On May spectral sequence and the algebraic transfer II

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 compatible. We also prove that the two Sq0 family ni ∈ Ext5,36·2 i A (Z2, Z2), i ≥ 0, and ki ∈ Ext7,36·2 i A (Z2, Z2), i ≥ 1, are in the image of the algebraic transfer.

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

compatible We also prove that the two Sq0-family n i∈ Ext5,36·2 i

A (Z/2, Z/2), i ≥ 0, and k i∈ Ext7,36·2 A i (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∗(BV s)]GL s (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 V s of rank s, for

degree 1, is both a module over the mod 2 Steenrod algebra as well as the general linear

group GL s = GL(V s) Moreover, these two module structures are compatible, so that one has

an induced action of GL s on the space Z/2 ⊗ A H∗(BV s) It is sometime more convenient to

consider the dual of (1.1) At rank s, it has the form

ϕ∗s : [P AH∗(BV s)]GL s→ Exts,s+∗ A (Z/2, Z/2), (1.2)

where P A H∗(BV s ) denote the subspace of the divided power algebra H∗(BV s) consisting of all

elements that are annihilated by all positive degree Steenrod squares, and M Gis 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 ϕ s 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

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 Ext∗,∗ A (Z/2, Z/2) groups, there is an action of the (big) Steenrod algebra (see Liulevicius [14]

or May [17]), where the operation Sq0is no longer the identity map In his thesis [12], Kameko

constructed an operation

Sq0: [P AHd (BV s)]GL s // [P AH 2d+s (BV s)]GL s ,

that corresponds to the operation Sq0on 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 E0A In section 4, we showed that

under the representation of the algebraic transfer in the E2terms 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

Theorem 1.1 (see also Corollary 5.3) The family {n i∈ Ext5,36·2 A i (Z/2, Z/2) : i ≥ 0} is

detected by the algebraic transfer (1.2)

Bruner [3] has shown that the relation k1= h2h5n0holds in Ext7,∗ A (Z/2, Z/2) Since it is well-known that the total transfer ϕ∗= ⊕s≥1 ϕ∗sis an algebra homomorphism (see Singer [26]), we

obtain an immediate corollary

Corollary 1.2 The family {k i∈ Ext7,36·2 i

A (Z/2, Z/2) : i ≥ 1} is in the image of the

seventh algebraic transfer

We do not know whether k0∈ Ext5,36

A (Z/2, Z/2) also belongs to the image of ϕ∗5 or not 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

of the main results of this paper that the two families n i , i ≥ 0 and k j , 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 E0A 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 E0A

The Steenrod algebra is filtered by powers of its augmentation ideal ¯A by setting FpA = A if

p ≥ 0 and FpA = ( ¯ A) ⊗−p if p < 0 Let E0A = ⊕p,qE0

p,q A, where E0

the associated graded algebra According to a well-known theorem of Milnor and Moore [19],

algebra of its restricted Lie algebra of its primitive elements In this case, the primitives are

the Milnor generators P j i (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 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 k j |j ≥ 0, k ≥ 1} Moreover,

(i) P i

j , P ` k  = δ i,k+`P j+` k for i ≥ k;

(ii) ξ(P k j ) = 0, where ξ is the restriction map (of its restricted Lie algebra structure).

Here, δ i,k+` is the usual Kronecker delta An element θ ∈ F p A but θ 6∈ F p−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) =P

binary expansion of m.

In particular, the weight of P i

j is just the subscript j In fact, May’s argument identifies

j)a ij in the associated graded, where r i=P a ij2j is the binary

expansion of r i In the language of Priddy’s theory of Koszul resolution [24], Theorem 2.1

states that E0A is a Koszul algebra with Koszul generators {P k j |j ≥ 0, k ≥ 1} and quadratic

relations:

P j i P ` k = P ` k P j i if i 6= k + `, P j i P ` i−` + P ` i−` P j i + P j+` i−` = 0, P j i P j i = 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 E0A, H∗(E0A), is isomorphic

to the homology of the complexR, where R is the polynomial algebra over Z/2 generated by

{R i

j |i ≥ 0, j ≥ 1} each of degree 2 i(2j− 1) and with the differential is given by

δ(R i j) =

j−1

X

k=1

R i k R i+k j−k

Moreover, cup products in H∗(E0A) correspond to products of representative cycles inR

We will need a more general version of the above theorem for the cohomology of E0A 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 R i

j of

length s If M is a right E0A-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

s,t

RR s t ⊗ mP s

for all R ∈Rs and all m ∈ M According to Priddy [24, Section 4], there exists a natural

isomorphism:

Θ : H s,t(R ⊗ M) // Exts,t

2.2 The May spectral sequence

We will be working with the cohomology version of the May spectral sequence Let A∗ be

the dual of A and let ¯ A∗= ( ¯A)∗ Then A∗ admits a filtration where F p A∗= 0 if p ≥ 0 and

F p A∗= ( ¯A/F p−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

p < 0, we have F p M∗⊆ F p−1 M∗ Thus (E0M )∗ ∼= E0M∗= ⊕p,q E p,q0 M∗, where E p,q0 M∗=

(F p M∗/F p+1 M∗)p+q , is a bigraded comodule over the associated graded coalgebra E0A∗ Let

¯

C(A; M ) be the cobar construction with the induced filtration:

F p C¯n (A∗; M∗) =XF p1A¯∗⊗ · · · ⊗ F p n A¯∗⊗ F p0M∗,

where the sum is taken over all sequences {p0, , pn } such that n +Pn

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:

0 →F

p+1 C(A; M )¯

p C(A; M )¯

p C(A; M )¯

F p+1 C(A; M )¯ → 0.

On the other hand, E p,q,t1 (M∗) is isomorphic to ¯C p+q (E0A; E0M ) −q,q+t as Z/2-trigraded vector space Under this identification, δ1is exactly the canonical differential of the cobar construction

¯

C∗(E0A; E0M ) Hence E2p,q,t (M ) ∼ = H p+q (E0A∗; E0M∗)−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 (E r, δr ) converging to E0H∗(A; M∗) and

having as its E2-term E2p,q,t (M ) = H p+q (E0A; (E0M )∗)−q,q+t Each δ r is a homomorphism

δ r : E r p,q,t (M ) −→ E r p+r,q−r+1,t (M ).

When M = Z/2, we will write E r for E r (M ) It is also well-known that E r (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 thatR 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 E2-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 E2ψ s in co-Koszul complex of E0H∗(BV s),

which will be denoted by E1ψ s

We begin with some notations For an s-dimensional Z/2-vector space V s, it is well-known

that H∗(BV s ) is isomorphic to the polynomial algebra P s = Z/2[x1, , xs], where each

generator x i is of degree 1 Dually, H∗(BV s ) is the divided power algebra H s = Γ(a1, , as),

where a i is the dual of x i For simplicity, we will write (i1, , i s ) for the monomial a (i1)1 a (i s)

s Let ˆP1 be the unique A-module extension of P1 by formally adding a generator x−11 of degree

-1 and require that Sq n (x−11 ) = x n−11 and let ˆH1 be the dual of ˆP1 There is a fundamental

short exact sequence of A-modules:

0 → Σ−1Z/2 → Hˆ1→ H1→ 0, passing to the associated graded, we have a similar short exact sequence of E0A-modules and

after tensoring this sequence with R ⊗ M for some right E0A-module M , we have a short

exact sequence of differential modules

R ⊗ M ⊗ Σ−1

Z/2 // R ⊗ M ⊗ E0Hˆ1 // R ⊗ M ⊗ E0H1.

Using the isomorphism (2.2), the connecting homomorphism of this short exact sequence can

be identified with

Exts−1,t E0 A (Z/2, M ⊗ E0H1) // Exts,t+1

E0 A (Z/2, M ).

Since there is a canonical isomorphism E0Hs∼= (E0H1)⊗s , we can construct and compose s

similar connecting homomorphisms so as to obtain a map

Extk,t E0 A (Z/2, M ⊗ E0Hs) // Extk+s,t+s

E0 A (Z/2, M ).

In particular, when M = Z/2 and k = 0, we obtains the E2-level of the algebraic transfer

E2ψ s: Ext0,t E0 A (Z/2, E0H s) // Exts,t+s

E0 A (Z/2, Z/2).

As we have noted, this map is induced by a chain level map

E1ψs : E0Hs // Rs.

We can describe this map explicitly

Proposition 3.1 The version of the algebraic transfer in E2-term of May spectral sequence is induced by the map

E1ψs : E0H∗(BV s) −→Rs,

given by

E1ϕs (a (n1 )

1 a (n s)



R k1

t1 R k s

t s , ni= 2k i(2t i − 1) − 1, 1 ≤ i ≤ s,

Proof Suppose R ⊗ m ⊗ a (n1)1 a (n s)

s is a nontrivial summand of a cycle x ∈ R ⊗ M ⊗ H s

It can be pulled back to the same element inR ⊗ M ⊗ H s−1⊗ ˆH1 Since δ(x) = 0, it comes

fromR ⊗ M ⊗ H s−1⊗ Σ−1Z/2 On the other hand, we have that a (n) P i

j = a(−1) if and only

if n = 2 i(2j− 1) − 1 Thus from the formula (2.1), we see that the connecting homomorphism

sends R ⊗ m ⊗ a (n1 )

1 a (n s)

s to zero if n s does not have the form 2i(2j− 1) − 1 for some

i ≥ 0, j ≥ 1; and to RR i j ⊗ m ⊗ a (n1 )

1 a (n s−1)

s−1 if n s= 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) = R0⊗ (1, 3, 3) = δ1(2, 3, 3) ∈ E2−1,2,8 (P3), so x is a cycle in

E −2,2,82 (P3) For r ≥ 3, E r −2+r,∗,∗ = 0, so δ r (x) = 0 for all r ≥ 3; therefore, x is a permanent

cycle

Using (3.1), we obtain

E1ψ3(x) = R11R11R30+ R11R02R12= R11(R02R21+ R03R11), this latter element is called h1h0(1) in the E2terms of the May spectral sequence (see Tangora

[29, Appendix 1]), and is a representation of c0in the 8-stem

Example 3.3 We see that the element ¯d0, 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, ¯d0 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, ¯d0 is a permanent cycle because δ r, r ≥ 5, is trivial Again from (3.1), we obtain

E1ψ4(X) = (R0R1+ R0R1)2, which is a representation of d0 in the E1-term of May spectral sequence Since ¯d0is a permanent cycle, it is a representation of the pre-image of d0under the algebraic transfer in the MSS

We end this section with two simple properties of the maps E rψs First of all, sinceR is a

commutative algebra, it is clear that E1ψs factors through the coinvariant ring [P E0AE0Hs]Σs The reader who is familiar with the algebraic transfer may wonder about the action of the

full general linear group GL s Unfortunately, in general this action does not preserve the May

filtration For example, if f = x2x5∈ F−2P2 and σ ∈ GL2, such that σ(x1) = x1+ x2 and

σ(x2) = x2, then we have σ(f ) = x2x5+ x7∈ F0P2

Secondly, the direct sum ⊕s≥1H∗(BV s) 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

E r ψ = ⊕ s≥1 E r ψ s: ⊕s E r ∗,∗ (P s) // E ∗,∗

r ,

is an algebra homomorphism

4 The squaring operations

In [21], Nakamura constructed a squaring operation on the MSS for the trivial module:

Sq0: E r p.q // E p,q

which is multiplicative in the E1page and therefore satisfies the Cartan formulas in higher E r

page (when elements are suitably represented in the E2term) The purpose of this section is to

introduce a similar squaring operation, defined for any r, s ≥ 1, which is also denoted as Sq0:

Sq0: E r p,q (P s) // E p,q

r (P s ),

so that it commutes with Nakamura’s Sq0via the map of spectral sequences E p,q

r (P s) // E p,q+s

r

constructed in the previous section

We begin with a description of Nakamura squaring operation in the complexR of (2.1), this

is reminiscent to the construction of Sq0 in Ext∗,∗ A (Z/2, Z/2) from an endomorphism of the

lambda algebra as in Tangora [29] Define an algebra map θ :R // R by setting θ(R i

j ) = R j i+1

By direct inspection, we see that θ commutes with the coboundary map δ, and thus induces

an endormorphism on Ext∗,∗ E0 A (Z/2, Z/2).

Proposition 4.1 The endomorphism θ induces Nakamura’s squaring operation

Sq0: E r p,q // E p,q

r

Proof According to Priddy [24], R i

j is represented in the cobar resolution by [ξ2i

j ] On the

other hand, in the cobar complex for E0A, the squaring operation has an explicit form

Sq0[α1| |α n ] = [α21| |α2

n ],

so it maps [ξ2j i ] to [ξ j2i+1] and the result follows immediately

On H∗(BV s), 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

a (t1)1 a (t s)

s 7→ a (2t1+1)1 a (2t s+1)

One quickly verifies that this endomorphism of H∗(BV s) commutes with the action of the

Steenrod algebra, in the sense that for all a ∈ H∗(BV s),

(θa)P t s = θ(aP t s−1 ) if s > 0, and (θa)P t0= 0.

Moreover, Vakil [30] observed that the map a (n) 7→ 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

R ⊗ H∗(BV s ), which is again denoted as Sq0, by setting

Sq0: R ⊗ a 7→ Sq0(R) ⊗ θ(a), for all R ∈ R, a ∈ H∗(BV s)

Lemma 4.2 The endormorphism Sq0onR ⊗ H∗(BV s ) commutes with the coboundary δ

of (2.1)

Proof We already know that Sq0 and δ commutes on R Also, (θa)R0

t = 0 and (θa)R s t=

θ(aR s−1 t ), we have

δSq0(R ⊗ a) =δ(Sq0R ⊗ θa)

t

=Sq0δR ⊗ θa +XSq0(RR s−1 t ) ⊗ θ(aR s−1 t )

=Sq0(δR ⊗ a +XRR s−1 t ⊗ aR s−1 t ) = Sq0δ(R ⊗ a).

The proof is complete

It follows that there exists an induced endomorphism Sq0 on E p,q

r (P s ) for all s, r ≥ 1 Our next result shows that this endomorphism commutes with Nakamura’s Sq0via the MSS transfer

Erψs, thus justifies for our choice of notation

Proposition 4.3 There exists a commutative diagram of maps between spectral sequences:

E p,q

E r ϕ s //

E r p,q (P s)

E p,q

r (P s)

Sq0

E r p,q (P s) E r ϕ s // E E r p,q+s p,q+s

r

E p,q+s

Sq0

Proof It sufices to show that there exists a commutative diagram at E1 page

E0H∗(BV s) E1ϕ s //Rs.

E0H∗(BV s)

E0H∗(BV s)

θ

E1ϕ s

// Rs

Rs.

Sq0

This can be verified directly from the formula (3.1) Note that if n i= 2k i(2t i− 1) − 1 then

2n i+ 1 = 2k i+1(2t i− 1) − 1

In particular, we have an induced map

θ : P E0A E0H d (BV s ) → P E0A E0H 2d+s (BV s ),

that fits in the following

Proposition 4.4 The representation of Kameko’s squaring operations, Sq0, and

Naka-mura’s squaring operations commute with each other through E ϕs In other words, the

following diagram commutes

P E0 A E0Ht (BV s)

θ

E2ϕ s // Exts,s+t

E0 A (Z/2, Z/2)

Sq0

P E0 A E0H 2t+s (BV s) E2ϕ s// Exts,s+t

E0 A (Z/2, Z/2).

Proof The assertion is implied directly from the formula of Sq0 and (3.1)

The operation θ commutes with the action of the symmetric group Σ s on E0H∗(BV s)

as well as its subspace P E0A E0H∗(BV s) This is essentially direct from the definition

Furthermore, E2ψsis Σs-equivariant sinceR is commutative Thus in the commutative diagram

of Proposition 4.4, we can replace P E0A E0H∗(BV s) by its Σs -coinvariant (P E0A E0H∗(BV s))Σs Our last result can be considered as an analogue to N H V Hưng’s analysis of the squaring

map on the space (P AHs)GL s [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 : (PE0AE0H∗(BV s))Σs // (P E0AE0H∗(BV s))Σs ,

is a monomorphism

Proof Suppose x ∈ P E0A E0Hs such that θ(x) = 0 in (P E0A E0H∗(BV s))Σs This means

that there exist z σ ∈ (P E0A E0H∗(BV s))Σs such that

σ∈Σ s zσσ + zσ.

We say that a monomial a i1

1 a i s

s is odd if all exponents i t are odd Otherwise, we say that

it is non-odd The left hand side of the above equation contains all odd monomials Each z σ can be written as the sum z σ0 + z σ00 where z σ0 consists of all non-trivial odd monomials in z σ

We first claim that both z σ0 and z σ00are E0A-annihilated.

Lemma 4.6 If x = y + z ∈ P E0A E0Hs where y is the sum of odd monomials summands

of x, then y and z belongs to P E0A E0Hs

Proof First of all, note that E0A is multiplicatively generated by P1s , s ≥ 0, so in order

to prove that y is E0A-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= ySq1= 0 If s > 1, then since P1sis a derivative,

and |P s| = 2s is even, we see that yP s , if non-zero, consists of only odd monomials while zP s consists of only non-odd monomials Because xP s = 0, we must have yP s = zP s= 0 for all

s > 0 The lemma is proved.

We now continue the proof of Proposition 4.5 We have a decomposition in P E0AE0Hs

θx = X(z σ0σ + z σ0) + X(z σ00σ + z σ00).

The second summand must vanish since it contains non-odd monomials The first summand

can be written as θx0 for some x0 of the form

x0 =X(y σ σ + y σ ), where y σ is such that θy σ = z σ0 Since θ is obviously a monomorphism on P E0A E0H s, it follows

that x = x0 and so x is trivial in (P E0A E0H s)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) ∈ P AH3where 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 : (P A H s)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 n i , 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 E r for some r ≥ 2 if its projection to a non-zero element in the E r A permanent cycle is an element killed by δ r for all r First of all, we need

a technical lemma

Lemma 5.1 If ¯x ∈ E p,−p,t1 (P s ) is a permanent cycle such that E1ψ s(¯x) represents an

element x ∈ Ext s,s+t A (Z/2, Z/2) in the E1-term of the May spectral sequence, then x is in

the image of the algebraic transfer

Proof Since E1ψs(¯x) is a representation of x in the E1-term of the May spectral sequence and ¯x survives to E∞∗,∗,∗ (P s ), 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

A (Z/2, Z/2) is in the image of the algebraic transfer.

The fact that n0 and n1= Sq0n0 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 Sq0-family {n i, i ≥ 0} starting with n0are indecomposable in Ext5,∗ A (Z/2, Z/2) Since Kameko’s squaring operation and the classical squaring operation Sq0 commute with each other through the algebraic transfer, we have the following immediate corollary

Corollary 5.3 The family of indecomposable elements n i∈ Ext5,36·2 A i (Z/2, Z/2), i ≥ 0,

are in the image of the algebraic transfer