⊕p,q Ep,q
p
∗
p+1
∗
0 ∗
(F M /F
M )p+q , is a bigraded comodule over the associated graded coalgebra E A . Let
¯
C(A;
M ) be the cobar construction with the induced filtration:
F p C¯ n (A∗ ; M ∗ ) =

F p1 A¯∗ ⊗ · · · ⊗ F pn A¯∗ ⊗ F p0 M ∗ ,
n

where the sum is taken over all sequences {p0 , . . . , pn } such that n + i=0 pi ≥ p. This filtration
respects the differential, and in the resulting spectral sequence, we have
E1p,q,t (M ∗ ) = F p C¯ p+q (A; M ) F p+1 C¯ p+q (A; M ) t .
Here p is the filtration degree, p + q is the homological degree and t is the internal degree. The
differential δ1 of this spectral sequence is the connecting homomorphism of the short exact
sequence:
¯
¯
¯
F p+1 C(A;
M)

F p C(A;
M)
F p C(A;
M)
0 → p+2
→ p+2
→ p+1
→ 0.
¯
¯
¯
F
F
F
C(A;
M)
C(A;
M)
C(A;
M)
On the other hand, E1p,q,t (M ∗ ) is isomorphic to C¯ p+q (E 0 A; E 0 M )−q,q+t as Z/2-trigraded vector
space. Under this identification, δ1 is exactly the canonical differential of the cobar construction
C¯ ∗ (E 0 A; E 0 M ). Hence E2p,q,t (M ) ∼
= H p+q (E 0 A∗ ; E 0 M ∗ )−q,q+t and we can summarize the
result in the following theorem.
Theorem 2.4 (May [16]). Let M be an A-module of finite type and positively graded.
There exists a third-quadrant spectral sequence (Er , δr ) converging to E 0 H ∗ (A; M ∗ ) and


ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER II

Page 5 of 13

having as its E2 -term E2p,q,t (M ) = H p+q (E 0 A; (E 0 M )∗ )−q,q+t . Each δr is a homomorphism
δr : Erp,q,t (M ) −→ Erp+r,q−r+1,t (M ).
When M = Z/2, we will write Er for Er (M ). It is also well-known that Er (M ) is a differential
Er -module. In his thesis, May have also demonstrated how to compute all the differentials,
at least in principle, using the so-called imbedding method (see Tangora [29, Section 5]). The
reason is that R is a quotient of the cobar complex, and the differentials comes from that of the
cobar complex as well. We shall use this method in the proof of the main theorem in Section 5.

3. The algebraic transfer
In [6, 7], we constructed a representation of the dual of the algebraic transfer in the E 2 -term
of the homology May spectral sequence. It turns out that the cohomology version that we are
going to present has better behaviour because of the algebra structure on Ext groups. Since
our construction is appropriate dual to the construction in [7], we will be very brief.
In this section, we construct the representation of E2 ψs in co-Koszul complex of E 0 H∗ (BVs ),
which will be denoted by E1 ψs .
We begin with some notations. For an s-dimensional Z/2-vector space Vs , it is well-known
that H ∗ (BVs ) is isomorphic to the polynomial algebra Ps = Z/2[x1 , . . . , xs ], where each
generator xi is of degree 1. Dually, H∗ (BVs ) is the divided power algebra Hs = Γ(a1 , . . . , as ),
(i )
(i )
where ai is the dual of xi . For simplicity, we will write (i1 , . . . , is ) for the monomial a1 1 . . . as s .
−1
Let Pˆ1 be the unique A-module extension of P1 by formally adding a generator x1 of degree
n−1
-1 and require that Sq n (x−1
and let Hˆ1 be the dual of Pˆ1 . There is a fundamental
1 ) = x1

short exact sequence of A-modules:
0 → Σ−1 Z/2 → Hˆ1 → H1 → 0,
passing to the associated graded, we have a similar short exact sequence of E 0 A-modules and
after tensoring this sequence with R ⊗ M for some right E 0 A-module M , we have a short
exact sequence of differential modules
R ⊗ M ⊗ Σ−1 Z/2

/ R ⊗ M ⊗ E 0 H1 .

/ R ⊗ M ⊗ E 0 Hˆ1

Using the isomorphism (2.2), the connecting homomorphism of this short exact sequence can
be identified with
/ Exts,t+1
Exts−1,t
(Z/2, M ⊗ E 0 H1 )
(Z/2, M ).
0
0
E A

E A

Since there is a canonical isomorphism E Hs ∼
= (E H1 ) , we can construct and compose s
similar connecting homomorphisms so as to obtain a map
0

Extk,t
(Z/2, M ⊗ E 0 Hs )

E0 A

0

⊗s

/ Extk+s,t+s
(Z/2, M ).
E0 A

In particular, when M = Z/2 and k = 0, we obtains the E2 -level of the algebraic transfer
E2 ψs : Ext0,t
(Z/2, E 0 Hs )
E0 A

/ Exts,t+s
(Z/2, Z/2).
E0 A

As we have noted, this map is induced by a chain level map
E1 ψs : E 0 Hs
We can describe this map explicitly.

/ Rs .


Page 6 of 13

PHAN HOÀNG CHƠN AND LÊ MINH HÀ

Proposition 3.1. The version of the algebraic transfer in E2 -term of May spectral
sequence is induced by the map
E1 ψs : E 0 H∗ (BVs ) −→ Rs ,
given by
(n1 )

E1 ϕs (a1

s)
. . . a(n
)=
s

(n )

Rtk11 . . . Rtkss , ni = 2ki (2ti − 1) − 1, 1 ≤ i ≤ s,
0,
otherwise.

(3.1)

(n )

Proof. Suppose R ⊗ m ⊗ a1 1 . . . as s is a nontrivial summand of a cycle x ∈ R ⊗ M ⊗ Hs .
It can be pulled back to the same element in R ⊗ M ⊗ Hs−1 ⊗ Hˆ1 . Since δ(x) = 0, it comes
from R ⊗ M ⊗ Hs−1 ⊗ Σ−1 Z/2. On the other hand, we have that a(n) Pji = a(−1) if and only
if n = 2i (2j − 1) − 1. Thus from the formula (2.1), we see that the connecting homomorphism
(n )
(n )
sends R ⊗ m ⊗ a1 1 . . . as s to zero if ns does not have the form 2i (2j − 1) − 1 for some

(ns−1 )
(n )
i
i ≥ 0, j ≥ 1; and to RRj ⊗ m ⊗ a1 1 . . . as−1
if ns = 2i (2j − 1) − 1. The required formula
can now be easily obtained by induction.
Example 3.2. Let x = (1, 1, 6) + (1, 2, 5) + (1, 4, 3) ∈ E1−2,2,8 (P3 ). It is easy to check
that δ1 (x) = 0 ∈ E1−1,2,8 (P3 ), so x is a cycle in the E1 -term and survives to a nontrivial
element in E2−2,2,8 (P3 ). Now δ2 (x) = R10 ⊗ (1, 3, 3) = δ1 (2, 3, 3) ∈ E2−1,2,8 (P3 ), so x is a cycle in
E2−2,2,8 (P3 ). For r ≥ 3, Er−2+r,∗,∗ = 0, so δr (x) = 0 for all r ≥ 3; therefore, x is a permanent
cycle.
Using (3.1), we obtain
E1 ψ3 (x) = R11 R11 R30 + R11 R20 R21 = R11 (R20 R21 + R30 R11 ),
this latter element is called h1 h0 (1) in the E2 terms of the May spectral sequence (see Tangora
[29, Appendix 1]), and is a representation of c0 in the 8-stem.
Example 3.3. We see that the element d¯0 , which is represented by the cycle X = x +
(13)x + (23)x ∈ E1−4,4,14 (P4 ), where
x = (2, 2, 5, 5) + (1, 1, 6, 6) + (2, 1, 6, 5) + (1, 2, 5, 6) + (4, 4, 3, 3)
+ (4, 2, 5, 3) + (2, 4, 3, 5) + (4, 1, 6, 3) + (1, 4, 3, 6),
is a permanent cycle. Indeed, since δ2 (X) = δ1 (y + (13)y + (23)y), where y = (3, 2, 6, 3) +
(2, 3, 3, 6), X is a cycle in E2−2,3,14 (P4 ); therefore, d¯0 survives to the E3−4,4,14 and, in the
E3 -term, it is represented by X + Y , where Y = y + (13)y + (23)y.
By inspection, we have
δ3 (X + Y ) = δ1 (Z);
δ4 (X + Y + Z) = δ1 (3, 3, 3, 5),
where Z = (5, 1, 5, 3) + (3, 5, 1, 5).
Therefore, d¯0 is a permanent cycle because δr , r ≥ 5, is trivial. Again from (3.1), we obtain
E1 ψ4 (X) = (R20 R21 + R30 R11 )2 , which is a representation of d0 in the E1 -term of May spectral
sequence. Since d¯0 is a permanent cycle, it is a representation of the pre-image of d0 under the
algebraic transfer in the MSS.

We end this section with two simple properties of the maps Er ψs . First of all, since R is a
commutative algebra, it is clear that E1 ψs factors through the coinvariant ring [PE 0 A E 0 Hs ]Σs .
The reader who is familiar with the algebraic transfer may wonder about the action of the


ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER II

Page 7 of 13

full general linear group GLs . Unfortunately, in general this action does not preserve the May
filtration. For example, if f = x21 x52 ∈ F−2 P2 and σ ∈ GL2 , such that σ(x1 ) = x1 + x2 and
σ(x2 ) = x2 , then we have σ(f ) = x21 x52 + x72 ∈ F0 P2 .
Secondly, the direct sum ⊕s≥1 H∗ (BVs ) has an algebra structure with concatenation product.
Stardard argument as in Singer [26] shows that
Proposition 3.4. For each r ≥ 1, the total homomorphism
Er ψ = ⊕s≥1 Er ψs : ⊕s Er∗,∗ (Ps )

/ Er∗,∗ ,

is an algebra homomorphism.

4. The squaring operations
In [21], Nakamura constructed a squaring operation on the MSS for the trivial module:
Sq 0 : Erp.q

/ Erp,q ,

r ≥ 1,

which is multiplicative in the E1 page and therefore satisfies the Cartan formulas in higher Er

page (when elements are suitably represented in the E2 term). The purpose of this section is to
introduce a similar squaring operation, defined for any r, s ≥ 1, which is also denoted as Sq 0 :
Sq 0 : Erp,q (Ps )

/ Erp,q (Ps ),

/ Erp,q+s
so that it commutes with Nakamura’s Sq 0 via the map of spectral sequences Erp,q (Ps )
constructed in the previous section.
We begin with a description of Nakamura squaring operation in the complex R of (2.1), this
is reminiscent to the construction of Sq 0 in Ext∗,∗
A (Z/2, Z/2) from an endomorphism of the
/ R by setting θ(Ri ) = Ri+1 .
lambda algebra as in Tangora [29]. Define an algebra map θ : R
j
j
By direct inspection, we see that θ commutes with the coboundary map δ, and thus induces
an endormorphism on Ext∗,∗
(Z/2, Z/2).
E0 A
Proposition 4.1. The endomorphism θ induces Nakamura’s squaring operation
/ Erp,q .
Sq 0 : Erp,q
i

Proof. According to Priddy [24], Rji is represented in the cobar resolution by [ξj2 ]. On the
other hand, in the cobar complex for E 0 A, the squaring operation has an explicit form
Sq 0 [α1 | . . . |αn ] = [α12 | . . . |αn2 ],
i

so it maps [ξj2 ] to [ξj2

i+1

] and the result follows immediately.

On H∗ (BVs ), there is also a squaring map constructed by Kameko [12] in his thesis which
has been extremely useful in the study of the hit problem (See for example Sum [28]). It is
given explicitly as follows.
(t )

(2t1 +1)

s)
a1 1 . . . a(t
→ a1
s

s +1)
. . . a(2t
.
s

One quickly verifies that this endomorphism of H∗ (BVs ) commutes with the action of the
Steenrod algebra, in the sense that for all a ∈ H∗ (BVs ),
(θa)Pts = θ(aPts−1 ) if s > 0, and (θa)Pt0 = 0.
Moreover, Vakil [30] observed that the map a(n) → a(2n+1) respects May’s filtration on
H∗ (BZ/2). This is clearly true for higher rank s > 1 as well. Define an endomorphism on

Page 8 of 13

PHAN HOÀNG CHƠN AND LÊ MINH HÀ

R ⊗ H∗ (BVs ), which is again denoted as Sq 0 , by setting
Sq 0 : R ⊗ a → Sq 0 (R) ⊗ θ(a),
for all R ∈ R, a ∈ H∗ (BVs ).
Lemma 4.2.
of (2.1).

The endormorphism Sq 0 on R ⊗ H∗ (BVs ) commutes with the coboundary δ

Proof. We already know that Sq 0 and δ commutes on R. Also, (θa)Rt0 = 0 and (θa)Rts =
θ(aRts−1 ), we have
δSq 0 (R ⊗ a) =δ(Sq 0 R ⊗ θa)
=δSq 0 R ⊗ θa +

Sq 0 (R)Rts ⊗ (θa)Rts

=Sq 0 δR ⊗ θa +

Sq 0 (RRts−1 ) ⊗ θ(aRts−1 )

=Sq 0 (δR ⊗ a +

RRts−1 ⊗ aRts−1 ) = Sq 0 δ(R ⊗ a).

The proof is complete.
It follows that there exists an induced endomorphism Sq 0 on Erp,q (Ps ) for all s, r ≥ 1. Our
next result shows that this endomorphism commutes with Nakamura’s Sq 0 via the MSS transfer

Er ψs , thus justifies for our choice of notation.
Proposition 4.3.
sequences:

There exists a commutative diagram of maps between spectral
Erp,q (Ps )

Er ϕ s

Sq 0

Sq 0


Erp,q (Ps )

/ Erp,q+s

Er ϕs


/ Erp,q+s .

Proof. It sufices to show that there exists a commutative diagram at E1 page.
E 0 H∗ (BVs )

E1 ϕs

Sq 0

θ


E 0 H∗ (BVs )

/ Rs

E1 ϕs


/ Rs .

This can be verified directly from the formula (3.1). Note that if ni = 2ki (2ti − 1) − 1 then
2ni + 1 = 2ki +1 (2ti − 1) − 1.
In particular, we have an induced map
θ : PE 0 A E 0 Hd (BVs ) → PE 0 A E 0 H2d+s (BVs ),
that fits in the following.
Proposition 4.4. The representation of Kameko’s squaring operations, Sq 0 , and Nakamura’s squaring operations commute with each other through E2 ϕs . In other words, the


ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER II

Page 9 of 13

following diagram commutes
E2 ϕs

PE0 A E 0 Ht (BVs )

Sq 0

θ


PE0 A E 0 H2t+s (BVs )

/ Exts,s+t
(Z/2, Z/2)
E0 A

E2 ϕs


/ Exts,s+t
(Z/2,
Z/2).
E0 A

Proof. The assertion is implied directly from the formula of Sq 0 and (3.1).
The operation θ commutes with the action of the symmetric group Σs on E 0 H∗ (BVs )
as well as its subspace PE 0 A E 0 H∗ (BVs ). This is essentially direct from the definition.
Furthermore, E2 ψs is Σs -equivariant since R is commutative. Thus in the commutative diagram
of Proposition 4.4, we can replace PE 0 A E 0 H∗ (BVs ) by its Σs -coinvariant (PE 0 A E 0 H∗ (BVs ))Σs .
Our last result can be considered as an analogue to N. H. V. Hưng’s analysis of the squaring
map on the space (PA Hs )GLs [10], where he showed that after (s − 2) iteration, the squaring
map becomes an isomorphism on its range. For the associated graded, the situation is much
simpler.
Proposition 4.5. For each s ≥ 1, the induced map
θ : (PE 0 A E 0 H∗ (BVs ))Σs

/ (PE 0 A E 0 H∗ (BVs ))Σs ,

is a monomorphism.
Proof. Suppose x ∈ PE 0 A E 0 Hs such that θ(x) = 0 in (PE 0 A E 0 H∗ (BVs ))Σs . This means
that there exist zσ ∈ (PE 0 A E 0 H∗ (BVs ))Σs such that
θx =

zσ σ + zσ .
σ∈Σs

We say that a monomial ai11 . . . aiss is odd if all exponents it are odd. Otherwise, we say that
it is non-odd. The left hand side of the above equation contains all odd monomials. Each zσ
can be written as the sum zσ + zσ where zσ consists of all non-trivial odd monomials in zσ .
We first claim that both zσ and zσ are E 0 A-annihilated.
Lemma 4.6. If x = y + z ∈ PE 0 A E 0 Hs where y is the sum of odd monomials summands
of x, then y and z belongs to PE 0 A E 0 Hs .
Proof. First of all, note that E 0 A is multiplicatively generated by P1s , s ≥ 0, so in order
to prove that y is E 0 A-annihilated, we just have to check that yP1s = 0 for all s ≥ 0. Since all
monomials in y are odd, it is clear that yP10 = ySq 1 = 0. If s > 1, then since P1s is a derivative,
and |P1s | = 2s is even, we see that yP1s , if non-zero, consists of only odd monomials while zP1s
consists of only non-odd monomials. Because xP1s = 0, we must have yP1s = zP1s = 0 for all
s > 0. The lemma is proved.
We now continue the proof of Proposition 4.5. We have a decomposition in PE 0 A E 0 Hs
θx =

(zσ σ + zσ ) +
σ∈Σs

(zσ σ + zσ ).
σ∈Σs

Page 10 of 13

PHAN HOÀNG CHƠN AND LÊ MINH HÀ

The second summand must vanish since it contains non-odd monomials. The first summand
can be written as θx for some x of the form
x =

(yσ σ + yσ ),

where yσ is such that θyσ = zσ . Since θ is obviously a monomorphism on PE 0 A E 0 Hs , it follows
that x = x and so x is trivial in (PE 0 A E 0 Hs )Σs .
It should be noted that the statement of Lemma 4.6 is not true for the original hit problem.
For example, consider the element x = (135) + (223) + (124) ∈ PA H3 where by (abc) we mean
the sum of all monomials that are permutations of (a, b, c). Then x = y + z where y = (135)
contains only odd monomials, but y is not A-annihilated. We do not know whether the similar
endomorphism on θ : (PA Hs )Σs remains a monomorphism. However, in light of the Singer’s
conjecture that ϕ∗s is always a monomorphism and current knowledge of Exts,∗
A (Z/2, Z/2) for
s ≤ 5, we believe that such an example will not be easy to find.

5. Proof of the main results
In this section we use our version of the algebraic transfer on the E2 -term of the May
spectral sequence to show that the family ni , i ≥ 0, belongs to the image of the algebraic
tranfer. It should be noted that this detection result is in degree that goes far beyond our
current knowledge of the hit problem.
An element in x ∈ E1 is said to survives to Er for some r ≥ 2 if its projection to a non-zero
element in the Er . A permanent cycle is an element killed by δr for all r. First of all, we need

a technical lemma.
Lemma 5.1. If x
¯ ∈ E1p,−p,t (Ps ) is a permanent cycle such that E1 ψs (¯
x) represents an
element x ∈ Exts,s+t
(Z/2, Z/2) in the E1 -term of the May spectral sequence, then x is in
A
the image of the algebraic transfer.
Proof. Since E1 ψs (¯
x) is a representation of x in the E1 -term of the May spectral sequence
∗,∗,∗
and x
¯ survives to E∞
(Ps ), under E∞ ϕs , the image of x
¯ is the presentation of x in the
E∞ -term of May spectral sequence. Thus, we have the assertion of the lemma.
With stardard notation of known nontrivial elements in the cohomology of the Steenrod
algebra [29], the following theorem is our main result.
Theorem 5.2. The element n0 ∈ Ext5,36
(Z/2, Z/2) is in the image of the algebraic transfer.
A
The fact that n0 and n1 = Sq 0 n0 are indecomposable elements of Ext5,∗
A (Z/2, Z/2) goes
back to Tangora [29]. Recently, completing a program initiated by Lin [13], Chen [5] proved
that the whole Sq 0 -family {ni , i ≥ 0} starting with n0 are indecomposable in Ext5,∗
A (Z/2, Z/2).
Since Kameko’s squaring operation and the classical squaring operation Sq 0 commute with
each other through the algebraic transfer, we have the following immediate corollary.
i

Corollary 5.3. The family of indecomposable elements ni ∈ Ext5,36·2
(Z/2, Z/2), i ≥ 0,
A
are in the image of the algebraic transfer.


ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER IIPage

11 of 13

According to Tangora [29], there exists an indecomposable element k ∈ Ext7,36
(Z/2, Z/2)
A
and a relation k1 = h2 h5 n0 ∈ Ext7,72
(Z/2,
Z/2),
therefore
we
have
A
i

Corollary 5.4.
algebraic transfer.

The elements ki ∈ Ext5,36·2
(Z/2, Z/2), i ≥ 1, are in the image of the
A

We do not know whether k0 also belongs to the image of the transfer or not.

−6,6,31
Proof of Theorem 5.2. We shall find a permanent cycle in E∞
(P5 ) which is represented
6,−6,31
(P5 ), to be described explicitly, such that under the E1 -version of
by an element X ∈ E1
algebraic transfer, the image of X is R12 (R30 )2 (R12 R22 + R31 R12 ). This image is known, according
to Tangora [29], to be a representative of n0 in the E1 -term of the MSS. Elements in E2 and
its projection (if exists) to Er will be written using the same letter.
Let X = x + (23)x + (243)x, where

x = (3, 3, 6, 6, 13) + (3, 5, 6, 6, 11) + (3, 6, 3, 5, 14) + (3, 6, 5, 3, 14)
+ (3, 9, 6, 6, 7) + (3, 10, 5, 6, 7) + (3, 10, 6, 5, 7) + (5, 10, 3, 6, 7)
+ (5, 10, 6, 3, 7) + (6, 9, 3, 6, 7) + (6, 9, 6, 3, 7)
+ (6, 10, 3, 5, 7) + (6, 10, 5, 3, 7),
and (23), (243) are the usual permutations in cycle form. By direct computation, X is a cycle
in E1−6,6,31 (P5 ). X is also a cycle in E2−6,6,31 (P5 ) since δ2 (X) belongs to image of δ1 . Infact, in
E1 (P5 ), we have an explicit formula
δ2 (X) = δ1 (y + (23)y + (243)y),
where
y = (5, 7, 6, 6, 7) + (6, 7, 5, 6, 7) + (6, 7, 6, 5, 7) + (5, 9, 5, 5, 7) + (5, 3, 5, 5, 13) + (5, 5, 5, 5, 11).
Hence, n
¯ 0 survives to E3−6,6,31 (P5 ); and in the E3 -term, n
¯ 0 has a representation X + Y , where
Y = y + (23)y + (243)y.
As δ3 (X + Y ) = δ1 (z + (23)z + (243)z), for z = (3, 9, 5, 3, 11) + (3, 9, 3, 5, 11) + (9, 9, 3, 3, 7) +
(9, 5, 3, 3, 11) + (9, 3, 3, 3, 13), X + Y is a cycle in the E3 -term; therefore, n
¯ 0 is nontrivial in
E4−6,6,31 (P5 ) and it is represented by X + Y + Z, where Z = z + (23)z + (243)z.
We can check that, δ4 (X + Y + Z) = δ1 (t + (23)t + (243)t + (7, 5, 3, 5, 11) + (7, 3, 5, 3, 13)),

where
t = (3, 7, 5, 5, 11) + (3, 11, 5, 5, 7) + (3, 13, 3, 5, 7) + (3, 13, 5, 3, 7)
+ (7, 9, 3, 5, 7) + (9, 7, 3, 5, 7) + (3, 7, 9, 5, 7) + (5, 7, 9, 3, 7).
It implies X + Y + Z is a cycle in E4−6,6,31 (P5 ). Hence, n
¯ 0 survives to the E5 -term, and in the
E5 -term, n
¯ 0 is represented by X + Y + Z + T , where T = t + (23)t + (243)t + (7, 5, 3, 5, 11) +
(7, 3, 5, 3, 13).
By similar argument, we have
δ5 (X + Y + X + T ) = δ1 (R) ∈ E5−1,2,31 ;
δ6 (X + Y + Z + T + R) = δ1 (S) ∈ E60,1,31 ,
where
R = (5, 7, 5, 7, 7) + (3, 3, 3, 11, 11) + (3, 3, 11, 3, 11) + (3, 11, 3, 3, 11);
S = (7, 7, 3, 3, 11) + (7, 3, 7, 3, 11) + (7, 3, 3, 7, 11) + (3, 7, 3, 7, 11).


Page 12 of 13 ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER II
Since, for r > 6, Er−6+r,∗,∗ (P5 ) = 0, δr (¯
n0 ) = 0, r > 6. In other words, n
¯ 0 is a permanent cycle.
Finally, using (3.1), we obtain E1 ψ5 (¯
n0 ) = R12 (R30 )2 (R12 R22 + R31 R12 ).
The proof is complete.
Acknowledgements. We would like to thank T. W. Chen for sending us an early version of
[5]. The second author is a Junior Associate at the ICTP, Trieste, Italy. The final version of
this paper was completed while both authors were visiting the Vietnam Institute for Advanced
Study in Mathematics. We thanks the VIASM for support and hospitality. Both authors are
supported by a NAFOSTED grant No. 101.01-2011.33.

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ON MAY SPECTRAL SEQUENCE AND THE ALGEBRAIC TRANSFER IIPage

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Department of Mathematics - Application
Saigon University,
273 An Duong Vuong, District 5,
Ho Chi Minh city, Vietnam.

Department of Mathematics-Mechanics
and Informatics,
Vietnam National University - Hanoi,
334 Nguyen Trai Street, Hanoi, Vietnam.