The algebraic transfer is an important tool to study the cohomology of the Ste-enrod algebra.. In this study, we will construct a version of the algebraic transfer in E2-term of May spe
Trang 1Phan 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
Ste-enrod algebra In this study, we will construct a version of the algebraic transfer in E2-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 ofϕsin 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 alge-bra 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 V s denote an s-dimensionalF2-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 = GL(V ), and the two actions commute For each s≥ 1, Singer [26] constructed anF2-linear map:
ϕs: TorA
s ,s+∗ (F2, F2) → [F2⊗A H∗(BVs)] G L s ,
from the homology of the Steenrod algebra to the elements ofF2⊗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: phchon@ctu.edu.vn; phchon.ctu@gmail.com 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: minhha@vnu.edu.vn
Mathematics Subject Classification (2000): Primary 55P47, 55Q45, 55S10, 55T15.
DOI: 10.1007/s00229-011-0487-0
Trang 2be 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
ϕ s∗: [P H∗(BVs)]G L s → Exts ,s+∗
A (F2, F2), from the coinvariant elements of P H∗(BVs ) to the cohomology of the Steenrod algebra.
Moreover, the “total” dual algebraic transferϕ∗= ⊕ϕ s∗is an algebra homomorphism Cal-culations by Singer [26] for s ≤ 2 and by Boardman [3] for s = 3 showed that ϕ s∗is an isomorphism This shows that the algebraic transfer is highly nontrivial, and an interesting problem is to determine elements of Exts ,s+∗
A (F2, F2) that are detected by the dual algebraic
transfer
In rank 4, it was already known to Singer thatϕ4∗is an isomorphism in a range It is from these preliminary calculation that he conjectured thatϕ∗s is a monomorphism for all
s≥ 1 In [5], Bruner, Hà and Hu’ng showed that the entire family of elements{g i : 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 p iare
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
{p i : 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 Ext5,5+∗
A (F2, F2), namely Ph1, that is not detected by the dual algebraic transfer Another
element, commonly denoted as Ph2is 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, h1Ph1as well as h0Ph2are not in the image ofϕ6∗; h21Ph1is not in the image
ofϕ7∗ 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 E0A, the associated graded of the Steenrod algebra, to E0Ext∗,∗
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 transferϕsin the May spec-tral 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 n0i , 0 ≤ n ≤ 5 and
h n0j, 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
trans-fer in the bar construction and the version of the algebraic transtrans-fer in the E2-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
Trang 32 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 E0A was also reworked in the framework of Priddy’s theory of Koszul
resolutions [24]
2.1 The associated graded algebra E0A
The Steenrod algebra is filtered by powers of its augmentation ideal ¯A by setting: Fp A=
A if p ≥ 0 and F p A = ( ¯A) −p if p < 0 Let E0A = ⊕p ,q E0,q A where E0,q A =
(Fp A /Fp−1A ) 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], E0A is isomorphic
to the universal enveloping algebra of its restricted Lie algebra of primitive elements, which
in this case, are the Milnor generators P k j ([19]) The following result from May’s thesis remains unpublished, but well-known
Theorem 2.1 (May [17]) E0A is a primitively generated Hopf algebra It is isomorphic
to the universal enveloping algebra of the restricted Lie algebra of its primitive elements
{P j
k | j ≥ 0, k ≥ 1} Moreover,
(1)
P i j , P k
= δ i ,k+ P k j + for i ≥ k;
(2) ξ(P j
k ) = 0, where ξ is the restriction map (of its restricted Lie algebra structure).
Here,δi ,k+ is the usual Kronecker delta The set of primitive elements R = {P j
k | j ≥
0, k ≥ 1} is equipped with a natural total order, given by P i
j < P k
if i < k or i = k, j < .
An elementθ ∈ Fp A but p−1A 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 i a i j where r i =j a i j2j is the binary expansion of r i
In fact, May has showed that Sq (R) is the sum of(P i
j ) a i j and terms of strictly greater
weight, so they can be identified in the associated graded E0A In particular, the weight of
P i j is just its subscript j In the language of Priddy’s theory of Koszul resolutions [24], The-orem2.1implies that E0A is a Koszul algebra with Koszul generators {P j
k | j ≥ 0, k ≥ 1}
and quadratic relations:
P i j P k = P k
P i j if i i j P i −
+ P i −
P i j + P i −
j + = 0, P i
j P i j = 0.
The following theorem is first proved in May’s thesis, but see also [24] for a modern treat-ment
Theorem 2.3 (May [17], Priddy [24]) H∗(E0A ) is the homology of the complex R, where
R is a polynomial algebra over F2generated by {R i , j |i ≥ 0, j ≥ 1} of degree 2 i (2 j − 1),
and with the differential is given by
δ(Ri , j ) =
j−1
k=1
R i ,k R i +k, j−k
Cup product in H∗(E0A) correspond to products of representative cycles in R.
Trang 4It 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 i j In fact, ¯X is embedded in the bar construction for E0A by sending γn(P i
j ) to
{P i
j |P i
j | |P i
j } (n factors)
and the product in ¯X corresponds to the shuffle product (see [2], pp 40) Under this embed-ding, 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 coho-mology 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 pM = A · M = M if p ≥ 0, andp Fp M = 0 The associated graded
module E0M=p ,q E0,q M where E0,q M = (F p M/Fp−1M) p +q is a bigraded
mod-ule over the associated graded algebra E0A Let B (A; M) be the usual bar construction with
the induced filtration:
Fp B(A; M) =Fp1 ¯A ⊗ · · · ⊗ F p n ¯A ⊗ F p0M
where the sum is taken over all(n + 1)-tuples (p0, , pn) such that n +n
i=0p i ≤ p.
This filtration respects the differential, and in the resulting spectral sequence, we have
E1,q,t (M) = F p B p +q (A; M) F p−1B 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 d1of the spectral sequence is the connecting homomorphism of the homology
of the following short exact sequence:
0→ F p−1B(A; M)
F p−2B(A; M)→
F p B (A; M)
F p−2B(A; M)→
F p B (A; M)
F p−1B(A; M) → 0.
On the other hand, E1,q,t (M) is isomorphic to B p +q (E0A ; E0M ) −q,q+tasF2-trigraded
vector space Under this identification, d1is exactly the canonical differential of the bar
construction B∗(E0A ; E0M ).
Hence E2,q,t (M) ∼ = H p +q (E0A ; E0M ) −q,q+tand 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 E2-term is E2,q,t (M) =
Hp +q (E0A ; E0M) −q,q+t
The differentials d r : E r
p ,q,t (M) −→ E r
p −r,q+r−1,t (M) are F2-linear maps
Trang 53 The algebraic transfer
Let V s be an s-dimensionalF2-vector space The mod 2 cohomology of BV sis a polynomial
algebra which we will write as P s = F2[x1, , xs ] where x iare in degree 1
The geometric stable transferπ∗(BVs )+ //π∗(S0) admits an algebraic analogue at the E2level of the Adams spectral sequence:
ϕs: TorA
s ,s+t (F2, F2) //TorA
0,t (F2, Ps ) ∼ = (F2⊗A P s )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 r
p ,q,t (F2) //E r p ,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 ˆP1be the unique A-module extension of P1= F2[x1] obtained by formally adding
a generator x−1
1 of degree−1 and requiring that Sq n (x1−1) = x n−1
1 Let u : A → ˆP1be the
uniquely determined A-map that sends an operator θ to θ(x1−1), and let ψ1be its restriction
to the augmentation ideal ¯A Clearly, ψ1maps onto P1 Letψs : ¯A ⊗s → P sbe defined by the following recursive formula:
ψs ({θs | |θ1}) =
|θ
s |>0
θ s(x s−1)θ s (ψs−1({θs−1| |θ1})), (3.1)
where we use the “bar notation” for elements of ¯A ⊗sand standard notation for the coproduct
θ ⊗ θ It is sometimes more convenient to use another form of Eq (3.1):
ψs ({θs | |θ1}) = θ s (x s−1ψs−1({θs−1| |θ1})) + x s−1θs ψs−1({θs−1| |θ1}).(3.2)
In [23], it is shown thatψs is a chain-level representation of the algebraic transfer
Theorem 3.1 (Nam [23], Section 5.1) For each s ≥ 1, ψ s is a chain-level representation
of the algebraic transfer
ϕs: TorA
s ,s+t (F2, F2) → Tor A
0,t (F2, Ps ) ∼ = (F2⊗A P s )t
Singer [26] showed that the image of ϕs actually lies in the G L s-invariant subspace of
F2⊗A P s
We extendψs to a chain homomorphism ˜ψs : B∗(A; F2) // B ∗−s (A; Ps ) between
the bar constructions Let
˜ψ s ({θn | |θ1}) = {θ n | |θ s+1} ⊗ ψ s ({θs | |θ1}). (3.3)
Trang 6Proposition 3.2 The map ˜ ψs is a chain homomorphism.
Proof We have
˜ψ s (∂({θn | |θ1})) =
n−1
i =s+1 {θ n | |θ i+1θi | |θ s+1}ψ s({θs | |θ1})
+
s
j=1
{θ n | |θ s+2}ψ s ({θs+1| |θ j+1θ j | |θ1}).
(3.4)
Since,
ψs ({θs+1| |θ j+1θ j | |θ1})
= θ s+1(x s−1 θ j+1θ j (x−1j θ1(x1−1) ) )
= θ s+1(x s−1 θ j+1(x−1j θ j (θ j−1(x−1j−1 θ1(x1−1) ))) )
+ θ s+1(x s−1 θ j+1(θj (x−1j θ1(x1−1) )) ),
where the action ofθi (x−1y ) is understood as the right hand side of (3.1) Then the second term on the right hand side of (3.4) equals
{θ n | |θ s+2}θ s+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 ˜ψs 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) → Fp B ∗−s (A; Ps ).
Thus, ˜ ψs induces a map between May spectral sequences:
E r ψs : E r
p ,q,t (F2) → E r
p ,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, ˜ψ1: B∗(A; F2) → B∗−1(A; P1) We need to
prove that if
Sq2i1 Sq2ik ∈ F −k+1 B1(A, F2) = F −k+1 ¯A,
then,
Sq2i1 Sq2ik (x1−1) ∈ F −k+1 B0(A, P1) = F −k+1 P1.
But,
Sq2i1 Sq2ik
(x1−1) = Sq2i1
Sq2ik−1 (x2ik−1
1 ), and x2ik−1
1 ∈ F0P1while Sq2i1 Sq2ik−1 ∈ F −k+1 ¯A The assertion follows.
The second statement follows immediately from the former
Trang 73.2 A description of E2ψ s
Proposition3.3allows us to use the May spectral sequence to study properties of the chain-level representationψs of the s-th algebraic transfer We are going to give an explicit formula for the maps in E2page:
E2ψs: TorE0 A
u ,v (F2, F2) → Tor E0 A
u −s,v−s (F2, E0Ps ).
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
0 //P P11 ι1 // ˆP ˆP11 π1 //FF22 //0,
whereι1is the inclusion andπ1is the obvious quotient map Note thatπ1has degree 1
There is a corresponding short exact sequence of E0A-modules:
0 //E E00P P11 E0ι1// E E00ˆP ˆP11 E0π1//FF22 //0,
and this in turn induces a short exact sequence of the bar constructions:
0−→ B∗(E0A ; E0P1) −→ B∗(E0A ; E0ˆP1) −→ B∗(E0A; F2) −→ 0.
Tensoring with E0M, where M is any A-module of finite type and bounded-below, we obtain
0−→ B∗(E0M ; E0P1) −→ B∗(E0M ; E0ˆP1) −→ B∗(E0M; F2) −→ 0.
The connecting homomorphism of this exact sequence is
E2ψ1(M) : Tor E0 A
s ,s+∗ (E0M, F2) → Tor E0 A
s −1,s−1+∗ (E0M, E0P1).
More generally, we obtain
E2(ψ1× P k−1)(M) : Tor E0 A
s ,s+∗ (E0M , E0P k−1) → Tor E0 A
s −1,s−1+∗ (E0M , E0P k ).
Splicing these maps when 0≤ k ≤ s together, we obtain
E2ψs (M) : Tor E0 A
s ,s+∗ (E0M , F2) → Tor E0 A
0,∗ (E0M , E0P s ).
When M= F2, E2ψs = E2ψs (F2) is the E2-level of the algebraic transfer in the May spectral sequence The following is the main theorem of this section
Proposition 3.4 The E2page of the algebraic transfer, E2ψs , is induced by the chain-level map
E1ψs (M): E0M ⊗ (E0A ) ⊗s → E0M ⊗ E0Ps , given inductively as follows.
E1ψs (M)(m{θs | |θ1}) =
|θ |>0
θ s (E1ψs−1(M)(m{θs−1| |θ1}))θ s (x s−1).
Trang 8Proof It is sufficient to prove that E1(ψ1× P s−1)(M) is given by
m {θ s }(x i1
1 x i s−1
|θ s |>0
m ⊗ θ s (x i1
1 x i s−1
s−1)θ s (x s−1).
Indeed, for any cycle x = m{θ s }(x i1
1 x i s−1
s−1) ∈ B(E0M , E0P s−1),
∂(x) = θs (m) ⊗ x i1
1 x i s−1
s−1+ m ⊗ θ s(x i1
1 x i s−1
s−1) = 0.
Then pre-image of x under E0πs is
x = m{θ s }(x i1
1 x i s−1
s−1x s−1).
Therefore,
∂(x ) = θs (m) ⊗ x i1
1 x i s−1
s−1x s−1+ m ⊗ θ s (x i1
1 x i s−1
s−1)x−1s
|θ
s |>0
m ⊗ θ s (x i1
1 x i s−1
s−1)θ s (x s−1)
|θ s |>0
m ⊗ θ s (x i1
1 x i s−1
s−1)θ s (x s−1).
(3.5)
The proof is complete
Example 3.5 For s = 1, the chain level of the transfer, E1ψ1: E0
−p,q A → E0
−p,q−1 P1,
for p ≥ 0, q > p, sends
{P i
p+1} → P i
p+1(x1−1) = x2i (2 p+1−1)−1
Sq2i +p−1
x2i +p−1
−p,q−1 P1 The non-trivial cycles in E0A are {P i
1} ∈ E0
0,2 i , and E1ψ1({P i
1}) = x2i−1
1 ∈ E0
0,∗ P1= P1
Since the A-generators of P1are exactly those of the form x2i−1
1 , i ≥ 0, thus E2ψ1is an isomorphism
Example 3.6 For s= 2, we have
{P i2
j2|P i1
j1} → E1ψ1({P i1
j1})P i2
j2(x2−1) = x2i1 (2 j1 −1)−1
1 x2i2 (2 j2 −1)−1
The nontrivial cycles in(E0A )⊗2are{P i
1|P1j }+{P1j |P i
1}( j > i +1) and {P i
j |P i
j } for j > i
(see [29, page 49]) Their corresponding images in E0P2are
{P i
1|P j
1} + {P j
1|P i
1} → x2i−1
1 x2j−1
2 + x2j−1
1 x2i−1
{P i
j |P i
j } → x2i (2 j −1)−1
1 x2i (2 j −1)−1
By induction we obtain the following result which is extremely useful for computation
in later sections
Corollary 3.7.
E1ψs ({P i s
j s | |P i1
j1}) = x2i1 (2 j1 −1)−1
1 x2is (2 js −1)−1
Proof The results follows immediately from the fact that P t s are primitive in E0A.
Trang 94 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 P sif it is a linear combination of elements in the
images of positive Steenrod squares The quotient of P sby this subspace is exactly the range
of the algebraic transferF2⊗ P s The hit problem asks for a construction of aF2-basis for the space of “nonhit” elementsF2⊗ P s 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 ele-ments 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 P sin homological degree 0, we have the edge homo-morphism
E2,−p,t (Ps ) ∼ = H0(E0A , E0P s )p ,−p+t = (F2⊗E0A (E0P s ))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⊗
Ps ) t Thus, the hit problem for E0Ps, considered as a module over the restricted Lie
alge-bra E0A whose structure is completely known, should be helpful as a first step toward
understanding the general structure ofF2⊗ P s
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 E0P1, 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 ∈ P s We denote by E ( f ), E r ( f ) and [ f ] the corresponding classes of f in E1(Ps ) = E0Ps , 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(x7
1x213x133 + x9
1x112 x313+ x8
1x212x313) = x7
1x132 x313, because x17x213x133 is in filtration−4 while the latter two monomials are in filtrations −5 and−9 respectively
Lemma 4.1 Let f ∈ P s 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 E2( f ) = E2(ess( f )) and thus E∞( f ) = E∞(ess( f )) Suppose on
the contrary that ess( f ) is hit in P s Write ess( f ) =i Sq t i f i It follows that there exists
r > 0 such that E r ( f ) = 0 in E r
Example 4.2 Consider g = x3y5+ x5y3 ∈ P3 in degree 8 Since g = Sq2(x3y3) +
Sq4(x2y2), we have that [g] = 0 in F2⊗A P s On the other hand, E (g) ∈ E0
−1,9 (P2) is nontrivial However, when passing to the E2terms, we have
g = Sq2(x3y3) + x4y4,
Trang 10and x4y4∈ F−4P2is trivial in E −1,90 P2, so E (g) = Sq2E0(x3y3) ∈ E0
−1,3 A · E0
0,6 P2in
E2(Ps) Thus E(g) is trivial in E2(P2).
One may wonder whether the converse is true, that is if f is a homogeneous polynomial such that E ( f ) is trivial in E2(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 = x7
1x213x313∈ P3, it is not difficult to check that m is nonhit in P3 On the other hand,
m = Sq2(x7
1x211x313) + x9
1x211x313+ x8
1x212x313+ x7
1x212x143 + x8
1x112 x314, where x19x211x313∈ F−5P3and the last three monomials are in lower filtrations Therefore,
E (m) = Sq2E (x7
1x211x313) ∈ E0
−4,37 P3.
So E (m) = 0 in E2(Ps ).
The point of Example4.3is 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
polyno-mial is hit (that is, it is trivial inF2⊗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 = x1x22x32+ x2
1x2x32+ x2
1x22x3 = Sq2(x1x2x3), so n is hit in
P3 It can be checked by direct inspection that E (n) is nontrivial in E2(P3) because it is
E0A-nonhit in E −2,70 (P3) The element n does not survive to E3because the above formula
shows that n = d2(n ), where n = Sq2⊗ (x1x2x3) ∈ E2
0,1,5 (P3).
The following is the main result of this section
Proposition 4.5 Let f ∈ P s 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 P s and there does not exist any non-hit polynomial g ∈ F q P s , 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 existθr −1,i ∈ ¯A and f r −1,i ∈ P ssuch that
E r−1(ess( f )) =
i
E r−1(θr −1,i f r −1,i ) ∈ E r−1(Ps ).
In the E r−2-term, we have
E r−2(ess( f )) =
i
E r−2(θr −1,i f r −1,i ) + f r −2∈ E r−2(Ps),
where f
r−2is in the image of d r−2 Thus, there existθr −2,i ∈ ¯A and f r −2,i ∈ P ssuch that
f
i
E r−2(θr −2,i f r −2,i ) ∈ E r−2(Ps ),
and we can write
E r−2(ess( f )) =
i ,r−2≤k≤r−1
E r−2(θki f ki ) ∈ E r−2(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... (F2) is the E2-level of the algebraic transfer in the May spectral sequence The following is the main theorem of this section
Proposition 3.4 The E2page... description of E2ψ s
Proposition3.3allows us to use the May spectral sequence to study properties of the chain-level representationψs of the s-th algebraic