THE WEAK CONJECTURE ON SPHERICAL CLASSES
NGUY^E~N H. V. HU_.NG
Abstract.Let A be the mod 2 Steenrod algebra. We construct a chain-level
representation of the dual of Singer's algebraic transfer, Tr*k: TorAk(F*
*2, F2) !
F2 F2[x1, ..., xk], which maps Singer's invariant-theoretic model of th*
*e dual
A
of the Lambda algebra, ^k, to F2[x11, ..., xk1] and is the inclusion of*
* the
Dickson algebra, Dk ^k, into F2[x1, ..., xk]. This chain-level repres*
*entation
allows us to confirm the weak conjecture on spherical classes (see [9]),*
* assuming
the truth of (1) either the conjecture that the Dickson invariants of at*
* least
k = 3 variables are homologically zero in TorAk(F2, F2), (2) or a conjec*
*ture
on A-decomposability of the Dickson algebra in ^k. We prove the conject*
*ure
in item (1) for k = 3 and also show a weak form of the conjecture in ite*
*m (2).
1.Introduction and statement of results
This paper continues our study of spherical classes that started in [9]. To m*
*ake
the paper self-contained, we first recall certain background of the classical c*
*onjec-
ture on spherical classes, which has been given in the introduction of our paper
[9].
We are interested in the following conjecture on spherical classes in Q0S0, i*
*.e.
elements belonging to the image of the Hurewicz homomorphism
H : ßs*(S0) ~=ß*(Q0S0) ! H*(Q0S0) .
Here and throughout the paper, the coefficient ring for homology and cohomology
is always F2, the field of two elements.
Conjecture 1.1. (conjecture on spherical classes). There are no spherical class*
*es
in Q0S0, except the Hopf invariant one and the Kervaire invariant one elements.
Some topologists believe that the conjecture is due to I. Madsen, while some
others say it is due to E. Curtis. (See Curtis [6] and Wellington [24] for a di*
*scussion.)
Let Ek be an elementary abelian 2-group of rank k. It is also viewed as a k-
dimensional vector space over F2. So, the general linear group GLk = GL(k, F2)
acts on Ek and therefore on H*(BEk) in the usual way. Let Dk be the Dickson
algebra of k variables, i.e. the algebra of invariants
Dk := H*(BEk)GLk ~=F2[x1, . .,.xk]GLk ,
where Pk = F2[x1, . .,.xk] is the polynomial algebra on k generators x1, . .,.x*
*k,
each of dimension 1. As the action of the (mod 2) Steenrod algebra, A, and that
of GLk on Pk commute with each other, Dk is an algebra over A.
____________
1This work was supported in part by the National Research Program, N01.4.2.
21991 Mathematics Subject Classification. Primary 55P47, 55Q45, 55S10, 55T15.
3Key words and phrases. Spherical classes, Loop spaces, Adams spectral seque*
*nces, Steenrod
algebra, Invariant theory, Dickson algebra, Algebraic transfer.
1
2 NGUY^E~N H. V. HU_.NG
One way to attack Conjecture 1.1 is to study the Lannes-Zarati homomorphism
'k : Extk,k+iA(F2, F2) ! (F2 Dk)*i,
A
which is compatible with the Hurewicz homomorphism (see [13], [14, p. 46]). The
domain of 'k is the E2-term of the Adams spectral sequence converging to ßs*(S0*
*) ~=
ß*(Q0S0). According to Madsen's theorem [16], which asserts that Dk is dual to
the coalgebra of Dyer-Lashof operations of length k, the range of 'k is a submo*
*dule
of H*(Q0S0). By compatibility of 'k and the Hurewicz homomorphism we mean
'k is a "liftingö f the latter from the "E1 -level" to the "E2-level".
The Hopf invariant one and the Kervaire invariant one elements are respective*
*ly
represented by certain permanent cycles in Ext1,*A(F2, F2) and Ext2,*A(F2, F2),*
* on
which '1 and '2 are non-zero (see Adams [1], Browder [5], Lannes-Zarati [14]).
Therefore, Conjecture 1.1 is a consequence of the following one.
Conjecture 1.2. 'k = 0 in any positive stem i for k > 2.
It is well known that the Ext group has intensively been studied, but remains
very mysterious. In order to avoid the shortage of our knowledge of the Ext gro*
*up,
we want to restrict 'k to a certain subgroup of the Ext group which (1) is large
enough and worthwhile to pursue and (2) could be handled more easily than the
Ext group itself. To this end, we combine the above data with Singer's algebraic
transfer.
Singer defined in [22] the algebraic transfer
T rk : F2 P Hi(BEk) ! Extk,k+iA(F2, F2) ,
GLk
where P H*(BEk) denotes the submodule consisting of all A-annihilated elements
in H*(BEk). It is shown to be an isomorphism for k 2 by Singer [22] and for
k = 3 by Boardman [3]. Singer also proved in [22] that it is not an isomorphism
for k = 5, and conjectured that T rk is a monomorphism for any k.
Restricting 'k to the image of T rk, we stated in [9] the following conjectur*
*e.
Conjecture 1.3. (weak conjecture on spherical classes).
'k . T rk : F2 P H*(BEk) ! P (F2 H*(BEk)) := (F2 Dk)*
GLk GLk A
is zero in positive dimensions for k > 2.
In other words, there are no spherical classes in Q0S0, which can be detected
by the algebraic transfer, except the Hopf invariant one and the Kervaire invar*
*iant
one elements.
In [9], we have proved that the inclusion of Dk into Pk is a chain-level repr*
*esen-
tation of T r*k. '*k: F2 Dk ! (F2 Pk)GLk. So, we get the following result.
A A
Theorem 1.4. The weak conjecture on spherical classes is equivalent to the con-
jecture that the homomorphism
jk : F2 (PkGLk) ! (F2 Pk)GLk
A A
induced by the identity map on Pk is zero in positive dimensions for k > 2.
Let D+k, A+ be respectively the submodules of Dk and A consisting of all ele-
ments of positive dimensions. Then the above conjecture on jk can equivalently *
*be
stated as follows.
THE WEAK CONJECTURE ON SPHERICAL CLASSES 3
Conjecture 1.5. If k > 2, then
D+k A+ . Pk.
In [9], we have got a proof of this conjecture for k = 3.
To introduce a new approach, we need to summarize Singer's invariant-theoretic
description of the lambda algebra [21]. According to Dickson [7], one has
Dk ~=F2[Qk,k-1, ..., Qk,0],
where Qk,idenotes the Dickson invariant of dimension 2k - 2i. Singer set k =
Dk[Q-1k,0], the localization of Dk given by inverting Qk,0, and defined ^kto b*
*e a
certain ön t too large" submodule of k. He also equiped ^ = k ^kwith a
differential @ : ^k! ^k-1and a coproduct. Then, he showed that the differenti*
*al
coalgebra ^ is dual to the lambda algebra of the six authors of [4]. Thus, Hk(*
* ^) ~=
T orAk(F2, F2). (Originally, Singer used the notation +kto denote ^k. However*
*, by
D+k, A+ we always mean the submodules of Dk and A respectively consisting of all
elements of positive dimensions, so Singer's notation +kwould make a confusion
in this paper. Therefore, we prefer the notation ^kto +k.)
One of the main results of this paper is to construct a homomorphism Tk : ^k!
F2[x11, ..., xk1] with the following properties.
Theorem 3.2 The homomorphism Tk : ^k! F2[x11, ..., xk1] maps the submodule
of all cycles in ^kto Pk and is a chain-level representation of T r*k: T orAk(*
*F2, F2) !
F2 Pk. Moreover, its restriction to Dk ^kis the inclusion of Dk into Pk.
A
Note that every q 2 D+kis a cycle in the differential module ^k. By means of
this theorem, the weak conjecture on spherical classes is an immediate conseque*
*nce
of the following conjecture.
Conjecture 1.6. If q 2 D+k, then [q] = 0 in T orAk(F2, F2) for k > 2.
This is a corollary of the following conjecture. (See Proposition 4.2 for a p*
*roof.)
Conjecture 4.1 Let Ker@k be the submodule of all cycles in ^k. Then, for k > 2,
D+k A+ . Ker@k.
We get the following result, which is weaker than the above conjecture.
Theorem 4.3 For k > 2,
D+k A+ . ^k.
We also have the following theorem.
Theorem 4.8 Conjecture 1.6 is true for k = 3.
Conjecture 1.5 is related to the difficult problem of determination of F2 Pk.
A
This problem has first been studied by F. Peterson [18], R. Wood [26], W. Singe*
*r [22],
S. Priddy [19]... who show its relationships to several classical problems in H*
*omo-
topy Theory. F2 Pk has explicitly been computed for k 3. The cases k = 1 and
A
4 NGUY^E~N H. V. HU_.NG
2 are not difficult, while the case k = 3 is very complicated and was solved by*
* M.
Kameko [12]. There is also another approach, the qualitative one, to the proble*
*m.
By this we mean giving conditions on elements of Pk to show that they go to zero
in F2 Pk, i. e. belong to A+ . Pk. Peterson's conjecture, which was established
A
by Wood [26], claims that F2 Pk = 0 in dimension d such that ff(d + k) > k.
A
Here ff(n) denotes the number of ones in the dyadic expansion of n. Recently, W.
Singer, K. Monks, J. Silverman... have refined the method of R. Wood to show
that many more monomials in Pk are in A+ . Pk. (See Silverman [20] and its refe*
*r-
ences.) Conjecture 1.5 presents a large family, whose elements are predicted to*
* be
in A+ . Pk.
The paper is organized as follows. Section 2 is a preliminary on invariant th*
*eory
and Singer's algebraic transfer. Sections 3 and 4 are respectively devoted to p*
*rove
Theorems 3.2, 4.3 and 4.8. Finally, in Section 5, we propose a way to prove the
classical conjecture on spherical classes.
The author would like to thank the referee for helpful suggestions, which have
improved the exposition of the paper.
2. Preliminary
This section begins with a few words on invariant theory and ends with a brief
sketch of Singer's definition of the algebraic transfer.
Let Tk be the Sylow 2-subgroup of GLk consisting of all upper triangular k x *
*k-
matrices with 1 on the main diagonal. The Tk-invariant ring, Mk = PkTk, is call*
*ed
the M`ui algebra. In [17], M`ui showed that
PkTk= F2[V1, ..., Vk],
where Y
Vi= (~1x1 + . .+.~i-1xi-1+ xi).
~j2F2
Then, the Dickson invariant Qk,ican inductively be defined by
Qk,i= Q2k-1,i-1+ Vk . Qk-1,i,
where, by convention, Qk,k= 1 and Qk,i= 0 for i < 0.
Let S(k) Pk be the multiplicative subset generated by all the non-zero line*
*ar
forms in Pk. Let k be the localization: k = (Pk)S(k). Using the results of
Dickson [7] and M`ui [17], Singer noted in [21] that
k := ( k)Tk = F2[V1 1, ..., Vk 1],
k := ( k)GLk = F2[Qk,k-1, ..., Qk,1, Qk1,0].
Further, he set
v1 = V1, vk = Vk=V1. .V.k-1 (k 2),
so that k-2 k-3
Vk = v21 v22 . .v.k-1vk (k 2).
Then, he obtained
k = F2[v11, ..., vk1],
with dimvi= 1 for every i.
Singer defined ^kto be the submodule of k = Dk[Q-1k,0] spanned by all mono-
mials fl = Qik-1k,k-1.Q.i.0k,0with ik-1, ..., i1 0, i0 2 Z, and i0 + dimfl *
*0.
THE WEAK CONJECTURE ON SPHERICAL CLASSES 5
In the remaining part of this section, we briefly sketch Singer's definition *
*of T r*k.
Let P1 = F2[x] with |x| = 1. Let ^P F2[x, x-1] be the submodule spanned
by all powers xi with i -1. The canonical A-action on P1 is extended to an
A-action on F2[x, x-1] (see Adams [2], Wilkerson [25]). P^ is an A-submodule of
F2[x, x-1]. We have a short-exact sequence of A-modules
2.1. 0 ! P1 '!P^!ß F2 ! 0 ,
where ' is the inclusion and ß is given by ß(xi) = 0 if i 6= -1 and ß(x-1) = 1.*
* Let
e1 be the corresponding element in Ext1A(F2, P1). Then, Singer defined
2.2. ek := e1 . . .e1 2 ExtkA(F2, Pk) (k times).
Now, T r*k: T orAk(F2, F2) ! T orA0(F2, Pk) F2 Pk is defined by
A
2.3. T r*k(z) := ek \ z .
Note that, since ß raises internal degree by one, T r*klowers it by k.
We need to relate T r*kwith connecting homomorphisms. Let N, P, Q, R be (left)
A-modules. From MacLane [15, p. 229], one has
f g = (f R) O (N g)
for f 2 Ext*A(N, P ), g 2 Ext*A(Q, R). Hence
2.4. ek = (e1 Pk-1) O . .O.(e1 P1) O e1 .
Cap and Yoneda products are related by the formula
(h O f) \ z = h \ (f \ z)
for z 2 T orA*(M, N), f 2 Ext*A(N, P ), h 2 Ext*A(P, Q).
Suppose f 2 Ext1A(N3, N1) is represented by the short-exact sequence of left
A-modules 0 ! N1 ! N2 ! N3 ! 0. Let (f) : T orAs(M, N3) ! T orAs-1(M, N1)
be the connecting homomorphism associated with this short-exact sequence, for
any right A-module M. Then one verifies easily
(f)(z) = f \ z
for any z 2 T orAs(M, N3).
Consequently, we get
2.5. T r*k= (e1 Pk-1) O . .O. (e1 P1) O e1 .
(See Singer [22, p. 498].)
Now we prepare to construct a chain-level representation of T r*k.
For every left A-module N, Singer defined in [21] a chain complex ^N: It is
given in homological dimension k by ( ^N)k = ^k N as an F2-vector space and
its differential @ : ( ^N)k ! ( ^N)k-1 is defined by
@(va11. .v.akk y) := (va11. .v.ak-1k-1 Sqak+1(y)) ,
for y 2 N. Then he proved that
Hk( ^N) ~=T orAk(F2, N).
This isomorphism is natural in N. Further, if (f) : 0 ! N1 ! N2 ! N3 ! 0 is any
short-exact sequence of left A-modules, then 0 ! ^N1 ! ^N2 ! ^N3 ! 0 is
a short-exact sequence of chain complexes, from which a chain-level representat*
*ion
of the connecting homomorphism (f) : T orAk(F2, N3) ! T orAk-1(F2, N1) can be
computed.
6 NGUY^E~N H. V. HU_.NG
3.A chain-level representation of the dual of the algebraic
transfer
Following Singer [21], the homomorphism
@k : F2[v11, ..., vk1] ! F2[v11, ..., vk1-1]
æ a1 ak-1
@k(va11. .v.akk) := v10. .v.k-1 ifoakt=h-1,erwise
maps ^kto ^k-1. Moreover, it is a differential on ^ = k ^kwith H*( ^) ~=
T orA*(F2, F2). This is an isomorphism of bigraded modules. Here the bidegree of
^ is given by X
bideg(va11. .v.akk) := (k, k + ai).
Note that, for every q 2 Dk, its expansion
X
q = va11. .v.akk
(a1,...,ak)
always has ak 0 in any term of the sum. Therefore, @k(q) = 0. This means
that every Dickson invariant is a cycle in the complex ^, whose homology is
T orA*(F2, F2).
Now we construct a homomorphism, denoted Tk, as follows.
Definition 3.1. The homomorphism Tk : F2[v11, ..., vk1] ! F2[x11, ..., xk1] is
defined by
i j
Tk(va11. .v.akk) := Sqa1+1 x-11. .S.qak-1+1(x-1k-1Sqak+1(x-1k)) . . .,
where a1, ..., ak are arbitrary integers. Here, we mean Sqi= 0 for i < 0.
Let Ker@k be the submodule of all cycles in ^k. Then, the goal of this secti*
*on
is to prove the following theorem.
Theorem 3.2. The homomorphism Tk satisfies the following two properties:
(i)Tk(Ker@k) Pk. Furthermore, Tk| ^k: ^k! F2[x11, ..., xk1] is a chain-
level representation of
T r*k: Hk( ^) ~=T orAk(F2, F2) ! F2 Pk .
A
(ii)Tk|Dk is the inclusion of Dk into Pk.
The first part of Theorem 3.2 is shown by the following lemma.
Lemma 3.3. Tk maps Ker@k to Pk and is a chain-level representation of T r*k.
Proof. In order to construct a chain-level representation for T r*k, following *
*2.5, we
first construct a chain-level representation for
(e1) : T orAk(F2, F2) ! T orAk-1(F2, P1),
the connecting homomorphism associated with the short-exact sequence 2.1. Let
us consider the induced short-exact sequence
0 ! ^P1 '! ^P^!ß ^F2 ! 0 .
A lifting of a cycle z 2 ^ = ^F2 over the chain map ß is given by the chain
z x-1k2 ^P^, where we are writing P1 = F2[xk], ^P= Span{xik|i -1}. The
THE WEAK CONJECTURE ON SPHERICAL CLASSES 7
boundary @(z x-1k) pulls back z x-1kunderP' to a cycle in ^P1, which represe*
*nts
(e1)(z) in T orAk-1(F2, P1). If z = va11. .v.akk, then by (4.1) of [21],
X X a
@(z x-1k) = @( va11. .v.akk x-1k) = va11. .v.k-1k-1 Sqak+1(x-1k) .
This means that the map given by
va11. .v.akk7! va11. .v.ak-1k-1 Sqak+1(x-1k)
is a chain-level representation of (e1).
We similarly construct a chain-level representation for the homomorphism (e1
P1) : T orAk-1(F2, P1) ! T orAk-2(F2, P2), which is the connecting homomorphism
associated with the short-exact sequence of chain complexes
0 ! ^P2 '!P1 ^(P^ P1) ß!P1 ^P1 ! 0 .
Here we are writing P1 = F2[xk], P2 = F2[xk-1, xk] and ^P= Span{xik-1|i -1}.
By the argument similar to the one given above, the map
va11. .v.ak-1k-1 xik7! @(va11. .v.ak-1k-1 x-1k-1xik) = va11. .v.ak-2k-2 Sqak*
*-1+1(x-1k-1xik)
is a chain-level representation of (e1 P1). So the composite map
i j
va11. .v.akk7! va11. .v.ak-2k-2 Sqak-1+1 x-1k-1Sqak+1(x-1k)
sends a cycle in ^P1 to a cycle in ^P2 and it is a chain-level representation*
* of
the composite homomorphism (e1 P1) O (e1).
By repeating the above argument, it is easy to see that the map Tk given by
Definition 3.1 is a chain-level representation of T r*k= (e1 Pk-1) O . .O. (*
*e1
P1) O (e1). In particular, we have
Tk(Ker@k) ^0Pk = Pk.
For the convenience of the latter use, Tk is thought of as a morphism from
F2[v11, ..., vk1] to F2[x11, ..., xk1].
The lemma follows.
Remark 3.4. Since the chain-level representation Tk has been extended to a ho-
momorphism, whose domain is F2[v11, ..., vk1], the computation of Tk(Vi) makes
sense as one can see in Lemmas 3.7 and 3.8 below.
The second part of Theorem 3.2 is proved by a number of lemmata.
Let M be a graded A-module, which is concentrated in non-negative dimensions.
We are concerned with the Steenrod homomorphism d*P : M ! F2[x 1] M given
by
|u|X
d*P (u) = d*Px(u) = x|u|-i Sqi(u) ,
i=0
where u 2 M, x is of dimension 1 and F2[x 1] supports the canonical A-action as
discussed in Section 2. Actually, Steenrod defined his Sqi by using d*P , which*
* is
constructed by means of cohomology of the weath products (see [23, Ch. VII]).
Suppose additionally that M is an A-algebra. Then, by the Cartan formula, d*P
is a homomorphism of algebras.
8 NGUY^E~N H. V. HU_.NG
Lemma 3.5. Let M be a unstable A-algebra. Suppose ff, fi 2 M, and a |ff|, b
|fi|. Then
Sqa+b+1(x-1 fffi) = Sqa+1(x-1 ff)Sqb+1(x-1 fi).
Proof. Recall that Sqi(x-1) = xi-1 for every integer i. From the unstability of*
* M
and a |ff|, one gets
|ff|X
Sqa+1(x-1 ff) = xa-i Sqi(ff) = xa-|ff|d*P (ff).
i=0
As d*P is an algebra homomorphism, the lemma follows.
Applying repeatedly this lemma to M = Pk, we obtain
Lemma 3.6. Suppose v = va11. .v.akkand v0= vb11. .v.bkksatisfy the conditions:
ai ai+1+ . .+.ak 0, bi bi+1+ . .+.bk 0,
for 1 i k. Then
Tk(v . v0) = Tk(v) . Tk(v0) .
Furthermore, the both sides are elements of Pk.
Proof. Set øi(v) = Sqai+1(x-1i. .S.qak+1(x-1k)). It is easy to see that øk(vv0)*
* =
øk(v)øk(v0) in Pk. Suppose inductively that øi+1(vv0) = øi+1(v)øi+1(v0) in Pk.
Then, applying Lemma 3.5 with a = ai, b = bi, ff = øi+1(v), fi = øi+1(v0), we g*
*et
øi(vv0) = øi(v)øi(v0) in Pk.
Thus, the lemma follows as Tk = ø1.
Lemma 3.7. The restriction of Tk to the M`ui algebra Mk = F2[V1, ..., Vk] is a
homomorphism of algebras.
Proof. Recall that Vi = v2i-21v2i-32.v.i.-1vi. So, one easily verifies that eve*
*ry el-
ement v 2 Mk is a sum of monomials va11. .v.akk, which satisfy the conditions of
Lemma 3.6. The lemma follows from the previous one.
Lemma 3.8.
Tk(Vi) = Vi (1 i k).
Therefore, Tk|Mk is the inclusion of Mk into Pk.
Proof. This is proved by induction on k. It is easy for k = 1. Suppose in-
ductivelyithat-it2hasibeen-shown3for k - 1. Then, using the expansion Vi =
v21 v22 . .v.i-1vi, one has
i i-2 j
Tk(Vi) = @ v21 x-11Tk-1(Vi-1(x2, ..., xi))
i-2+1i -1 j
= Sq2 x1 Vi-1(x2, ..., xi) (by inductive hypothesis)
i j
= d*Px1 Vi-1(x2, ..., xi) (since |Vi-1| = 2i-2)
= Vi(x1, x2, ..., xi).
The last equality is showed in M`ui [17, Lemma 5.3], (see also [8]).
The lemma is proved.
THE WEAK CONJECTURE ON SPHERICAL CLASSES 9
Since Dk Mk, Lemmas 3.7 and 3.8 show that Tk|Dk is the inclusion of Dk into
Pk.
Theorem 3.2 is completly proved.
4.Homological classes induced by Dickson invariants
According to Singer [21], ^ksupports a canonical A-action as follows. The
usual action of A on Pk commutes with the action of GLk. So, it induces an
action of A on Dk = PkGLk= F2[Qk,k-1, ..., Qk,0]. This is extended to an action
of A on k = Dk[Q-1k,0]. By definition [21], ^kis the submodule of k spanned
by all monomials fl = Qik-1k,k-1.Q.i.0k,0with -1 < i0 < +1, 0 i1, ..., ik-1 a*
*nd
0 i0+ dimfl. By means of the Hai-Hu_.ng formula for the A-action on Dk (see [*
*8]
or 4.4 below), it is easy to verify that ^kis an A-submodule of k.
From Singer [21], the A-action on ^ = k ^kcommutes with the differential of
^.
Now we show that the weak conjecture on spherical classes follows from the
following one.
Conjecture 4.1. Let Ker@k be the submodule of all cycles in ^k. Then,
D+k A+ . Ker@k,
for k > 2.
Proposition 4.2. Conjecture 4.1 implies Conjecture 1.6, which in turn implies t*
*he
weak conjecture on spherical classes.
P
Proof. By Conjecture 4.1, for a given q 2 D+k, we have q = iSqi(fli) with some
i > 0 and fli2 Ker@k.
On the other hand, from Singer [21,PTh. 1.3], the induced A-action on Hk( ^) =
T orAk(F2, F2) is trivial. So [q] = iSqi[fli] = 0 in T orAk(F2, F2). By Theor*
*em 3.2,
[q] = T r*[q] = 0 in F2 Pk for every q 2 D+k, or equivalently, D+k A+ . Pk, f*
*or
A
k > 2. The proof is complete.
Here is an alternative proof for the fact that Conjecture 4.1 implies the weak
conjecture on spherical classes.
By Theorem 3.2, Tk(Ker@k) Pk, Tk(Dk) = Dk.
Since Tk is an A-homomorphism, Conjecture 4.1 implies
D+k A+ . Pk,
for k > 2. That means Conjecture 1.5 holds. Hence, the weak conjecture on
spherical classes is proved.
The following theorem is a weak form of Conjecture 4.1.
Theorem 4.3. If k > 2, then
D+k A+ . ^k.
In order to prove the theorem, we need the Hai-Hu_.ng formula for the action *
*of
A on Dk (see [8]):
8 j r
>> Qk,tQk,r2 i = 2k j
>:Qk,j i = 2 - 2 ,
0 otherwise.
10 NGUY^E~N H. V. HU_.NG
In particular, s+j-1 s s
Sq2 (Q2k,j) = Q2k,j-1,
for 0 < j < k.
The following remark is an immediate consequence of 4.4 and the Cartan formul*
*a.
Remark 4.5. (on jump steps). If Sqi(Q2sk,j) 6= 0, then either i = 0 or i 2s+j*
*-1.
Furthermore, if SqiX = Q2sk,j-1for some X 2 Dk, then either i = 0 or i 2s+j-1.
Proof of Theorem 4.3. We use techniques of the proof in [11] for the fact that
F2 k = 0 ,
A
for k > 1.
We follow the notations of [10]. For a given non-negative integer i, we denot*
*e by
s(i) the number with 2s(i)being the first missing 2-power in the dyadic expansi*
*on
of i. In other words, i 2s(i)- 1 (mod 2s(i)+1).
We prove the theorem by a downward induction.
Let QI = Qik-1k,k-1.Q.i.0k,0, I = (ik-1, . .,.i0) with ij 0 if j 1 and i0*
* 2 Z, be
a monomial in k. Set fj(I) = s(ij)+j -1 for j 1 and f(I) = min{fj(I)| j 1}.
Suppose fj(I) = f(I) with j 2, then s(ij-1) + j - 2 = fj-1(I) fj(I) =
s(ij) + j - 1, so s(ij-1) s(ij) + 1. In particular, ij-1 2s(ij). Hence, app*
*lying
4.4 we get
s(ij),i -2s(ij),...,iI)L
4.6. Sq2f(I)Q(ik-1,...,ij+2 j-1 = Q + Q .0
We show that s(`j-1) s(ij) and thus fj-1(L) = s(`j-1)+j -2 < s(ij)+j -1 =
fj(I) = f(I). Hence f(L) < f(I) for any L in the sum. Let ffm (a) denote the
coefficient of 2m in the dyadic expansion of a, and s = s(ij). There are two ca*
*ses.
In the first case, ffs(`j-1) = ffs(ij-1) = 1. Then, using the remark on jump st*
*eps,
we easily verify that Sq2f(I)acts only to send Q2sk,jto Q2sk,j-1. So QL = QI, b*
*ut
not an extra term. In the second case, ffs(`j-1) = ffs(ij-1 - 2s) = 0. Then, *
*by
definition, s(`j-1) s = s(ij).
It should be noted that if QI 2 Dk, then the "killer"
s(ij),i -2s(ij),...,i )
QJ = Q(ik-1,...,ij+2 j-1 0
and thus any extra term QL is in Dk.
Next we consider the case f1(I) = f(I) and fj(I) > f(I) if 1 < j < k. Similar*
*ly
as in 4.6, we have
4.7. Sq2f(I)Q(ik-1,...,i1+2s(i1),i0-2s(i1))= QI + QL.
We show that, for any L in the sum, there exists j (1 j < k) such that
fj(L) < f1(I), therefore f(L) < f(I). Suppose the contrary that fj(L) f1(I) or
equivalently s(`j)+j-1 s(i1) for 1 j < k. As s(`1) s(i1), then `1 = i1+2s*
*(i1).
Otherwise, by the remark on jump steps, Sq2f(I)= Sq2s(i1)acts only to send
Q2s(i1)k,1to Q2s(i1)k,0, thus QL = QI, but not an extra term. Suppose inductiv*
*ely
that `1 = i1 + 2s(i1), `2 = i2, ..., `j-1 = ij-1. We show `j = ij (1 j < k).
Indeed, if `j 6= ij, then combining s(`j) + j - 1 s(i1), s(ij) + j - 1 > s(i1)
with the remark on jump steps, we easily verify that actually s(`j) + j - 1 = s*
*(i1)
s(`j) 2s(`j)
and Sq2f(I)= Sq2s(i1)acts only to send Q2k,j to Qk,j-1. This contradicts the
THE WEAK CONJECTURE ON SPHERICAL CLASSES 11
hypothesis `j-1 = ij-1 (or the hypothesis `1 = i1 + 2s(i1)if j = 2). Consequent*
*ly,
we get `1 = i1 + 2s(i1), `2 = i2, ..., `k-1 = ik-1. So Sq2f(I)= Sq2s(i1)acts on*
*ly to
increase the power of Qk,0. However, dimQk,0= 2k - 1 6 | 2s(i1)if k > 1. This i*
*s a
contradiction.
Now we must show that if QI 2 Dk, then the "killer"
s(i1),i -2s(i1))
QJ = Q(ik-1,...,i1+2 0
and every "killer" needed in the procedure of using 4.6 and 4.7 to kill extra t*
*erms
QL or to kill extra terms of extra terms ... are all in ^k. Suppose QK is such*
* a
"killer". Let s = s(i1). As f = f(I) decreases in the procedure, then
k0 i0 - 2s - 2s-1- . .-.1 .
On the other hand, combining f(I) = s with the two facts that the dimension of
any extra term equals to dimQI and f(extra term) < f(I), we have
dimQK dimQI - 2f(I)= dimQI - 2s.
Thus
k0 + dimQK (i0 - 2s - 2s-1- . .-.1) + (dimQI - 2s)
= (i0 - 2s+1+ 1) + (dimQI - 2s)
dimQI - 3 . 2s + 1 .
From s(i1) = s, it implies i1 2s - 1. Then we get
dimQI dim(Qi1k,1) (2s - 1)(2k - 2) .
Hence
k0 + dimQK (2s - 1)(2k - 2) - 3 . 2s + 1
= (2s - 1)(2k - 2) - 3 . (2s - 1) - 3 + 1
= (2s - 1)(2k - 5) - 2
2k - 7 > 0 (as k > 2),
except for s = 0. (This case is handled by the next step.) Consequently, QK 2 *
*^k.
To start the induction, assume QI 2 Dk with f(I) = 0. Then f1(I) = f(I) = 0
as, by definition, fj(I) > 0 for 1 < j < k. Then s = s(i1) = 0, and i1 0 (mod*
* 2).
Thus
Sq1Q(ik-1,...,i1+1,i0-1)= QI.
The "killer" Q(ik-1,...,i1+1,i0-1)is in ^k, because
(i0 - 1) + dimQ(ik-1,...,i1+1,i0-1)= i0 + dimQI - 2 2k-1 - 2 > 0,
for k > 2. The first inequality holds as there exists at least one ij 6= 0 and
dim(Qk,j) 2k-1 for 0 j < k.
The theorem is proved.
The following theorem establishes Conjecture 1.6 for k = 3.
Theorem 4.8. Let k = 3. Then, for every q 2 D+3, [q] = 0 in H3( ^) ~=
T orA3(F2, F2).
12 NGUY^E~N H. V. HU_.NG
Proof. Let <., .> denote the dual pairing between T orA3(F2, F2) and Ext3A(F2, *
*F2),
and also the one between (F2 P3)GL3 and F2 P H*(BE3). Here, by P H*(BE3)
A GL3
we mean the submodule consisting of all A-annihilated elements of H*(BE3).
By Boardman [3], T r3 : F2 P H*(BE3) ! Ext3A(F2, F2) is an isomorphism.
GL3
In particular, we have
<[q], Ext3A(F2, F2)>= <[q], T r3(F2 P H*(BE3))>
GL3
=
GL3
= <[q], F2 P H*(BE3)> (by Theorem 3.2)
GL3
= 0,
because [q] = 0 in (F2 P3)GL3, by Theorem 3.2 of our paper [9].
A
Hence [q] = 0 in T orA3(F2, F2), for every q 2 D+3. The theorem is proved.
5. Final remarks
Remark 5.1. The weak conjecture on spherical classes is actually equivalent to
the fact that for every q 2 D+kand any k > 2, [q] 2 T orAk(F2, F2) vanishes on
the image of Singer's k-th transfer, T rk(F2 P H*(BEk)) ExtkA(F2, F2). This
GLk
observation can be read off from the proof of Theorem 4.8.
Remark 5.2. Conjecturei1.6, anditherefore Conjecture 4.1, is false when k = 1 or
2. Indeed, [Q21-1,0] and [Q22-1,1] are non-zero in T orA*(F2, F2). They are res*
*pectively
dual to the Adams element hi2 Ext1A(F2, F2) and its square h2i2 Ext2A(F2, F2).
One easily verifies this assertion by combining Theorem 3.2 with Theorem 2.1 of*
* [9]
and Proposition 5.3 of [14]. The only Hopf invariant one elements are represent*
*ed
by hi for i = 1, 2, 3 (see Adams [1]). Furthermore, the only Kervaire invariant*
* one
elements are represented by h2i, wherever h2iis a permanent cycle in the Adams
spectral sequence for spheres (see Browder [5]).
Let '*k: F2 Dk ! T orAk(F2, F2) be the dual of the Lannes-Zarati homomor-
A
phism, which is compatible with the Hurewicz one H : ß*(Q0S0) ! H*(Q0S0).
In [9] we have proved that the inclusion Dk Pk is a chain-level representation
of T r*k. '*k: F2 Dk ! F2 Pk. This together with Theorem 3.2 lead us to the
A A
following conjecture.
Conjecture 5.3. The inclusion Dk ^kis a chain-level representation of '*k.
This conjecture and Conjecture 1.6 imply the classical conjecture on spherical
classes.
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Department of Mathematics, Vietnam National University, Hanoi
334 Nguy^e~n Tr~ai Street, Hanoi, Vietnam
E-mail address: nhvhung@vnu.edu.vn
Note added in proof: Recently, we have established Conjecture 5.3.