DSpace at VNU: On May spectral sequence and the algebraic transfer

manuscripta math. 138, 141–160 (2012)

© Springer-Verlag 2011

Phan Hoàng Cho’n, Lê Minh Hà

On May spectral sequence and the algebraic transfer
Received: 30 September 2010 / Revised: 27 July 2011
Published online: 4 October 2011
Abstract. The algebraic transfer is an important tool to study the cohomology of the Steenrod algebra. In this study, we will construct a version of the algebraic transfer in E 2 -term
of May spectral sequence and use this version to study the image of the algebraic transfer.
By this method, we obtain the description of the image of ϕs in some degrees.

1. Introduction
This article explores the May spectral sequence as a tool for understanding the algebraic
transfer, defined by Singer [26]. We work exclusively at the prime 2, and let A denote
the mod 2 Steenrod algebra [27,19]. The cohomology algebra, Ext∗,∗
A (F2 , F2 ), is a central
object of study in algebraic topology because it is the initial page of the Adams spectral
sequence converging to stable homotopy groups of the spheres [1]. This cohomology algebra has been intensively studied, see Lin [15] and Bruner [4] for the most recent results, but
its structure remains largely mysterious. One approach to better understand the structure of
Ext∗,∗
A (F2 , F2 ) was proposed by Singer [26] where he introduced an algebra homomorphism
from a certain subquotient of the divided power algebra to the cohomology of the Steenrod
algebra, which can be seen as an algebraic formulation of the the stable transfer B(Z/2)s+
/ S0 .
Let Vs denote an s-dimensional F2 -vector space. Its mod 2 cohomology is a polynomial
algebra on s generators, each in degree 1. H ∗ (BVs ) admits a left action of the Steenrod
algebra as well as a right action of the automorphism group G L s = G L(V ), and the two
actions commute. For each s ≥ 1, Singer [26] constructed an F2 -linear map:
A

ϕs : Tor s,s+∗
(F2 , F2 ) → [F2 ⊗ A H ∗ (BVs )]G L s ,

from the homology of the Steenrod algebra to the elements of F2 ⊗ A H ∗ (BVs ) invariant
under the group action. This is called the rank s algebraic transfer. Dually, let P H∗ (BVs )
Junior Associate at the Abdus-Salam ICTP
P. H. Cho’n: Department of Mathematics, College of Science, Cantho University,
3/2 St., Ninh Kieu, Cantho, Vietnam. e-mail: ;
Current address:
P. H. Cho’n: Department of Mathematics and Application, Saigon University, 273 An Duong
Vuong, District 5, Ho Chi Minh city, Vietnam
L. M. Hà (B): Department of Mathematics-Mechanics-Informatics, Vietnam National
University, Hanoi, 334 Nguyen Trai St., Thanh Xuan, Hanoi, Vietnam.
e-mail:
Mathematics Subject Classification (2000): Primary 55P47, 55Q45, 55S10, 55T15.

DOI: 10.1007/s00229-011-0487-0


142

P. H. Cho’n, L. M. Hà

be the subspace of H∗ (BVs ) consisting of all elements that are annihilated by all positive
degree Steenrod squares, then there is an induced action of G L s on P H∗ (BVs ), and we have
a map
(F2 , F2 ),
ϕs∗ : [P H∗ (BVs )]G L s → Ext s,s+∗
A
from the coinvariant elements of P H∗ (BVs ) to the cohomology of the Steenrod algebra.

Moreover, the “total” dual algebraic transfer ϕ ∗ = ⊕ϕs∗ is an algebra homomorphism. Calculations by Singer [26] for s ≤ 2 and by Boardman [3] for s = 3 showed that ϕs∗ is an
isomorphism. This shows that the algebraic transfer is highly nontrivial, and an interesting
(F2 , F2 ) that are detected by the dual algebraic
problem is to determine elements of Exts,s+∗
A
transfer.
In rank 4, it was already known to Singer that ϕ4∗ is an isomorphism in a range. It is
from these preliminary calculation that he conjectured that ϕs∗ is a monomorphism for all
s ≥ 1. In [5], Bruner, Hà and Hu’ng showed that the entire family of elements {gi : i ≥ 1} is
not in the image of the transfer, thus refuting a question of Minami concerning the so-called
new doomsday conjecture [21]. Here we are using the standard notation of elements in the
cohomology of the Steenrod algebra as was used in [29,15,4].
One of the main results of this paper is the proof that all elements in the family pi are
in the image of the rank 4 algebraic transfer. Combining the results of Hu’ng [11], Hà [10]
and Nam [23], we obtain a complete picture of the behaviour of the rank 4 dual transfer.
It should be noted that in [12], Hu’ng and Qu`ynh claimed to have a proof that the family
{ pi : i ≥ 0} is detected by the dual algebraic tranfer, but the details have not appeared. Our
method is completely different.
Very little information is known when s ≥ 5. Singer gave an explicit element in
(F2 , F2 ), namely Ph 1 , that is not detected by the dual algebraic transfer. Another
Ext 5,5+∗
A
element, commonly denoted as Ph 2 is also not detected (See Qu`ynh [25]). Using the lambda
algebra, we are able to prove in [7,8] several non-detection results in even higher rank. For
example, h 1 Ph 1 as well as h 0 Ph 2 are not in the image of ϕ6∗ ; h 21 Ph 1 is not in the image
of ϕ7∗ . Often, these results are available because it is possible to compute the domain of the
algebraic transfer in the given bidegree.
Besides the lambda algebra, a relatively efficient tool to compute the cohomology of the
Steerod algebra is the May spectral sequence, which passes from the cohomology of E 0 A,
the associated graded of the Steenrod algebra, to E 0 Ext ∗,∗

A (F2 , F2 ), an algebra associated to
the cohomology of A. Much of the known information about the cohomology of the Steenrod
algebra has been obtained by using this technique.
In this article, we construct a representation of the algebraic transfer ϕs in the May spectral sequence and apply this description to study the algebraic transfer. Using this method,
we recover, with much less computation, results in [25], [7], [8] and [12]. Moreover, our
method can also be applied, as illustrated in the case of the elements h n0 i, 0 ≤ n ≤ 5 and
h n0 j, 0 ≤ n ≤ 2, to degrees where computation of the domain of the algebraic transfer seems
out of reach at the moment.
This article is the detailed version of the note with the same title [6].

Organization of the paper. The first two sections are preliminaries. In Sect. 2, we recall
basic facts about May spectral sequence for an arbitrary A-module M. In Sect. 3, we present
the algebraic transfer. Detailed information about the representation of the algebraic transfer in the bar construction and the version of the algebraic transfer in the E 2 -term of May
spectral sequence are also given. The structure of E ∞
p,− p,∗ (Ps ) is presented in Sect. 4. Three
final sections contains the main results of this article.


May spectral sequence and algebraic transfer

143

2. May spectral sequence
In this section, we recall May’s results on the construction of his spectral sequence. The
original papers [17] and [18] are our main references. May’s chain complex to compute
the cohomology of E 0 A was also reworked in the framework of Priddy’s theory of Koszul
resolutions [24].

2.1. The associated graded algebra E 0 A
The Steenrod algebra is filtered by powers of its augmentation ideal A¯ by setting: F p A =

¯ − p if p < 0. Let E 0 A = ⊕ p,q E 0p,q A where E 0p,q A =
A if p ≥ 0 and F p A = ( A)
(F p A/F p−1 A) p+q be the associated graded algebra. It is clearly a primitively generated
Hopf algebra, and according to a theorem due to Milnor and Moore [20], E 0 A is isomorphic
to the universal enveloping algebra of its restricted Lie algebra of primitive elements, which
j
in this case, are the Milnor generators Pk ([19]). The following result from May’s thesis
remains unpublished, but well-known.
Theorem 2.1. (May [17]). 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
j
{Pk | j ≥ 0, k ≥ 1}. Moreover,
k
P ji , P k = δi,k+ P j+
for i ≥ k;
j
(2) ξ(Pk ) = 0, where ξ is the restriction map (of its restricted Lie algebra structure).

(1)

j

Here, δi,k+ is the usual Kronecker delta. The set of primitive elements R = {Pk | j ≥
0, k ≥ 1} is equipped with a natural total order, given by P ji < P k if i < k or i = k, j < .
An element θ ∈ F p A but θ ∈ F p−1 A is said to have weight − p. The following result
determines the weight of a given Milnor generator Sq(R), where R = (r1 , r2 . . .).
Theorem 2.2. (May [17]). The weight of a Milnor generator Sq(R), where R = (r1 , r2 , . . .),
is w(R) = i, j iai j where ri = j ai j 2 j is the binary expansion of ri .
In fact, May has showed that Sq(R) is the sum of

(P ji )ai j and terms of strictly greater

weight, so they can be identified in the associated graded E 0 A. In particular, the weight of
P ji is just its subscript j. In the language of Priddy’s theory of Koszul resolutions [24], Thej

orem 2.1 implies that E 0 A is a Koszul algebra with Koszul generators {Pk | j ≥ 0, k ≥ 1}
and quadratic relations:
P ji P k = P k P ji if i = k + ,

i−
P ji P i− + P i− P ji + P j+
= 0,

P ji P ji = 0.

The following theorem is first proved in May’s thesis, but see also [24] for a modern treatment.
Theorem 2.3. (May [17], Priddy [24]). H ∗ (E 0 A) is the homology of the complex R, where
R is a polynomial algebra over F2 generated by {Ri, j |i ≥ 0, j ≥ 1} of degree 2i (2 j − 1),
and with the differential is given by
j−1

δ(Ri, j ) =

Ri,k Ri+k, j−k .
k=1

Cup product in H ∗ (E 0 A) correspond to products of representative cycles in R.


144

P. H. Cho’n, L. M. Hà

It is more convenient for our purposes to work with the homology version. The dual
complex, denoted as X¯ in [18], is an algebra with divided powers on the generators P ji . In
fact, X¯ is embedded in the bar construction for E 0 A by sending γn (P i ) to
j

{P ji |P ji | . . . |P ji } (n factors)
and the product in X¯ corresponds to the shuffle product (see [2], pp. 40). Under this embedding, the differential for X¯ is exactly the differential of the bar construction. This embedding
technique was successfully exploited by Tangora ([29], Chap. 5) to compute of the cohomology of the mod 2 Steenrod algebra, up to a certain range.

2.2. May spectral sequence
Let M be a left A-module of finite type and bounded-below. M admits a filtration, induced
from that of A, given by
F p M = F p A · M.
It is clear that F p M = A · M = M if p ≥ 0, and p F p M = 0. The associated graded
0
0
module E 0 M =
p,q E p,q M where E p,q M = (F p M/F p−1 M) p+q is a bigraded mod0
ule over the associated graded algebra E A. Let B(A; M) be the usual bar construction with
the induced filtration:
F p B(A; M) =

F p1 A¯ ⊗ · · · ⊗ F pn A¯ ⊗ F p0 M

n
where the sum is taken over all (n + 1)-tuples ( p0 , . . . , pn ) such that n + i=0
pi ≤ p.

This filtration respects the differential, and in the resulting spectral sequence, we have

E 1p,q,t (M) = F p B p+q (A; M) F p−1 B 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 d 1 of the spectral sequence is the connecting homomorphism of the homology
of the following short exact sequence:
0→

F p−1 B(A; M)
F p B(A; M)
F p B(A; M)
→
→
→ 0.
F p−2 B(A; M)
F p−2 B(A; M)
F p−1 B(A; M)

On the other hand, E 1p,q,t (M) is isomorphic to B p+q (E 0 A; E 0 M)−q,q+t as F2 -trigraded
vector space. Under this identification, d 1 is exactly the canonical differential of the bar
construction B∗ (E 0 A; E 0 M).
Hence E 2p,q,t (M) ∼
= H p+q (E 0 A; E 0 M)−q,q+t and we can summarize the situation
in the following theorem.
Theorem 2.4. ([18]). Let M be an A-module of finite type and bounded-below. There exists a
third-quadrant spectral sequence converging to H∗ (A; M), whose E 2 -term is E 2p,q,t (M) =
H p+q (E 0 A; E 0 M)−q,q+t .
The differentials d r : E rp,q,t (M) −→ E rp−r,q+r −1,t (M) are F2 -linear maps.

May spectral sequence and algebraic transfer

145

3. The algebraic transfer
Let Vs be an s-dimensional F2 -vector space. The mod 2 cohomology of BVs is a polynomial
algebra which we will write as Ps = F2 [x1 , . . . , xs ] where xi are in degree 1.
/ π∗ (S 0 ) admits an algebraic analogue at
The geometric stable transfer π∗ (BVs )+
2
the E level of the Adams spectral sequence:
A
ϕs : Tor s,s+t
(F2 , F2 )

/ Tor A (F2 , Ps ) ∼
= (F2 ⊗ A Ps )t .
0,t

This map was constructed by W. Singer in [26] and further investigation (see [3,5,10–
12,21,23]) shows that it is highly nontrivial. In this section, we will refine this algebraic
gadget by constructing a map between May spectral sequences:
E r ψs : E rp,q,t (F2 )

/ E rp,q−s,t−s (Ps ),

which “converges” to the algebraic transfer. More precisely, E ∞ ψs in homological degree
p + q = s coincides with the induced map of the algebraic transfer in the corresponding
associated graded modules.

3.1. The algebraic transfer and the bar construction
There are several ways to describe the algebraic transfer [3,8,14,23,26]. Each has its own
advantages and disadvantages. We choose to follow the presentation in [23] because we need
an explicit lift of the algebraic transfer on the bar construction.
Let Pˆ1 be the unique A-module extension of P1 = F2 [x1 ] obtained by formally adding
a generator x1−1 of degree −1 and requiring that Sq n (x1−1 ) = x1n−1 . Let u : A → Pˆ1 be the
uniquely determined A-map that sends an operator θ to θ (x1−1 ), and let ψ1 be its restriction
¯ Clearly, ψ1 maps onto P1 . Let ψs : A¯ ⊗s → Ps be defined by
to the augmentation ideal A.
the following recursive formula:
θs (xs−1 )θs (ψs−1 ({θs−1 | . . . |θ1 })),

ψs ({θs | . . . |θ1 }) =

(3.1)

|θs |>0

where we use the “bar notation” for elements of A¯ ⊗s and standard notation for the coproduct
(θ ) = θ ⊗ θ . It is sometimes more convenient to use another form of Eq. (3.1):
ψs ({θs | . . . |θ1 }) = θs (xs−1 ψs−1 ({θs−1 | . . . |θ1 })) + xs−1 θs ψs−1 ({θs−1 | . . . |θ1 }). (3.2)
In [23], it is shown that ψs is a chain-level representation of the algebraic transfer.
Theorem 3.1. (Nam [23], Section 5.1). For each s ≥ 1, ψs is a chain-level representation
of the algebraic transfer
A
A
(F2 , F2 ) → Tor 0,t
(F2 , Ps ) ∼
ϕs : Tor s,s+t
= (F2 ⊗ A Ps )t .

Singer [26] showed that the image of ϕs actually lies in the G L s -invariant subspace of
F2 ⊗ A Ps .
/ B∗−s (A; Ps ) between
We extend ψs to a chain homomorphism ψ˜ s : B∗ (A; F2 )
the bar constructions. Let
ψ˜ s ({θn | . . . |θ1 }) = {θn | . . . |θs+1 } ⊗ ψs ({θs | . . . |θ1 }).

(3.3)


146

P. H. Cho’n, L. M. Hà

Proposition 3.2. The map ψ˜ s is a chain homomorphism.
Proof. We have
ψ˜ s (∂({θn | . . . |θ1 })) =

n−1

{θn | . . . |θi+1 θi | . . . |θs+1 }ψs ({θs | . . . |θ1 })
i=s+1
s

+

(3.4)
{θn | . . . |θs+2 }ψs ({θs+1 | . . . |θ j+1 θ j | . . . |θ1 }).

j=1

Since,
ψs ({θs+1 | . . .|θ j+1 θ j | . . . |θ1 })
−1
= θs+1 (xs−1 . . . θ j+1 θ j (x −1
j . . . θ1 (x 1 ) . . . ) . . . )
−1
−1
= θs+1 (xs−1 . . . θ j+1 (x −1
j θ j (θ j−1 (x j−1 . . . θ1 (x 1 ) . . . ))) . . . )
−1
+ θs+1 (xs−1 . . . θ j+1 (θ j (x −1
j . . . θ1 (x 1 ) . . . )) . . . ),

where the action of θi (x −1 y) is understood as the right hand side of (3.1). Then the second
term on the right hand side of (3.4) equals
{θn | . . . |θs+2 }θs+1 (ψs ({θs | . . . |θ1 })).
Therefore,
ψ˜ s (∂({θn | . . . |θ1 })) = ∂(ψ˜ s ({θn | . . . |θ1 })).
The proof is complete.
Our next result shows that the chain map ψ˜ s just constructed respects the May filtration.
Proposition 3.3. For each s ≥ 1, ψ˜ s restricts to chain map:
F p ψ˜ s : F p B∗ (A; F2 ) → F p B∗−s (A; Ps ).
Thus, ψ˜ s induces a map between May spectral sequences:
E r ψs : E rp,q,t (F2 ) → E rp,q−s,t−s (Ps ), r ≥ 1.
Proof. Since the coproduct preserves May filtration, the general case follows at once, if we
can prove the theorem for the case s = 1, ψ˜ 1 : B∗ (A; F2 ) → B∗−1 (A; P1 ). We need to
prove that if
i

i

1
k
¯
Sq 2 . . . Sq 2 ∈ F−k+1 B1 (A, F2 ) = F−k+1 A,

then,
Sq 2 . . . Sq 2 (x1−1 ) ∈ F−k+1 B0 (A, P1 ) = F−k+1 P1 .
ik

i1

But,
Sq 2 . . . Sq 2 (x1−1 ) = Sq 2 . . . Sq 2
ik

i1

i1

i k−1

ik
(x12 −1 ),

¯ The assertion follows.
∈ F−k+1 A.
and x12 −1 ∈ F0 P1 while Sq 2 1 . . . Sq 2

The second statement follows immediately from the former.
ik

i

i k−1


May spectral sequence and algebraic transfer

147

3.2. A description of E 2 ψs
Proposition 3.3 allows us to use the May spectral sequence to study properties of the chainlevel representation ψs of the s-th algebraic transfer. We are going to give an explicit formula
for the maps in E 2 page:
0

0

E A
E A
E 2 ψs : Tor u,v
(F2 , F2 ) → Tor u−s,v−s
(F2 , E 0 Ps ).

Our construction is similar to Singer’s original construction of the algebraic transfer in [26].
Therefore, we will give only a sketch construction. Consider the short exact sequence of
A-modules

/ P1

0

ι1

/ Pˆ1

π1

/ F2

/ 0,

where ι1 is the inclusion and π1 is the obvious quotient map. Note that π1 has degree 1.
There is a corresponding short exact sequence of E 0 A-modules:

/ E 0 P1

0

E 0 ι1

/ E 0 Pˆ1

E 0 π1

/ F2

/ 0,

and this in turn induces a short exact sequence of the bar constructions:
0 −→ B∗ (E 0 A; E 0 P1 ) −→ B∗ (E 0 A; E 0 Pˆ1 ) −→ B∗ (E 0 A; F2 ) −→ 0.
Tensoring with E 0 M, where M is any A-module of finite type and bounded-below, we obtain
0 −→ B∗ (E 0 M; E 0 P1 ) −→ B∗ (E 0 M; E 0 Pˆ1 ) −→ B∗ (E 0 M; F2 ) −→ 0.
The connecting homomorphism of this exact sequence is
0

0

E A
E A
E 2 ψ1 (M) : Tor s,s+∗
(E 0 M, F2 ) → Tor s−1,s−1+∗
(E 0 M, E 0 P1 ).

More generally, we obtain
0

0

E A
E A
E 2 (ψ1 × Pk−1 )(M) : Tor s,s+∗
(E 0 M, E 0 Pk−1 ) → Tor s−1,s−1+∗
(E 0 M, E 0 Pk ).

Splicing these maps when 0 ≤ k ≤ s together, we obtain
0

0

E A
E A
(E 0 M, F2 ) → Tor 0,∗
(E 0 M, E 0 Ps ).
E 2 ψs (M) : Tor s,s+∗

When M = F2 , E 2 ψs = E 2 ψs (F2 ) is the E 2 -level of the algebraic transfer in the May
spectral sequence. The following is the main theorem of this section.
Proposition 3.4. The E 2 page of the algebraic transfer, E 2 ψs , is induced by the chain-level
map
E 1 ψs (M) : E 0 M ⊗ (E 0 A)⊗s → E 0 M ⊗ E 0 Ps ,
given inductively as follows.
θs (E 1 ψs−1 (M)(m{θs−1 | . . . |θ1 }))θs (xs−1 ).

E 1 ψs (M)(m{θs | . . . |θ1 }) =
|θs |>0


148

P. H. Cho’n, L. M. Hà

Proof. It is sufficient to prove that E 1 (ψ1 × Ps−1 )(M) is given by
i

s−1
)→
m{θs }(x1i 1 . . . xs−1

s−1
m ⊗ θs (x1i 1 . . . xs−1
)θs (xs−1 ).

i

|θs |>0

i

s−1
) ∈ B(E 0 M, E 0 Ps−1 ),
Indeed, for any cycle x = m{θs }(x1i 1 . . . xs−1

i

i

s−1
s−1
+ m ⊗ θs (x1i 1 . . . xs−1
) = 0.
∂(x) = θs (m) ⊗ x1i 1 . . . xs−1

Then pre-image of x under E 0 πs is
s−1 −1
x = m{θs }(x1i 1 . . . xs−1
xs ).

i

Therefore,
s−1 −1
s−1
xs + m ⊗ θs (x1i 1 . . . xs−1
)xs−1
∂(x ) = θs (m) ⊗ x1i 1 . . . xs−1

i

i

s−1
m ⊗ θs (x1i 1 . . . xs−1
)θs (xs−1 )

i

+
|θs |>0

=
|θs |>0

(3.5)

i s−1
m ⊗ θs (x1i 1 . . . xs−1
)θs (xs−1 ).

The proof is complete.
0
0
Example 3.5. For s = 1, the chain level of the transfer, E 1 ψ1 : E −
p,q A → E − p,q−1 P1 ,
for p ≥ 0, q > p, sends
2i (2 p+1 −1)−1

i
i
} → Pp+1
(x1−1 ) = x1
{Pp+1

i

i+ p−1

= Sq 2 . . . Sq 2

The non-trivial cycles in E 0 A are {P1i } ∈ E 0

0,2i

i+ p −1

x12

0
∈ E−

p,q−1 P1 .

i
0 P = P .
, and E 1 ψ1 ({P1i }) = x12 −1 ∈ E 0,∗
1
1

i
Since the A-generators of P1 are exactly those of the form x12 −1 , i ≥ 0, thus E 2 ψ1 is an
isomorphism.

Example 3.6. For s = 2, we have
2i 1 (2 j1 −1)−1 2i 2 (2 j2 −1)−1
x2
.

{P ji 2 |P ji 1 } → E 1 ψ1 ({P ji 1 })P ji 2 (x2−1 ) = x1
2
1
1
2

(3.6)

The nontrivial cycles in (E 0 A)⊗2 are {P1i |P1 } + {P1 |P1i }( j > i + 1) and {P ji |P ji } for j > i
j

j

(see [29, page 49]). Their corresponding images in E 0 P2 are
i
j
j
i
j
j
{P1i |P1 } + {P1 |P1i } → x12 −1 x22 −1 + x12 −1 x22 −1 ,

2i (2 j −1)−1 2i (2 j −1)−1
x2
.

{P ji |P ji } → x1

By induction we obtain the following result which is extremely useful for computation
in later sections.
Corollary 3.7.
2i 1 (2 j1 −1)−1

i

E 1 ψs ({P j s | . . . |P ji 1 }) = x1
s

1

2i s (2 js −1)−1

. . . xs

.

Proof. The results follows immediately from the fact that Pst s are primitive in E 0 A.


May spectral sequence and algebraic transfer

149

4. Two hit problems
The algebraic transfer is closely related to an important problem in algebraic topology called
the hit problem. Call a polynomial hit in Ps if it is a linear combination of elements in the
images of positive Steenrod squares. The quotient of Ps by this subspace is exactly the range
of the algebraic transfer F2 ⊗ Ps . The hit problem asks for a construction of a F2 -basis for
the space of “nonhit” elements F2 ⊗ Ps . For a comprehensive survey of the hit problem
and related questions in modular representation theory, we recommend Wood [33] and the
upcoming book [31] by Walker and Wood.
The hit problem in rank s ≤ 2 is relatively straightforward, but it appears to be very
difficult in general. In fact, the case s = 4 was completely analyzed just recently in a 200
page preprint by Sum [28]. Very little information is available for higher rank, except at a
certain “generic degrees” [9,23]. From the point of view of the algebraic transfer, the elements in generic degrees corresponds to the Adams subalgebra of Ext∗,∗
A (F2 , F2 ) generated
by the elements denoted as h i , i = 0, 1, . . . Our goal is to find more “exotic” elements in
the cohomology of the Steenrod algebra that are also detected by the algebraic transfer.
In the May spectral sequence for Ps in homological degree 0, we have the edge homomorphism
E 2p,− p,t (Ps ) ∼
= H0 (E 0 A, E 0 Ps ) p,− p+t = (F2 ⊗ E 0 A (E 0 Ps )) p,− p+t

E∞

p,− p,t (Ps ),

where the targets E ∞
p,− p,t (Ps ), when p varies, are associated graded components of (F2 ⊗
t
Ps ) . Thus, the hit problem for E 0 Ps , considered as a module over the restricted Lie algebra E 0 A whose structure is completely known, should be helpful as a first step toward
understanding the general structure of F2 ⊗ Ps .
The second hit problem should be simpler in principle. But there are several questions
that remain unsolved, even in the rank 1 case. In [30], Vakil described a recursive algorithm
to compute the filtration degree of a given element of E 0 P1 , but no closed formula was
obtained. We plan to investigate this second hit problem, and its deeper relationship with the
original hit problem elsewhere.
Given a homogeneous polynomial f ∈ Ps . We denote by E( f ), E r ( f ) and [ f ] the
corresponding classes of f in E 1 (Ps ) = E 0 Ps , E r (Ps ) and F2 ⊗ A Ps respectively. Note
that E( f ) is uniquely determined by those monomials of highest filtration degree, which we
will call the essential part of f , and denote as ess( f ). For example,
ess(x17 x213 x313 + x19 x211 x313 + x18 x212 x313 ) = x17 x213 x313 ,
because x17 x213 x313 is in filtration −4 while the latter two monomials are in filtrations −5
and −9 respectively.
Lemma 4.1. Let f ∈ Ps be a homogeneous polynomial. If E( f ) is a nontrivial permanent
cycle in the May spectral sequence for Ps , then ess( f ) is non-hit in Ps .
Proof. It is clear that E 2 ( f ) = E 2 (ess( f )) and thus E ∞ ( f ) = E ∞ (ess( f )). Suppose on
the contrary that ess( f ) is hit in Ps . Write ess( f ) = i Sq ti f i . It follows that there exists
r > 0 such that E r ( f ) = 0 in E r .
Example 4.2. Consider g = x 3 y 5 + x 5 y 3 ∈ P3 in degree 8. Since g = Sq 2 (x 3 y 3 ) +
0
(P2 ) is
Sq 4 (x 2 y 2 ), we have that [g] = 0 in F2 ⊗ A Ps . On the other hand, E(g) ∈ E −1,9
nontrivial. However, when passing to the E 2 terms, we have
g = Sq 2 (x 3 y 3 ) + x 4 y 4 ,

150

P. H. Cho’n, L. M. Hà

0
0
0 P in
P2 , so E(g) = Sq 2 E 0 (x 3 y 3 ) ∈ E −1,3
A · E 0,6
and x 4 y 4 ∈ F−4 P2 is trivial in E −1,9
2
E 2 (Ps ). Thus E(g) is trivial in E 2 (P2 ).

One may wonder whether the converse is true, that is if f is a homogeneous polynomial
such that E( f ) is trivial in E 2 (Ps ), is it necessarily true that f is hit in Ps ? The following
example implies that this is not the case.
Example 4.3. Let m = x17 x213 x313 ∈ P3 , it is not difficult to check that m is nonhit in P3 . On
the other hand,
m = Sq 2 (x17 x211 x313 ) + x19 x211 x313 + x18 x212 x313 + x17 x212 x314 + x18 x211 x314 ,
where x19 x211 x313 ∈ F−5 P3 and the last three monomials are in lower filtrations. Therefore,
0
P3 .
E(m) = Sq 2 E(x17 x211 x313 ) ∈ E −4,37

So E(m) = 0 in E 2 (Ps ).
The point of Example 4.3 is that in the class [m], one can choose a different representative
which is in lower filtration than m. Our last example shows that if a homogeneous polynomial is hit (that is, it is trivial in F2 ⊗ A Ps ), then we may have to wait for a while before this
element is killed in the spectral sequence.

Example 4.4. Consider n = x1 x22 x32 + x12 x2 x32 + x12 x22 x3 = Sq 2 (x1 x2 x3 ), so n is hit in
P3 . It can be checked by direct inspection that E(n) is nontrivial in E 2 (P3 ) because it is
0
(P3 ). The element n does not survive to E 3 because the above formula
E 0 A-nonhit in E −2,7

2
shows that n = d 2 (n ), where n = Sq 2 ⊗ (x1 x2 x3 ) ∈ E 0,1,5
(P3 ).

The following is the main result of this section.
Proposition 4.5. Let f ∈ Ps be a homogeneous polynomial in filtration degree p. Then f
is a nontrivial permanent cycle in the May spectral sequence for Ps if and only if ess( f ) is
non-hit in Ps and there does not exist any non-hit polynomial g ∈ Fq Ps , with q < p, such
that ess( f ) − g is hit.
Proof. Suppose f is trivial in E r (Ps ). Then E r −1 ( f ) = E r −1 (ess( f )) is in the image of
the differential d r −1 . Therefore, there exist θr −1,i ∈ A¯ and fr −1,i ∈ Ps such that
E r −1 (ess( f )) =

E r −1 (θr −1,i fr −1,i ) ∈ E r −1 (Ps ).
i

In the E r −2 -term, we have
E r −2 (ess( f )) =

E r −2 (θr −1,i fr −1,i ) + fr −2 ∈ E r −2 (Ps ),
i

where fr −2 is in the image of d r −2 . Thus, there exist θr −2,i ∈ A¯ and fr −2,i ∈ Ps such that
E r −2 (θr −2,i fr −2,i ) ∈ E r −2 (Ps ),

fr −2 =
i

and we can write
E r −2 (ess( f )) =

E r −2 (θki f ki ) ∈ E r −2 (Ps ).
i,r −2≤k≤r −1


May spectral sequence and algebraic transfer

151

Repeating this process, in the E 1 -term, we obtain

E 1 (ess( f )) =

r −1

E 1 (θki f ki ) ∈ E 1 (Ps ).

i,k=1

Note that E 1 (Ps ) = E 0 (Ps ), we finally have
r −1

ess( f ) =

θki f ki + g ∈ Ps ,
i,k=1

where g is some polynomial in Fq Ps , with q < p. Thus, g is hit in Ps iff so is ess( f ).
Conversely, if there exists g ∈ Fq Ps with q < p such that ess( f ) − g is hit in Ps then, in
E rp,∗,∗ Ps ,
E r ( f ) = E r (ess( f ) − g).
Since ess( f ) − g is hit in Ps , ess( f ) − g is a trivial permanent cycle and so is f .
Companion to Proposition 4.5 is the following key result, due to Wood [32], which will be
used frequently in our applications.
Theorem 4.6. (Wood [32]). Let f be a monomial of degree d in Ps with exactly r odd
exponents and suppose r < μ((d − r )/2). Then f is hit.
In the above theorem, μ(d) is the least number k for which it is possible to write d as a sum
k
(2 i − 1).
of k numbers d = i=1

5. First application: a non-detection result
In this section, we use the representation of the algebraic transfer in the E 2 -term of the May
spectral sequence to investigate its image. The first author Cho’n’s thesis (in preparation,
2010) contains a complete analysis of the algebraic transfer in rank up to 4. Notably, we
recover all previously known results, obtained by various different methods.
Here is our first main result.
Theorem 5.1. The following elements in the cohomology of the Steenrod algebra
(F2 , F2 );
(1) h 1 Ph 1 ∈ Ext 6,16
A
(F2 , F2 );
(2) h 20 Ph 2 ∈ Ext 7,18
A

(F2 , F2 ), 0 ≤ n ≤ 5;
(3) h n0 i ∈ Ext 7+n,30+n
A


152

P. H. Cho’n, L. M. Hà

(F2 , F2 ), 0 ≤ n ≤ 2,
(4) h n0 j ∈ Ext 7+n,33+n
A
are not detected by the algebraic transfer.
Note that according to Bruner [4], h 60 i = h 30 j = 0. Thus the last two results are best
possible. We would like to emphasize that the dimension of the above elements go far beyond
the current computational knowledge of the hit problem.
(F2 , F2 ) and Ph 2 ∈ Ext 5,16
(F2 , F2 ) are not in
Corollary 5.2. ([26,25]). Ph 1 ∈ Ext5,14
A
A
the image of the dual algebraic transfer.
The claims of this corollary are not new-they are due to Singer [26] and Qu`ynh [25] respectively. We believe that our proof is much less computational.
Proof (Proof Of Corollary 5.2). It is well-known that the subalgebra generated by the h i s is
in the image of the dual of the algebraic transfer. Thus, if Ph 1 or Ph 2 were detected, then
h 1 Ph 1 and h 20 Ph 2 would be as well, since the dual of the algebraic transfer is an algebra
homomorphism. This contradicts Theorem 5.1.
We are now ready to prove Theorem 5.1. The proof will be divided into 4 parts, corresponding
to the four nontrivial elements in the cohomology of the Steenrod algebra under consideration. The general strategy is as follows. Suppose an element in x ∈ Tor s,s+∗
A (F2 , F2 ) has

nontrivial image y, via the transfer ϕs in F2 ⊗ A Ps , we choose a representative x¯ for this
element in the E 1 (F2 ) term of the May spectral sequence. Because of naturality, its image y¯
in E 1 (Ps ) must be a nontrivial permanent cycle and represents y. We then show that this is
not possible using knowledge of the A-module structure of Ps and a combination of degree
arguments as well as computation in the May spectral sequence.

5.1. h 1 Ph 1 is not detected
According to Tangora [29], a representation for h 1 Ph 1 in the E 1 -term of the May spectral
2 R 4 . Its dual, therefore, contains
sequence is R1,1
0,2
1
(F2 ),
X = {P11 |P11 } ∗ {P20 |P20 |P20 |P20 } ∈ E −4,10,16

where ∗ denotes the shuffle product (see [2, pp 40]).
Corollary 3.7 allows us to find the image of X under E 1 ψ6 :
(E 1 ψ6 )(X ) = x1 x2 x32 x42 x52 x62 + all its permutations
= Sq 4 (x1 x2 x3 x4 x5 x6 ).
Therefore, (E 1 ψ6 )(X ) ∈ E 0 (P6 ) is hit in P6 . In the bar construction, we must have
(h 1 Ph 1 )∗ = X + x, where x ∈ F p B(F2 ) with p < −4. Thus, if h 1 Ph 1 is detected,
then
ψ6 ((h 1 Ph 1 )∗ ) = E 1 ψ6 (X ) + y,
where y ∈ F p P6 with p < −4, is nonhit in P6 . On the other hand, by direct computation,
we see that there are only two possible monomials (or their permutations) in F−5 P6 , namely
x14 x24 x32 x40 x50 x60 and x14 x22 x32 x42 x50 x60 . But these two monomials are clearly hit in P6 .


May spectral sequence and algebraic transfer

153

5.2. h 20 Ph 2 is not detected
The argument is similar to the first case, so we will only give a sketch proof. Again, from
[29], we know that a representation for h 20 Ph 2 :
∗

h 20 Ph 2 = {P10 }2 ∗ {P12 } ∗ {P20 }4 + {P10 }3 ∗ {P11 } ∗ {P20 }2 ∗ {P30 },
where {a}n denotes {a|a| . . . |a} (n factors). Its image under E 1 ψ7 is
x10 x20 x33 x42 x52 x62 x72 + x10 x20 x30 x41 x52 x62 x76 + all their permutations,
and again one can easily verify that this image is hit in P7 . The argument now follows a
similar line as in the first case, noting that there does not exist any non-hit polynomial in
F p P7 with p < −4 in degree 11.

5.3. h n0 i is not detected
It is sufficient to provide a proof for n = 5. According to Tangora ([29]), h 50 i is represented
6 R 2 R 2 (R
in the E 1 -term of the May spectral sequence by x = R0,1
0,2 0,3 1,1 R0,3 + R1,2 R0,2 ).
By direct inspection, we see that the dual of h 50 i has a representation in the E 1 -term of the
May spectral sequence as
x ∗ = {P20 }2 ∗ {P30 }3 ∗ {P11 } ∗ {P10 }6 + {P20 }3 ∗ {P30 }2 ∗ {P21 } ∗ {P10 }6 .
The image of this element under E 1 ψ12 is
6 x 6 x 6 + x 0 · · · x 0 x 2 x 2 x 2 x 5 x 6 x 6 + all their permutations.
x10 · · · x60 x71 x82 x92 x10
11 12
1
6 7 8 9 10 11 12

It can easily be checked, using Wood’s result ([32]), that this image is hit in P12 . Therefore,

ψ12 ((h 50 i)∗ ) = E 1 ψ12 (x ∗ ) + X,
for some X ∈ F p P12 in filtration p < −8 and degree 23. Using Wood’s result again, we
obtain that X is hit in P12 . Thus, ψ12 ((h 50 i)∗ ) is hit in P12 .

5.4. h n0 j is not detected
We need only to consider the case n = 2. According to Tangora [29], a representation of
3 R 2 R 2 R 2 . We compute its dual
h 20 j in the E 1 -term of the May spectral sequence is R0,1
0,2 0,3 1,2
in the bar construction, the result is
{P10 }3 ∗ {P20 }2 ∗ {P30 }2 ∗ {P21 }2 + {P10 }3 ∗ {P20 } ∗ {P21 } ∗ {P30 }3 ∗ {P11 } + x.
where x = {P10 }3 ∗ {P20 } ∗ {P30 }3 ∗ {P10 |P30 } ∈ F−9 B(A; F2 ).
The image under E 1 ψ9 of the first two summands is
x42 x52 x65 x75 x86 x96 + x4 x52 x65 x76 x86 x96 + all their permutations.
which are all hit in P9 by using Wood’s test for the case r = 2, d = 26. Let X ∈ F−9 P9
be the image of x. We claim that X also consists of monomials in which there are at most


154

P. H. Cho’n, L. M. Hà

2 odd exponents, and therefore X is also hit. Indeed, looking at Corollary 3.7, we see that
an odd exponent is obtained whenever there appears a P ji with i > 0. Since x only contains
monomials {a1 | . . . |a9 }, where ai is primitive in A, and the coproducts of P21 and P11 , in A,
are
(P21 ) = 1 ⊗ P21 + P20 ⊗ P20 + P21 ⊗ 1,
(P11 ) = 1 ⊗ P11 + P10 ⊗ P10 + P11 ⊗ 1.

6. Second application: p0 is in the image of the transfer

In this sections, we show that our method can also be used to detect elements in the image
of the algebraic transfer. This completes the proof of a conjecture in [11] which describe the
behaviour of the algebraic transfer in rank 4.
The following is our second main result.
(F2 , F2 ) is in the image of the fourth algebraic
Theorem 6.1. The element p0 ∈ Ext 4,37
A
transfer.
This result is announced [12], but the details have not appeared. All previous methods of
computation such as [26,23,10,11] have failed to work with p0 .
As an immediate corollary of the above theorem, the entire family of elements { pi }
belongs to the image of the dual algebraic transfer.
i

(F2 , F2 ), i ≥ 0 is in the image
Corollary 6.2. Every element in the family pi ∈ Ext 4,37·2
A
of the algebraic transfer.
Proof. it is well known that the cohomology of the Steenrod algebra (in fact, any cocommutative Hopf algebra) admits an action of a Steenrod algebra in which Sq 0 is an independent
operation, not equal to the identity ([16]). In the family, { pi }, one has Sq 0 ( pn ) = pn+1 .
On the other hand, it follows from work of Boardman [3] (see also Minami [21]) that there
exists a similar Sq 0 operation, first constructed by Kameko [13], that commutes with the
classical Sq 0 on the E xt group. Therefore, if p0 belongs to the image of the dual algebraic
transfer, so does the entire family { pi }.
2 so that its
Proof of Theorem 6.1 According to Tangora, p0 is represented by R0,1 R3,1 R1,3

dual p0∗ in the E 1 -term of May spectral sequence contains

p¯ 0 = {P13 } ∗ {P10 } ∗ {P31 |P31 } + {P13 |P13 } ∗ {P21 } ∗ {P40 }.

In fact, by direct inspection, a cycle representing p0∗ can be chosen as p0∗ = p¯0 +x + y +z +t,
where
x = {P12 |P12 } ∗ {P21 } ∗ {P40 } ∈ F−5 B(A; F2 );
y = {P20 |Sq(6)} ∗ {P31 |P31 } ∈ F−6 B(A; F2 );
z = {P40 |P30 } ∗ {P10 } ∗ {P31 } + {Sq(7)|P40 } ∗ {P10 } ∗ {P31 }
+ {P30 |Sq(4, 1)} ∗ {P13 } ∗ {P40 } + {P30 |Sq(7)} ∗ {P13 } ∗ {P40 }
+ {Sq(0, 3)|P30 } ∗ {P21 } ∗ {P40 } + {P30 |P20 |P22 } ∗ {P40 }
+ {P30 |Sq(3)|P22 } ∗ {P40 } + {P12 |P20 |P40 } ∗ {P40 } + {P20 |Sq(5)|P40 } ∗ {P31 }
+ {P20 |P12 |P40 |P40 } + {Sq(6)|P40 |P40 } ∗ {P10 } ∈ F−7 B(A; F2 );
t = {P10 |Sq(6, 3)} ∗ {P21 } ∗ {P40 } + {P10 |Sq(5, 3)|P30 } ∗ {P40 }
+ {P30 |P30 |Sq(2, 2)} ∗ {P40 } ∈ F−9 B(A; F2 ).


May spectral sequence and algebraic transfer

155

We want to compute the image of p0∗ under the chain level map ψ4 . To simplify notation, we
write (a, b, c, d) for the monomial x1a x2b x3c x4d ∈ P4 , and make use of a ∗ operation which
is similar to the shuffle product. For example,
(a, b, c) ∗ (d) = (a, b, c, d) + (a, b, d, c) + (a, d, b, c) + (d, a, b, c).
Note that we only care about the image modulo hit elements. The computation is greatly
simplified because of the following observation.
Lemma 6.3. Let f be a monomial of degree 33 in P4 .
(i) If f has exactly one odd exponent, then f is hit.
(ii) If f ∈ F−9 P4 , then f is hit.
Proof. Since μ( 33−1
2 ) = 2, the first statement follows immediately from Wood’s criterion (Theorem 4.6). So we need only be concerned with monomials which have exactly 3
odd exponents. For the second statement, there is simply no monomial with exactly 3 odd
exponents in degree 33 having filtration degree at most −8.

It follows from the above lemma that the image of t is hit in P4 . Modulo hit elements,
the image of p0∗ is
ψ4 ( p0∗ ) = X + X (12) + X (132) + X (1432) + Y,
where
X = x10 x27 x313 x413 + x10 x213 x37 x413 + x10 x213 x313 x47
+ x10 x213 x317 x43 + x10 x217 x313 x43 + x10 x217 x33 x413 ;
Y = (7, 7) ∗ (5) ∗ (14) + (16, 5, 7) ∗ (5) + (18, 3, 7) ∗ (5) + (20, 1, 7) ∗ (5)
+ (11, 3, 14) ∗ (5) + (11, 3) ∗ (5) ∗ (14) + (5, 2) ∗ (13, 13) + (17, 1, 2) ∗ (13)
+ (14, 9, 3, 7) + (9, 14, 3, 7) + (9, 3, 14, 7) + (7, 14, 3, 9) + (7, 9, 3, 14)
+ (14, 7, 3, 9) + (9, 3, 7, 14) + (9, 7, 3, 14) + (20, 1, 5, 7) + (16, 9, 1, 7)
+ (9, 16, 1, 7) + (5, 16, 9, 3) + (9, 5, 16, 3) + (18, 3, 9, 3) + (9, 3, 18, 3)
+ (9, 5, 14, 5) + (9, 5, 5, 14) + (5, 9, 5, 14).
(12), (132), (1432) are elements of the symmetric group S4 permuting the four variables
of P4 . The notation X (12) means it is obtained from X by applying the permutation (12).
Since
X = Sq 6 (x27 x313 x47 ) + Sq 4 (x27 x313 x49 + x29 x313 x47 + x213 x313 x43 )
+ x213 x37 x413 + x217 x33 x413 (mod A¯ P4 )
= Sq 2 (x213 x37 x411 + x217 x33 x411 ) + Sq 4 (x213 x35 x411 ) (mod A¯ P4 ),
we see that all four summands X, X (12), X (132) and X (1432) are hit in P4 . We can also
simplify Y . Observe that


156

P. H. Cho’n, L. M. Hà
(7, 7) ∗ (5) ∗ (14) = Sq 2 ((7, 7) ∗ (3) ∗ 14))
+(7) ∗ (9) ∗ (3) ∗ (14) + (7, 7) ∗ (3) ∗ (16) + others;
(7, 7) ∗ (3) ∗ (16) = Sq 8 ((7, 7) ∗ (3) ∗ (8)) + Sq 4 ((11, 11) ∗ (3) ∗ (4))
+(13, 13) ∗ (3) ∗ (4) + others;

(7) ∗ (9) ∗ (3) ∗ (14) = Sq 4 ((7) ∗ (5) ∗ (3) ∗ (14)) + Sq 2 ((7) ∗ (5, 5) ∗ (14))
+(11) ∗ (5) ∗ (3) ∗ (14) + (7) ∗ (5) ∗ (3) ∗ (18) + others;
(7) ∗ (5) ∗ (3) ∗ (18) = Sq 8 ((7) ∗ (5) ∗ (3) ∗ (10)) + Sq 4 ((7) ∗ (9) ∗ (3) ∗ (10))
+Sq 2 ((11) ∗ (5, 5) ∗ (10)) + (13) ∗ (5) ∗ (3) ∗ (10) + others,
where “others” means monomials that are either hit or in filtration degree strictly less
than −5. This shows that
Y = (3, 5) ∗ (11) ∗ (14) + Y2 (modulo hit),
1
where Y2 is in F p P4 with p < −5. Thus, E(Y ) = E((3, 5) ∗ (11) ∗ (14)) ∈ E −5,5,33
(P4 ).

Lemma 6.4. Y1 = (3, 5) ∗ (11) ∗ (14) is a nontrivial permanent cycle in the May spectral
sequence for P4 .
The proof of this lemma is elementary but rather technical, so we postpone it to Sect. 7.
Assume that Lemma 6.4 is proved, then ψ4 ( p0∗ ) is non-hit in P4 . Thus, p0 is in the
image of the fourth algebraic transfer. The proof is complete.

7. Proof of lemma 6.4
In this final section, we prove Lemma 6.4. Our strategy is to prove that Y1 represents a
nontrivial permanent cycle in the May spectral sequence for P4 .
Proposition 7.1. Y1 is nonhit in E 0 P4 , considered as an E 0 A-module. In other words, it is
a nonzero element in term E 2 = H0 (E 0 A, E 0 P4 ).
Proof. We prove this by contradiction. Assume that there exist A1 , B1 , C1 , D1 , E 1 ∈
1
, such that
E −4,4,∗
Y1 = P10 (A1 ) + P11 (B1 ) + P12 (C1 ) + P13 (D1 ) + P14 (E 1 ).

(7.1)

Such an expression is called a hit presentation of Y1 . Here, we are using a simple fact that
can be verified directly from the relations in Theorem 2.1 that the P1i generate E 0 A. We also
assume that this presentation is reduced, that is, one can not extract a smaller hit presentation
of Y1 . It is convenient to visualize the process in terms of “lighting flash”. For example, everytime we choose a monomial in A1 to hit a certain monomial, the operation P10 potentially
can create extra monomials, which we again need to hit by some other monomials... The
process will stop once we find that besides the monomials in Y1 , there are no other extra
monomials. The pair of a monomial and an arrow chosen must not be repeated. A forced
presentation means one in which the choice of the first monomial dictates unique choices of
the remaining monomials.
We begin with some simple observations.
Lemma 7.2. If A1 , B1 , C1 , D1 and E 1 , provide a hit presentation of Y1 , then C1 does
not contain any permutations of (3, 12, 7, 7) and B1 does not contain any permutations of
(3, 10, 7, 11).


May spectral sequence and algebraic transfer

157

Proof. If C1 contained (3, 12, 7, 7), then the right hand side of (7.1) must contain
(3, 16, 7, 7), but there is no other way to hit (3, 16, 7, 7) to eliminate it from Y1 . Note
that while expression P10 (3, 15, 7, 7) contains (3, 16, 7, 7) in A-module P4 , it is trivial in
E 0 A-module E 0 P4 because of the filtration discrepancy. This also explain why the hit problem for E 0 Ps should be simpler. The statement for B1 follows easily since P11 (3, 10, 7, 11)
contains (3, 12, 7, 11) and the only other way to hit (3, 12, 7, 11) is with P12 (3, 12, 7, 7),
which is not possible.
2
.
Lemma 7.3. Y1 = (5, 3) ∗ (11) ∗ (14) in E −5,5,33

Proof. In fact, we have the following identity:

(3, 5, 11, 14) + (3, 5, 14, 11) = (5, 3, 11, 14) + (5, 3, 14, 11) + P10 [(3, 5, 11, 13)
+ (3, 5, 13, 11) + (5, 3, 11, 13) + (5, 3, 13, 11)

(7.2)

+ (5, 5, 11, 11)] + P11 [(3, 6, 11, 11) + (6, 3, 11, 11)].

Now consider the following list of 4 pairs of elements in degree 33.
•
•
•
•

a = (3, 5, 11, 14) + (3, 5, 14, 11), a = (5, 3, 11, 14) + (5, 3, 14, 11);
b = (5, 6, 11, 11), b = (6, 5, 11, 11);
c = (9, 6, 7, 11) + (6, 9, 7, 11), c = (9, 6, 11, 7) + (6, 9, 11, 7);
d = (10, 5, 7, 11) + (5, 10, 7, 11), d = (10, 5, 11, 7) + (5, 10, 11, 7).

Lemma 7.4. The above 8 elements induces equivalent classes in E 2 .
Proof. Because of symmetry and Lemma 7.3, we need only to show that a, b, c and d are
equal in E 2 . This follows from the following explicit formula.
a = P10 [(3, 5, 11, 13) + (3, 5, 13, 11)] + P11 (3, 6, 11, 11) + P12 (3, 4, 11, 11) + b. (7.3)
b = P10 [(5, 9, 7, 11)] + P12 (5, 6, 7, 11) + c.
(7.4)
c = P10 [(5, 9, 7, 11) + (9, 5, 7, 11)] + d.

(7.5)

The proof of Proposition 7.1 follows from the following key observation.
Lemma 7.5. In any partial hit presentation of a, there appears an even number of elements

in the list above (including a).
It follows from this lemma that all these 8 elements are “inseparable” in the sense that everytime we try to find a hit presentation for one element by adding arrows to the process, we
always create extra terms which is exactly another element of the list.
We will provide a detailed argument for b. The argument for other elements are similar
and will be omitted. Clearly, there are four ways to hit b:
P10 (5, 5, 11, 11); P12 (5, 6, 7, 11); P12 (5, 6, 11, 7); P11 (3, 6, 11, 11).
The first choice ends up with b . For the second choice, note that
b = P12 (5, 6, 7, 11) + (9, 6, 7, 11) + (5, 10, 7, 11).
For the extra terms, if we use either P10 (9, 5, 7, 11) or P10 (5, 9, 7, 11), we arrive at either c
or d. The only other way to hit them is with P12 [(9, 6, 7, 7) + (5, 10, 7, 7)] (we cannot use


158

P. H. Cho’n, L. M. Hà

P11 (3, 10, 7, 11) because of Lemma 7.2.), but then this will give us either c or d . The third
choice is similar to the second, by symmetry. Finally,
P11 (3, 6, 11, 11) = P12 (3, 4, 11, 11) + (3, 6, 13, 11) + (3, 6, 11, 13).
Now we can hit (3, 6, 13, 11) with either P01 (3, 5, 13, 11) or P12 (3, 6, 7, 13). So there are
four cases:
•
•

P10 [(3, 5, 11, 13) + (3, 5, 13, 11)] or P12 [(3, 6, 7, 13) + (3, 6, 13, 7)];
P01 (3, 5, 11, 13) + P12 (3, 6, 13, 7) or P01 (3, 5, 13, 11) + P12 (3, 6, 7, 13).

The first choice yields extra term a. The second yields (3, 10, 7, 13) + (3, 10, 13, 7) for
which we have only one possible continuation by
P10 [(3, 9, 7, 13) + (3, 9, 13, 7)] + P12 [(3, 5, 14, 7) + (3, 5, 7, 14)]

(7.6)

and this again ends with a.
In the third case, the extra term is (3, 5, 11, 14) + (3, 10, 13, 7) and again for the latter
monomial (3, 10, 13, 7), there is only one continuation with P01 (3, 9, 13, 7)+ P12 (3, 5, 14, 7)
and we obtain (3, 5, 14, 11).
Proposition 7.6. [Y1 ] is a nontrivial cycle in the May spectral sequence E r (P4 ).
2
Proof. Suppose there exists x ∈ E −3,4,33
such that d 2 (x) = [Y1 ]. Write B for the bar
0
0
construction B(E A, E P4 ) and let

Z rp = {z|z ∈ F p B, ∂z ∈ F p−r B}.
Since ∂(θ1 θ2 ⊗ m) = ∂(θ1 ⊗ θ2 m) we can assume that x =
such that P1i (m i ) ∈ F−5 B.
Thus, we can write

i
i P1 ⊗ m i where m i ∈ F−3 B

2
,
Y1 = P10 (m 0 ) + P11 (m 1 ) + P12 (m 2 ) + P13 (m 3 ) + P14 (m 4 ) ∈ E −5,5,33

and by direct inspection, we see that there are no choices for m 0 , while the m i , i > 0 might
be a combination (of permutations) of monomials in the collection (Ci ) below.
(C1 ) = {(1, 1, 14, 15), (1, 7, 8, 15), (4, 5, 7, 15)}

(C2 ) = {(1, 3, 10, 15), (2, 3, 9, 15), (3, 3, 8, 15)}
(C3 ) = {(1, 2, 3, 19), (3, 7, 7, 8), (5, 6, 7, 7)}
(C4 ) = {(1, 1, 1, 14), (1, 1, 2, 13), (1, 1, 4, 11), (1, 1, 7, 8),
(1, 3, 3, 10), (1, 4, 5, 7), (2, 3, 3, 9), (3, 3, 3, 8), (3, 3, 5, 6)}.
It is straightforward to verify that P14 (m) ∈ F−6 (E 0 P4 ) for all m ∈ (C4 ). Also, direct
computation shows that P11 (m 1 ) and P12 (m 2 ) are all E 0 A-hit in E 0 P4 , for example
P11 (1, 1, 14, 15) = (2, 2, 14, 15) + (1, 1, 16, 15) = P10 (1, 2, 14, 15) + P12 (1, 1, 12, 15).
2 for all i = 0, 1, 2, 4. For C , we have P 3 (1, 2, 3, 19) ∈
Thus P1i (m i ) are all trivial in E −5
3
1
F−6 . There are two monomials left, namely (3, 7, 7, 8) and (5, 6, 7, 7). On the other hand,

P13 (3, 7, 7, 8) = (3, 8, 11, 11) + (3, 16, 7, 7) = P12 (3, 4, 11, 11) + (3, 16, 7, 7),


May spectral sequence and algebraic transfer

159

2 . Similarly, we have
so P13 (3, 7, 7, 8) = (3, 16, 7, 7) in E −5

P13 (5, 6, 7, 7) = (9, 10, 7, 7) + (5, 6, 11, 11).
2
. Indeed,
In fact, we claim that P13 (3, 8, 7, 7) = P13 (5, 6, 7, 7) in E −5,5,33

P13 ((3, 8, 7, 7) + (5, 6, 7, 7)) =P13 ((3, 6, 9, 7) + (3, 6, 7, 9)) + P12 ((3, 12, 7, 7)
+ (5, 10, 7, 7) + (3, 6, 13, 7) + (3, 6, 7, 13)

+ (5, 6, 7, 11) + (5, 6, 11, 7) + (3, 4, 11, 11))
+ P11 ((3, 10, 11, 7) + (3, 10, 7, 11)).
So P13 (m 3 ) is a sum of permutations of (9, 10, 7, 7) + (5, 6, 11, 11), and we need to show
that it can never be equal to Y1 , which is the sum of 6 permutations of (5, 6, 11, 11), where
5 always precede 6. The proof now uses similar argument as in the proof of Lemma 7.3,
noting that elements in the “orbit” of (9, 10, 7, 7) are
•
•
•
•
•

(10,9,7,7),
(5,10,7,11) + (5,10,11,7),
(6,9,7,11) + (6,9,11,7),
(10,5,7,11) + (10,5,11,7),
(9,6,7,11) + (6,9,7,11).

3
. The fact that Y1 is nontrivial
This completes the proof that Y1 is non trivial in E −5,5,33
r
in E −5,5,33 , r = 4, 5 is proved by using similar argument. In fact, there are very few choices
of monomials in degree 33 that can jump filtration, and the problem becomes much simpler
than that at E 2 .

Finally, we would like to remark that R. Bruner has independently used computer software to calculate the space F2 ⊗ A P4 in degree 33. He found a unique G L 4 -invariant element,
and has also confirmed by Bruner (2009 personal communication) that our hand-computed
element Y is the same as his (though of different presentations in F2 ⊗ A P4 ).
Acknowledgments. The authors would like to thank R. R. Bruner, N. H. V. Hu’ng, W. H.

Lin and J. P. May for many enlightening e-mail exchanges and discussions. We are grateful
to the referee for helpful comments and corrections. This article is partially supported by
a Grant from the Vietnam National Foundation for Science and Technology Development
(NAFOSTED) and VNU Grant QG-10-02.

References
[1] Adams, J.F.: On structure and applications of the Steenrod algebra. Comment. Math.
Helv. 32, 180–214 (1958)
[2] Adams, J.F.: On the non-existence of elements of Hopf invariant one. Ann. of
Math. 72, 20–104 (1960)
[3] Boardman, J.M.: Modular representations on the homology of powers of real projective
space. Contemp. Math. 146, 49–70 (1993)
[4] Bruner, R.R.: The cohomology of the mod 2 Steenrod algebra: A computer calculation. WSU Research Report 37, 217 (1997)
[5] Bruner, R.R., Hà, L.M., Hu’ng , N.H.V.: On behavior of the algebraic transfer. Trans.
Amer. Math. Soc. 357, 473–487 (2005)


160

P. H. Cho’n, L. M. Hà

[6] Cho’n P.H., Hà L.M.: On May spectral sequence and the algebraic transfer. Proc. Japan
Acad., 86, Ser. A, 159–164 (2010)
[7] Cho’n, P.H., Hà, L.M.: Lambda algebra and the Singer transfer. C. R. Acad. Sci. Paris,
Ser. I 349, 21–23 (2011)
[8] Cho’n P.H., Hà L.M.: Lambda algebra and the Singer transfer, preprint (2010)
[9] Crabb, M.C., Hubbuck, J.R.: Representations of the homology of BV and the Steenrod
algebra II. Progr. Math. 136, 143–154 (1996)
[10] Hà, L.M.: Sub-Hopf algebra of the Steenrod algebra and the Singer transfer. Geo. Topo.
Mono. 11, 81–104 (2007)

[11] Hu’ng, N.H.V.: The cohomology of the Steenrod algebra and representations of the
general linear groups. Trans. Amer. Math. Soc. 357, 4065–4089 (2005)
[12] Hu’ng, N.H.V., Qu`ynh, V.T.N.: The image of the fourth algebraic transfer. C. R. Acad.
Sci. Paris, Ser. I 347, 1415–1418 (2009)
[13] Kameko M.: Products of projective spaces as Steenrod modules, Thesis, Johns Hopkins
University, Baltimore (1990)
[14] Lannes, J., Zarati, S.: Sur les foncteurs dérivés de la déstabilisation. Math. Z. 194(1),
25–59 (1987)
5,∗
[15] Lin, W.H.: Ext4,∗
A (Z/2, Z/2) and Ext A (Z/2, Z/2). Topo. App. 115, 459–496 (2008)
[16] Liulevicius A.: The factorization of cyclic reduced powers by secondary cohomology
operations, Mem. Amer. Math. Soc. 42, (1962)
[17] May J.P.: The cohomology of restricted Lie algebras and of Hopf algebras; applications
to the Steenrod algebra PhD, Princeton University, Princeton (1964)
[18] May, J.P.: The cohomology of restricted Lie algebra and of Hopf algebra. J. Algebra 3, 123–146 (1966)
[19] Milnor, J.: The Steenrod algebra and its dual. Ann. Math. 67, 150–171 (1958)
[20] Milnor, J., Moore, J.C.: On the structure of Hopf algebras. Ann. Math. 81,
211–264 (1965)
[21] Minami, N.: The iterated transfer analogue of the new doomsday conjecture. Trans.
Amer. Math. Soc. 351, 2325–2351 (1999)
[22] Mitchell, S.A.: Splitting B(Z/ p)n and BT n via modular representation theory. Math.
Z. 189, 1–9 (1985)
[23] Nam, T.N.: Transfert algébrique et action du groupe linéaire sur les puissances divisées
modulo 2. Ann. Inst. Fourier (Grenoble) 58, 1785–1837 (2008)
[24] Priddy, S.B.: Koszul resolutions. Trans. Amer. Math. Soc. 152, 39–60 (1970)
[25] Qu`ynh, V.T.N.: On behavior of the fifth algebraic transfer. Geo. Topo. Mono. 11,
309–326 (2007)
[26] Singer, W.M.: The transfer in homological algebra. Math. Z. 202, 493–523 (1989)
[27] Steenrod E.N., Epstein D.B.A.: Cohomology operations. Ann. Math. Studies 50 (1962)

[28] Sum N.: The hit problem for the polynomial algebra of four variables, preprint (2006)
[29] Tangora, M.C.: On the cohomology of the Steenrod algebra. Math. Z. 116, 18–64 (1970)
[30] Vakil, R.: On the Steenrod length of real projective spaces: finding longest chains in
certain directed graphs. Discrete. Math. 204, 415–425 (1999)
[31] Walker G., Wood R.M.W.: Polynomials and the Steenrod algebra, book in progress,
available at (2010)
[32] Wood, R.M.W.: Steenrod squares of polynomials and the Peterson conjecture. Math.
Proc. Camb. Phil. Soc. 150, 307–309 (1989)
[33] Wood, R.M.W.: Problems of the Steenrod algebra. Bull. London Math. Soc. 30,
449–517 (1998)