TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 353, Number 11, Pages 4447-4460
S 0002-9947(01)02766-0
Article electronically published on May 22, 2001
SPHERICAL CLASSES AND THE LAMBDA ALGEBRA
NGUY~^EN H. V. HU,NG
Abstract.Let ^ = L k ^kbe Singer's invariant-theoretic model of the
dual of the lambda algebra with Hk( ^) ~=TorAk(F2, F2), where A denotes
the mod 2 Steenrod algebra. We prove that the inclusion of the Dickson
algebra, Dk, into ^kis a chain-level representation of the Lannes-Zarat*
*i dual
homomorphism
'*k: F2A Dk ! TorAk(F2, F2) ~=Hk( ^) .
The Lannes-Zarati homomorphisms themselves, 'k, correspond to an associ-
ated graded of the Hurewicz map
H : is*(S0) ~=i*(Q0S0) ! H*(Q0S0) .
Based on this result, we discuss some algebraic versions of the classica*
*l con-
jecture on spherical classes, which states that Only Hopf invariant one *
*and
Kervaire invariant one classes are detected by the Hurewicz homomorphism.
One of these algebraic conjectures predicts that every Dickson element, *
*i.e. ele-
ment in Dk, of positive degree represents the homology class 0 in TorAk(*
*F2, F2)
for k > 2.
We also show that '*kfactors through F2 AKer@k, where @k : ^k!
^k-1denotes the differential of ^. Therefore, the problem of determini*
*ng
F2A Ker@k should be of interest.
1.Introduction and statement of results
Let Q0S0 be the basepoint component of QS0 = limn nSn. It is a classical
unsolved problem to compute the image of the Hurewicz homomorphism
H : ßs*(S0) ~=ß*(Q0S0) ! H*(Q0S0) .
Here and throughout the paper, homology and cohomology are taken with coeffi-
cients in F2, the field of two elements. The long-standing conjecture on spheri*
*cal
classes reads as follows.
Conjecture 1.1. The Hopf invariant one and the Kervaire invariant one classes
are the only elements in H*(Q0S0) detected by the Hurewicz homomorphism. (See
Curtis [5], Snaith and Tornehave [22] and Wellington [23] for a discussion.)
An algebraic version of this problem goes as follows. Let Pk = F2[x1, . .,.x*
*k]
be the polynomial algebra on k generators x1, . .,.xk, each of degree 1. Let t*
*he
____________
Received by the editors February 4, 1999 and, in revised form, November 4, 1*
*999.
2000 Mathematics Subject Classification. Primary 55P47, 55Q45, 55S10, 55T15.
Key words and phrases. Spherical classes, loop spaces, Adams spectral sequen*
*ces, Steenrod
algebra, lambda algebra, invariant theory, Dickson algebra.
The research was supported in part by the National Research Project, No. 1.4*
*.2.
cO2001 American Mathematical Socie*
*ty
4447
4448 NGUY~^EN H. V. HU,NG
general linear group GLk = GL(k, F2) and the mod 2 Steenrod algebra A both act
on Pk in the usual way. The Dickson algebra of k variables, Dk, is the algebra *
*of
invariants
Dk := F2[x1, . .,.xk]GLk .
As the action of A and that of GLk on Pk commute with each other, Dk is an
algebra over A. In [14], Lannes and Zarati construct homomorphisms
'k : Extk,k+iA(F2, F2) ! (F2 Dk)*i,
A
which correspond to an associated graded of the Hurewicz map. The proof of
this assertion is unpublished, but it is sketched by Lannes [12] and by Goerss *
*[7].
The Hopf invariant one and the Kervaire invariant one classes are respectively
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 [4], Lannes-Zarati [14]).
Therefore, we are led to the following conjecture.
Conjecture 1.2. 'k = 0 in any positive stem i for k > 2.
The present paper follows a series of our works ([8], [10], [11]) on this con*
*jec-
ture. To state our main result, we need to summarize Singer's invariant-theoret*
*ic
description of the lambda algebra [20]. According to Dickson [6], one has
Dk ~=F2[Qk,k-1, ..., Qk,0],
where Qk,idenotes the Dickson invariant of degree 2k - 2i. Singer sets k =
Dk[Q-1k,0], the localization of Dk given by inverting Qk,0, andLdefines ^kto b*
*e a
certain ön t too large" submodule of k. He also equips ^ = k ^kwith a
differential @ : ^k! ^k-1and a coproduct. Then, he shows that the differential
coalgebra ^ is dual to the lambda algebra of the six authors of [3]. Thus, Hk(*
* ^) ~=
T orAk(F2, F2). (Originally, Singer uses 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 degrees, so Singer's notation +kwould cause confusion in *
*this
paper. Therefore, we prefer the notation ^k.)
The main result of this paper is the following theorem, which has been conjec-
tured in our paper [10, Conjecture 5.3].
Theorem 3.9. The inclusion Dk ^kis a chain-level representation of the
Lannes-Zarati dual homomorphism
'*k: (F2 Dk)i! T orAk,k+i(F2, F2).
A
An immediate consequence of this theorem is the equivalence between Conjec-
ture 1.2 and the following one.
Conjecture 1.3. If q 2 D+k, then [q] = 0 in T orAk(F2, F2) for k > 2.
This has been established for k = 3 in [10, Theorem 4.8], while Conjecture 1.2
has been proved for k = 3 in [8, Corollary 3.5] .
From the view point of this conjecture, it seems to us that Singer's model of*
* the
dual of the lambda algebra, ^, is somehow more natural than the lambda algebra
itself.
The canonical A-action on Dk is extended to an A-action on ^k. This action
commutes with @k (see [20]), so it determines an A-action on Ker@k, the submodu*
*le
of all cycles in ^k. We also prove
SPHERICAL CLASSES AND THE LAMBDA ALGEBRA 4449
Proposition 4.1.'*kfactors through F2 Ker@k as shown in the commutative
A
diagram
*
F2 Dk ________________-T'orAk(F , F )
A k 2 2
@ _ _ `
@ i p
@
@R
F2 Ker@k ,
A
_ _
where iis induced by the inclusion Dk Ker@k, and p is an epimorphism induced
by the canonical projection p : Ker@k ! Hk( ^) ~=T orAk(F2, F2).
From this result, the problem of determining F2 Ker@k would be of interest.
A
The paper is divided into 4 sections.
In Section 2 we recollect some materials on invariant theory, particularly on
Singer's invariant-theoretic description of the lambda algebra and the Lannes-
Zarati homomorphism. Section 3 is devoted to prove Theorem 3.9. Finally, Section
4 is a discussion on factoring '*k.
The main results of this paper were announced in [9].
The author would like to thank Haynes Miller for introducing him to Stewart
Priddy's work [18] on exploiting an explicit homotopy equivalence between the b*
*ar
resolution of F2 over A and the dual of the lambda algebra. He also thanks the
referee for helpful suggestions, which led to improving the exposition of the p*
*aper.
2. Recollections on modular invariant theory
We start this section by sketching briefly Singer's invariant-theoretic descr*
*iption
of the lambda algebra.
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 shows 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 [6] and M`ui [17], Singer notes in [20] that
k := ( k)Tk = F2[V1 1, ..., Vk 1],
k := ( k)GLk = F2[Qk,k-1, ..., Qk,1, Qk1,0].
Further, he sets
v1 = V1, vk = Vk=V1. .V.k-1 (k 2),
4450 NGUY~^EN H. V. HU,NG
so that k-2 k-3
Vk = v21 v22 . .v.k-1vk (k 2).
Then, he obtains
k = F2[v11, ..., vk1],
with degvi= 1 for every i.
Singer defines ^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 + degfl *
*0. He also
shows in [20] that the homomorphism
@k : F2[v11, ..., vk1] ! F2[v11, ..., vk1-1],
æ j1 jk-1
@k(vj11. .v.jkk) := v10.,.v.k-1,ifojkt=h-1,erwise,
L
maps ^kto ^k-1. Moreover, it is a differential onP ^ = k ^k. This module is
bigraded by putting bideg(vj11. .v.jkk) = (k, k + ji).
Let be the (opposite) lambda algebra, in which the product in lambda symbols
is written in the order opposite to that used in [3]. It is also bigraded by pu*
*tting (as
in [19, p. 90]) bideg(~i) = (1, 1 + i). Singer proves in [20] that ^ is a diff*
*erential
bigraded coalgebra, which is dual to the differential bigraded lambda algebra *
* via
the isomorphisms
^k ! *k
vj11. .v.jkk7!(~j1. .~.jk)*.
Here the duality * is taken with respect to the basis of admissible monomials o*
*f .
As a consequence, one gets an isomorphism of bigraded coalgebras
H*( ^) ~=T orA*(F2, F2).
In the remaining part of this section, we recall the definition of the Lannes*
*-Zarati
homomorphism.
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 [24]). Then ^Pis an A-submodule
of F2[x, x-1]. One has a short-exact sequence of A-modules
2.1. 0 ! P1 '!P^!ß -1F2 ! 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( -1F2, P1).
Definition 2.2 (Singer [21]).(i)ek = e1___._.-.e1z____"2 ExtkA( -kF2, Pk).
k times
(ii)ek(M) = ek M 2 ExtkA( -kM, Pk M) , for M a left A-module.
Here M also means the identity map of M.
Following Lannes-Zarati [14], the destabilization of M is defined by
DM = M=EM,
where EM := Span{Sqix| i > degx, x 2 M}. They show that the functor associ-
ating M to DM is a right exact functor. Then they define Dk to be the kth left
derived functor of D. So one gets
Dk(M) = Hk(DF*(M)),
SPHERICAL CLASSES AND THE LAMBDA ALGEBRA 4451
where F*(M) is an A-free (or A-projective) resolution of M.
The cap-product with ek(M) gives rise to the homomorphism
ek(M) : Dk( -kM) ! D0(Pk M) Pk M
ek(M)(z) = ek(M) \ z .
Since F2 is an unstable A-module, one gets
Theorem 2.3 (Lannes-Zarati [14]).Let Dk Pk be the Dickson algebra of k vari-
ables. Then ffk := ek( F2) : Dk( 1-kF2) ! Dk is an isomorphism of internal
degree 0.
By definition of the functor D, one has a natural homomorphism, D(M) !
F2 M. Then it induces a commutative diagram
A
. . ._____- DFk(M) ________- DFk-1(M) ______-. . .
| |
| |
|ik |ik-1
| |
|? |?
. . ._____-F2 Fk(M) ________- F2 Fk-1(M) ______-. ...
A A
Here the horizontal arrows are induced from the differential in F*(M), and
ik[Z] = [1 Z]
A
for Z 2 Fk(M). Passing to homology, one gets a homomorphism
ik :F2 Dk(M) ! T orAk(F2, M)
2.4. A1 [Z] 7! [1 Z] .
A A
Taking M = 1-kF2, one obtains a homomorphism
ik : F2 Dk( 1-kF2) ! T orAk(F2, 1-kF2) .
A
Note that the suspension : F2 Dk ! F2 Dk and the desuspension
A A
~= A -k
-1 : T orAk(F2, 1-kF2) -! T ork (F2, F2)
are isomorphisms of internal degree 1 and (-1), respectively. This leads one to
Definition 2.5 (Lannes-Zarati [14]).The homomorphism 'k of internal degree 0
is the dual of
'*k= -1ik(1 ff-1k) : F2 Dk ! T orAk(F2, -kF2) .
A A
Remark 2.6.In Theorem 3.9 we also denote by '*kthe composite of the above '*k
~=
with the suspension isomorphism k : T orAk,i(F2, -kF2) -! T orAk,k+i(F2, F2).
We need to relate ffk = ek( F2) with connecting homomorphisms.
Suppose f 2 Ext1A(M3, M1) is represented by the short-exact sequence of left
A-modules 0 ! M1 ! M2 ! M3 ! 0. Let (f) : Ds(M3) ! Ds-1(M1) be the
connecting homomorphism associated with this short-exact sequence. Then one
easily verifies
(f)(z) = f \ z
for any z 2 Ds(M3).
4452 NGUY~^EN H. V. HU,NG
One has
2.7. ek( F2) = (e1( F2) Pk-1) O . .O.(e1( 3-kF2) P1) O e1( 2-kF2) .
Therefore, one gets
2.8. ffk = (e1( F2) Pk-1) O . .O. (e1( 3-kF2) P1) O e1( 2-kF2) .
(See Singer [21, p. 498].)
This formula will be useful to construct a chain-level representation of ffk.
3. A chain-level representation of the Lannes-Zarati homomorphism
Suppose again M is a left graded A-module. Let B*(M) be the bar resolution
of M over A. Recall that
Bk(M) = A I___._.-.Iz___" M (k 0),
k times
where I denotes the augmentationLideal of A and the tensor products are taken
over F2. The module B*(M) = kBk(M) is bigraded by assigning anPelement
a0 a1 . . .ak x with homological degree k and internal degree ki=0(degai) +
degx.
The differential dk : Bk(M) ! Bk-1(M) is defined by
dk(a0 a1 . . .ak x)= a0a1 . . .ak x + a0 a1a2 . . .ak x
+ . .+.a0 a1 . . .akx.
So dk preserves internal degree and lowers homological degree by 1.
The action of A on Bk(M) is given by
a(a0 a1 . . .ak x) = aa0 a1 . . .ak x,
for a 2 A.
Suppose additionally that N is a right graded A-module. As the bar resolution
is an A-free resolution, by definition one has
T orAk(N, M) := Hk(N B*(M)).
A
Since Dk F2[v1, ..., vk], every element q 2 Dk has an unique expansion
X j j
q = v11. .v.kk,
(j1,...,jk)
where j1, ..., jk arePnon-negative. We associate with q 2 Dk the following elem*
*ent
of internal degree ki=1ji+ 1:
Definition 3.1.
X
~q= Sqj1+1 . . .Sqjk+1 1-k1 2 Bk-1( 1-kF2).
(j1,...,jk)
Lemma 3.2. If q 2 Dk, then
~q2 EBk-1( 1-kF2) := Span{Sqix| i > degx, x 2 Bk-1( 1-kF2)}.
SPHERICAL CLASSES AND THE LAMBDA ALGEBRA 4453
Proof.From the definition of the A-action on the bar resolution, one has
Sqj1+1 . . .Sqjk+1 1-k1 = Sqj1+1(1 Sqj2+1 . . .Sqjk+1 1-k1).
Hence, it suffices to show that
j1 + 1 > (j2 + 1) + . .+.(jk + 1) + (1 - k) = j2 + . .+.jk,
for every term in the expansion of ~q.
Recall that Vi = v2i-21v2i-32.v.i.-1vi. So, one easily verifies that every el*
*ement
v 2 Mk = F2[V1, ..., Vk] is a sum of monomials vj11. .v.jkk, which satisfy the *
*condition
j1 j2 + . .+.jk.
The lemma follows from the fact that the Dickson algebra Dk is a subalgebra_of
the M`ui algebra Mk. |__|
Lemma 3.3. ~qis a cycle in the chain complex EB*( 1-kF2), for every q 2 Dk.
This is a consequence of the following lemma, which is actually an exposition*
* of
the Adem relations.
Lemma 3.4. The homomorphism
ßk,p: k ! Ak-1 = A . . .A (k - 1 times)
vj11. .v.jppvjp+1p+1.v.j.kk7!Sqj1+1 . . .Sqjp+1Sqjp+1+1 . . .Sqjk+1
vanishes on k k, for 1 p < k.
Proof.Consider the diagonal _ : k ! p-1 2 k-p-1 defined by
8
< vi 1 1, i < p,
_(vi) = : 1 vi-p+1 1, p i p + 1,
1 1 vi-p-1,p + 1 < i.
From Proposition 2.1 of Singer [20], one gets
_( k) p-1 2 k-p-1.
Define the homomorphism !t: t! At by
!t(vj11. .v.jtt) = Sqj1+1 . . .Sqjt+1.
Then one has
ßk,p= (!p-1 ß2,1 !k-p-1)_.
By Proposition 3.1 of Singer [20], the Adem relations yield
ß2,1( 2) = 0.
Hence, ßk,p( k) = 0 for 1 p < k. The lemma is proved. |___|
Proof of Lemma 3.3.First, we note that Sqjk+1( 1-k1) = 0 for any jk 0. Then,
by definition of the differential in the bar resolution, we get
k-1X
dk-1(~q) = (ßk,p id 1-kF2)(q 1-k1).
p=1
Since q 2 Dk k, Lemma 3.4 yields ßk,p(q) = 0. Thus dk-1(~q) = 0. The_lemma_
is proved. |__|
4454 NGUY~^EN H. V. HU,NG
For the convenience of the latter use, we define ~ßk,pas follows:
ß~k,p(Sqj1+1 . . .Sqjk+1) = Sqj1+1 . . .Sqjp+1Sqjp+1+1 . . .Sqjk+1
for 1 p < k.
Suppose as before that
X j j
q = v11. .v.kk2 Dk.
J=(j1,...,jk)
For a fixed (k - s)-index (js+1, ..., jk), we define J(js+1, ..., jk) to be the*
* set of all s-
indices (j1, ..., js)'s such that (j1, ..., js, js+1, ..., jk) occurs as a k-in*
*dex in the above
sum.
The following lemma is a slight generalization of Lemma 3.4.
P j1 j
Lemma 3.5. If q = Jv1 . .v.kk2 Dk, then
X j +1 j +1
~ßs,p Sq 1 . . .Sq s = 0
J(js+1,...,jk)
for 1 p < s k.
Proof.Let us consider the diagonal _2 : k ! s k-s given by
æ
_2(vi) = vi1 1, v1 i s,
i-s,s < i k.
According toPProposition 2.1 of Singer [20], _( k) s k-s. Since q 2 Dk k,
it implies J(js+1,...,jk)vj11. .v.jss2 s. Then, by Lemma 3.4, we have
X j +1 j +1 X j j
~ßs,p Sq 1 . . .Sq s = ßs,p v11. .v.ss= 0.
J(js+1,...,jk) J(js+1,...,jk)
The lemma is proved. |___|
By definition of the destabilization functor D, for any left A-module M, one *
*has
an exact sequence of chain complexes
0 ! EB*(M) iE!B*(M) jD!DB*(M) ! 0,
in which the bar resolution B*(M) is exact. Hence, by use of the induced long
exact sequence, the connecting homomorphism is an isomorphism
~=
@* : Dk(M) := Hk(DB*(M)) -! Hk-1(EB*(M)).
Take M = 1-kF2. The following lemma deals with the connecting isomorphism
~= 1-k
@* : Dk( 1-kF2) := Hk(DB*( 1-kF2)) -! Hk-1(EB*( F2)).
Let [~q] be the homology class of the cycle ~qin
Dk( 1-kF2) ~=Hk-1(EB*( 1-kF2)).
Lemma 3.6. If q 2 Dk, then
@*[1 ~q] = [~q].
P j1 j P
Proof.Suppose q = Jv1 . .v.kk. The element J1 Sqj1+1 . . .Sqjk+1
1-k1 2 Bk( 1-kF2) is a lifting over jD of its class modulo EBk( 1-kF2) in
DBk( 1-kF2). Let d denote the differential in B*( 1-kF2), we get
SPHERICAL CLASSES AND THE LAMBDA ALGEBRA 4455
X
d( 1 Sqj1+1 . . .Sqjk+1 1-k1)
J X
= 1 . Sqj1+1 . . .Sqjk+1 1-k1
J
k-1X X
+ 1 ~ßk,p( Sqj1+1 . . .Sqjk+1) 1-k1
p=1 J
X
+ 1 Sqj1+1 . . .Sqjk+1 1-k1.
J
By Lemma 3.4
X
~ßk,p( Sqj1+1 . . .Sqjk+1) = ßk,p(q) = 0.
J
On the other hand, Sqjk+1( 1-k1) = 0 for any jk 0. Therefore, we obtain
X X
d( 1 Sqj1+1 . . .Sqjk+1 1-k1) = Sqj1+1 . . .Sqjk+1 1-k1
J J
= iE (~q).
By definition of the connecting homomorphism, we have
@*[1 ~q] = [~q].
The lemma is proved. |___|
The following theorem deals with the isomorphism ffk : Dk( 1-kF2) ! Dk
treated in Theorem 2.3.
Theorem 3.7. If q 2 Dk, then
ffk[~q] = q.
Proof.We compute ffk by means of the following formula
ffk = (e1( F2) Pk-1) O . .O. (e1( 3-kF2) P1) O e1( 2-kF2)
= ffik . .f.fi2ffi1.
Here ffis stands for (e1( 1-k+sF2) Ps-1), for brevity.
Consider the short exact sequence representing e1( 2-kF2):
0 ! 2-kP1 '! 2-kP^!ß 1-kF2 ! 0.
Then the connecting homomorphism induced by this exact sequence is nothing but
ffi1 : Hk-1(EB*( 1-kF2)) ! Hk-2(EB*( 2-kP1)).
P
A lifting of ~q= JSqj1+1 . . .Sqjk+1 1-k1 over ß is
X
Sqj1+1 . . .Sqjk+1 2-kx-1k2 EB*( 2-kP^),
J
where we are writing P1 = F2[xk], ^P= Span{xik| i -1}. The boundary of this
element in EB*( 2-kP^) is pulled back under ' to a cycle in EB*( 2-kP1), which
4456 NGUY~^EN H. V. HU,NG
represents ffi1[~q]. That means
X
ffi1[~q]=[d( Sqj1+1 . . .Sqjk+1 2-kx-1k)]
J
k-1X X j +1 j +1 2-k -1
= ~ßk,p( Sq 1 . . .Sq k ) xk
p=1 J
X
+ Sqj1+1 . . .Sqjk-1+1 Sqjk+1( 2-kx-1k)
J
X j +1 j +1 2-k j +1 -1
= Sq 1 . . .Sq k-1 Sq k (xk ) ,
J
where the last equality follows from Lemma 3.4. Indeed,
X
~ßk,p( Sqj1+1 . . .Sqjk+1) = ßk,p(q) = 0.
J
Similarly, ffi2 : Hk-2(EB*( 2-kP1)) ! Hk-3(EB*( 3-kP2)) is the connecting ho-
momorphism induced by the short exact sequence representing e1( 3-kF2) P1:
0 ! 3-kP2 '!P1 3-k(P^ P1) ß!P1 2-kP1 ! 0.
Here we are writing P1 = F2[xk], P2 = F2[xk-1, xk], ^P= Span{xik-1|i -1}. A
lifting of Sqj1+1 . . .Sqjk-1+1 2-kSqjk+1(x-1k) over ß P1 is
Sqj1+1 . . .Sqjk-1+1 3-kx-1k-1Sqjk+1(x-1k).
Therefore, by an argument similar to the one given above, we get
X j +1 j +1 3-k -1 j +1 -1
ffi2ffi1[~q]=[d Sq 1 . . .Sq k-1 xk-1Sq k (xk ) ]
J
k-2XX j +1 j +1 3-k -1 j +1 -1
= ~ßk-1,p(Sq 1 . . .Sq k-1 ) xk-1Sq k (xk )
p=1 J
X
+ Sqj1+1 . . .Sqjk-2+1 Sqjk-1+1( 3-kx-1k-1Sqjk+1(x-1k))
J
X j +1 j +1 3-k j +1 -1 j +1 -1
= Sq 1 . . .Sq k-2 Sq k-1 (xk-1Sq k (xk ))
J
(by Lemma 3.5).
Repeating the above argument, we then have
ffk[~q]= ffik . .f.fi1[~q]
X
= [ Sqj1+1(x-11Sqj2+1(x-12. .S.qjk+1(x-1k) . .).) ].
J
By Theorem 3.2 of our paper [10], we get
X
[ Sqj1+1(x-11Sqj2+1(x-12. .S.qjk+1(x-1k) . .).) ] = [ q] = q.
J
The theorem is proved. |___|
This theorem has an immediate consequence as follows.
SPHERICAL CLASSES AND THE LAMBDA ALGEBRA 4457
Corollary 3.8.The homomorphism Dk ! EBk-1( 1-kF2), q 7! ~qis a chain-level
representation of the homomorphism
(1 ff-1k) : F2 Dk ! F2 Dk( 1-kF2).
A A A
Theorem 3.9. The inclusion Dk ^kis a chain-level representation of the Lannes-
Zarati dual homomorphism
'*k: (F2 Dk)i! T orAk,k+i(F2, F2).
A
Proof.Suppose again that
X j j
q = v11. .v.kk2 Dk.
J=(j1,...,jk)
By Corollary 3.8 and Lemma 3.6, we have
(1 ff-1k) :F2 Dk ! F2 Dk( 1-kF2)
A A A
[q] 7! [~q] @*[1 ~q].
From the definition of ik (see 2.4), we get
ik :F2 Hk(DB*( 1-kF2)) ! T orAk(F2, 1-kF2)
A
[1 ~q]7! [1 ~q].
Let us consider the desuspension
-1 : T orAk(F2, 1-kF2) ! T orAk(F2, -kF2),
P P
which sends [ J 1 Sqj1+1 . . .Sqjk+1 1-k1] to [ J 1 Sqj1+1 . . .
Sqjk+1 -k1]. Then the map
'*k= -1ik(1 ff-1k) : F2 Dk ! T orAk(F2, -kF2)
A A
is given by X
'*k[q] = [ 1 Sqj1+1 . . .Sqjk+1 -k1].
J
The canonical isomorphism
k : T orAk,i(F2, -kF2) ! T orAk,k+i(F2, F2)
is defined by the chain-level version
k(a0 a1 . . .ak -k1) = a0 a1 . . .ak 1.
By ambiguity of notation, the composite k'*kis also denoted by '*k(see Re-
mark 2.6). Hence
'*k: (F2 Dk)i ! T orAk,k+i(F2, F2)
A P
[q] 7! [ J 1 Sqj1+1 . . .Sqjk+1 1].
__
In [18], Priddy constructs the Koszul complex K *(A), a subcomplex of B*(F2),
which is isomorphic to the dual of the lambda algebra. More precisely, it is de*
*fined
as follows. Let be the (opposite) lambda algebra, in which the product in lam*
*bda
symbols is written in the order opposite to that used in [3]. (See Singer [20, *
*p. 687]
4458 NGUY~^EN H. V. HU,NG
__
for a precise definition of .) Then, according to Priddy [18, x7], K *(A) is *
*the
image of the monomorphism
* ! B*(F2)
(~j1. .~.jk)*7! 1 Sqj1+1 . . .Sqjk+1 1,
which is a homotopy equivalence. Here * denotes the dual of and the duality *
is taken with respect to the basis of admissible monomials of . Combining it w*
*ith
Singer's isomorphism
^ ! *
vj11. .v.jkk7!(~j1. .~.jk)*,
we get the following homotopy equivalence
^ ! B*(F2)
vj11. .v.jkk7!1 Sqj1+1 . . .Sqjk+1 1.
As a consequence, for any q 2 Dk, we obtain
X
'*k[q]= [ 1 Sqj1+1 . . .Sqjk+1 1]
XJ
= [ vj11. .v.jkk]
J
= [q].
It means that the inclusion Dk ^kis a chain-level representation of '*k._ The
theorem is completely proved. |__|
Corollary 3.10.Conjecture 1.2 is equivalent to Conjecture 1.3.
This follows immediately from Theorem 3.9.
We have proved Conjecture 1.2 for k = 3 in [8] and Conjecture 1.3 for k = 3 in
[10].
4. Factoring the Lannes-Zarati homomorphism
The purpose of this section is to prove the following proposition.
Proposition 4.1.'*kfactors through F2 Ker@k as shown in the commutative
A
diagram:
*
F2 Dk ________________-T'orAk(F , F )
A k 2 2
@ _ _ `
@ i p
@
@R
F2 Ker@k ,
A
_ _
where iis induced by the inclusion Dk Ker@k, and p is an epimorphism induced
by the canonical projection p : Ker@k ! Hk( ^) ~=T orAk(F2, F2).
Proof.The canonical projection
p : Ker@k ! T orAk(F2, F2) = Ker@k=Im@k+1
sends x to [x] = x + Im@k+1.
SPHERICAL CLASSES AND THE LAMBDA ALGEBRA 4459
By Theorem 5.15 of Singer [20], the action of A on Ker@k induces a trivial ac*
*tion
of A upon T orAk(F2, F2). Therefore, p induces the epimorphism
_p: F A
2 Ker@k ! T ork (F2, F2)
A
[x] 7! [x].
For any q 2 Dk, we have
_p. _i[q] = _p[q] = [q] = '*
_ k[q].
So, we get '*k= _p. i. The proposition is proved. |_*
*__|
In [10], we have stated the following conjecture.
Conjecture 4.2. D+k A+ . Ker@k for k > 2.
Obviously, this is stronger than Conjectures 1.2 and 1.3 and equivalent to the
following one.
_
Conjecture 4.3. The homomorphism i: F2 Dk ! F2 Ker@k, induced by the
A A
inclusion i : Dk ! Ker@k, is trivial for k > 2.
Based on the above discussion, we believe the following problem is something *
*of
interest.
Problem 4.4. Determine F2 Ker@k.
A
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Department of Mathematics, Vietnam National University, Hanoi, 334 Nguy^en Tr*
*~ai
Street, Hanoi, Vietnam
E-mail address: nhvhung@hotmail.com