DSpace at VNU: Spherical classes and the lambda algebra

We prove that the inclusion of the Dickson algebra, Dk, into rA is a chain-level representation of the Lannes-Zarati dual homomorphism A~~~~~~~A wk : F2 93 Dk -->Tor4 ]F2, IF2 _-Hk r

Spherical Classes and the Lambda Algebra

Author(s): Nguyễn H V Hung

Source: Transactions of the American Mathematical Society, Vol 353, No 11 (Nov., 2001), pp 4447-4460

Published by: American Mathematical Society

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Volume 353, Number 11, Pages 4447-4460

S 0002-9947(01)02766-0

Article electronically published on May 22, 2001

NGUYEN H V HUNG

dual of the lambda algebra with Hk(rA) - Tor 4(F2,F2), where A denotes

the mod 2 Steenrod algebra We prove that the inclusion of the Dickson

algebra, Dk, into rA is a chain-level representation of the Lannes-Zarati dual

homomorphism

A~~~~~~~A

wk : F2 (93 Dk >Tor4 (]F2, IF2) _-Hk (r/\

The Lannes-Zarati homomorphisms themselves, fOk, correspond to an associ-

ated graded of the Hurewicz map

H: 7r (SO)- r* (QoSO) -?H* (QoSO) Based on this result, we discuss some algebraic versions of the classical 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 Tor 4 (F2, F2)

for k > 2

We also show that (P* factors through k F2 (0 Ker&k, where 09k: rkA

~~~Ak

]p_ denotes the differential of rA Therefore, the problem of determining

F2 (0 Ker9k should be of interest

1 INTRODUCTION AND STATEMENT OF RESULTS

Let QoS0 be the basepoint component of QS? = limn QnSn It is a classical unsolved problem to compute the image of the Hurewicz homomorphism

H :7r (So) 7r* (Qo SO) -> H* (Qo SO) 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 spherical classes reads as follows

Conjecture 1.1 The Hopf invariant one and the Kervaire invariant one classes are the only elements in H* (QoSO) 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 [x, , Xk]

be the polynomial algebra on k generators x1, , Xk, each of degree 1 Let the

Received by the editors February 4, 1999 and, in revised form, November 4, 1999

2000 Mathematics Subject Classification Primary 55P47, 55Q45, 55S10, 55T15

Key words and phrases Spherical classes, loop spaces, Adams spectral sequences, Steenrod algebra, lambda algebra, invariant theory, Dickson algebra

The research was supported in part by the National Research Project, No 1.4.2

4447

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[XI - , XGk

As the action of A and that of GLk on Pk cominute with each other, Dk is an algebra over A In [14], Lannes and Zarati construct homomorphisms

S0k EExtk?i (2 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 Ext j *(F2, F2) and Ext2*(IF2, F2), on

which Pi and P2 are non-zero (see Adams [1], Browder [4], Lannes-Zarati [14])

Therefore, we are led to the following conjecture

Conjecture 1.2 0k 0 in any positive stem i for k > 2

The present paper follows a series of our works ([8], [10], [11]) on this conjec- ture To state our main result, we need to summarize Singer's invariant-theoretic description of the lambda algebra [20] According to Dickson [6], one has

where Qk,i denotes the Dickson invariant of degree 2 - 2 Singer sets Fk Dk[Q-17], the localization of Dk given by inverting Qk,o, and defines P' to be a certain "not too large" submodule of Fk He also equips rp = EkrkP with a differential 0: rFA rA_l and a coproduct Then, he shows that the differential coalgebra pA is dual to the lambda algebra of the six authors of [3] Thus, Hk (IA) A

TorA (IF2, F2) (Originally, Singer uses the notation F+ to denote PA However, by D+, A+ we always mean the submodules of Dk and A respectively consisting of all elements of positive degrees, so Singer's notation F+ would cause confusion in this paper Therefore, we prefer the notation FeA.)

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 c Pk is a chain-level representation of the Lannes-Zarati dual homomorphism

*

k (F2 XDk)i Tor k?A(F2,F2)-

An immediate consequence of this theorem is the equivalence between Conjec- ture 1.2 and the following one

Conjecture 1.3 If q C D+, then [q] = 0 in TorA (IF2,1F2) 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 poinit of this conjecture, it seems to us that Singer's model of the dual of the lambda algebra, pA, is somehow more natural than the lambda algebra itself

The canonical A-action on Dk is extended to an A-action on PA This action commutes with Ok (see [20]), so it determines an A-action on Ker&k, the submodule

of all cycles in PA We also prove

Proposition 4.1 so* factors through F2 OKer&k as shown in the commutative

A diagram

Tor (A2,F2)

F2 &Ker&k,

A where Z is induced by the inclusion Dk C Ker&k, and - is an epimorphism induced

by the canonical projection p: Ker&k -> Hkz (FA) TorA (F2, F2)

From this result, the problem of determining F2 0Ker&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 p

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 bar 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 paper

2 RECOLLECTIONS ON MODULAR INVARIANT THEORY

We start this section by sketching briefly Singer's invariant-theoretic description

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 diagolnal The Tk-invariant ring, Mk PT, is called the Miui algebra In [17], Miui shows that

where

Vi = (Aix, + *- +Ai-,xi-, + Xi)

Xj EF2

Then, the Dickson invariant Qk,i can inductively be defined by

Qk,i = + Vk * Qk-l,i,

where, by convention, Qk,k = 1 and Qk,i = 0 for i < 0

Let S(k) C Pk be the multiplicative subset generated by all the non-zero linear forms in Pk Let (Pk be the localization, (Pk = (Pk)s(k) Using the results of Dickson [6] and Miui [17], Singer notes in [20] that

Ak = (1k)T -

F- V1,- ) Vk?I]

r k ((.k = 2[Qk,k-Ii * , Qk,l, Qk,0 Further, he sets

VIV1 ,Vk 2=Vk/V1 Vk-1 (k > 2))

so that

Vk =2k-VI k Vk_lVk (k > 2)

Then, he obtains

Ak = F2 [V I, , v 1],

with deg vi 1 for every i

Singer defines F' to be the submodule of Fk = Dk [Qk-, spanned by all mono- mials Y Qkk-1

shows in [20] that the homomorphism

I k 2[ v V [VI vk-1]'

3Jk

f

Vj1',

if ik

I Vk, o, otherwise, maps FA to FA1 Moreover, it is a differential on r1 A

k rP This module is

bigraded by putting bideg(v' vik) - (k, k + Eii)

Let A 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 putting (as

in [19, p 90]) bideg(Ai) = (1, 1 + i) Singer proves in [20] that pA is a differential bigraded coalgebra, which is dual to the differential bigraded lambda algebra A via the isomorphisms

V 1 V3 I (Al A Here the duality * is taken with respect to the basis of admissible monomials of A

As a consequence, one gets an isomorphism of bigraded coalgebras

In the remaining part of this section, we recall the definition of the Lannes-Zarati homomorphism

Let P1 F2[x] with lxl = 1 Let P C 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 P is an A-submodule

of F2 [X, X-1] One has a short-exact sequence of A-modules

where t is the inclusion and wT is given by 1r(X') = 0 if i 7 -1 and wT(x-1) = 1 Let

el be the corresponding element in Ext (E1F2, F1)

Definition 2.2 (Singer [21]) (i) ek el 0 * 0 El E Extk(E-kIF2, Pk)

k times

(ii) ek(Mll) = ek 0 M C Ext (A k, Pk 0 Ml), 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{Sq'xl i > deg x, x C 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)),

where F* (M) is an A-free (or A-proj ective) resolution of M

The cap-product with ek (M) gives rise to the homomorphism

ek(M/l): Dk( M) ' DoPpk 0 M) -Pk 0 M

ek (M) (z) ek(M) n z

Since F2 is an unstable A-module, one gets

Theorem 2.3 (Lannes-Zarati [14]) Let Dk C Pk be the Dickson algebra of k vari- ables Then ak :_ ek(EF2): Dk(El-kF2) -> EDk 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

Here the horizontal arrows are induced from the differential in F* (M), and

ik [Z] = [1 ( Z]

A

for Z c Fk(M) Passing to homology, one gets a homomorphism

ik F 2 0Dk(M) TorA (F2, )

Taking M - El-kF2, one obtains a homomorphism

ikF: 2 0Dk (lk F2 )

TorkA (F2, lk F2)

A

Note that the suspension Z: F2 0 Dk -i F2 0 YDk and the desuspension

1

T?rko(2, lk F2) Tork (F2, kF2) are isomorphisms of internal degree 1 and (-1), respectively This leads one to Definition 2.5 (Lannes-Zarati [14]) The homomorphism pOk of internal degree 0

is the dual of

k= Elik(l 90ak)Z 2 0Dk Tor A(F2, Ek 2)

Remark 2.6 In Theorem 3.9 we also denote by fok the composite of the above yok with the suspension isomorphism Ek (o2rk : F2, E F2)

We need to relate ak = ek(EF2) with connecting homomorphisms

Suppose f E Ext' (M3, M) is represented by the short-exact sequence of left

A-modules 0 - M, -

M2 M3 * 0 O Let A(f): D,(M3) -* D.1(M1) be the

connecting homomorphism associated with this short-exact sequence Then one easily verifies

A(f)(z) = f nz

for any z C D,(M3)

4452 NGUYEN H V HUNG

One has

2.7 ek(ZF2) =(el(ZF2) Pk-) o * o (ei(Z3kF2) (8 P) 0 e(Z2-kF2)

Therefore, one gets

2.8 Ozk = A(el(ZF2) 0 P_) o * o A(ei(Z3kF2) ( Pl) o l (E2 kF2)

(See Singer [21, p 498].)

This formula will be useful to construct a chain-level representation of ak

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 A/I over A Recall that

Bk (MA) = A X I * (IAg[ I (k > O),

k times where I denotes the augmentation ideal of A and the tensor products are taken over IF2 The module B* (M/) = ? Bk (MA) is bigraded by assigning an element

ao ? a, O ak Ox with homological degree k and internal degree .0%(deg ai) + deg x

The differential dk: Bk(A/I) Bk-l (Ml) is defined by

dk(ao Oal, ak (x) aoal O* (*0ak ( x +ao (00ala2 00 ak (x

+* +aoaO 0a, 1 akx

So dk preserves internal degree and lowers homological degree by 1

The action of A on Bk(M) is given by

a(ao 0 a, 1 * ak 0 x)= aao 0a, 1 * ak x, for a C A

Suppose additionally that N is a right graded A-module As the bar resolution

is an A-free resolution, by definition one has

TorA (N, M) Hk(N 0 B* (M))

A

Since Dk c F2 [vl, , Vk], every element q C Dk has an unique expansion

q = q~~~ ~ Vli 1 Vk

(i1,-**ik)

where 1l, ,ik are non-negative We associate with q c Dk the following element

of internal degree Ek ij + 1:

Definition 3.1

q = S Sq'l+ Sqik+l

E

Z1l C Bk(1 IF2) (1 Jik)

Lemma 3.2 If q C Dk, then

q c EBk-1(ElIF2) :=Span{Sq2x| i > degx,x c Bk-1(El kF2)}

Proof From the definition of the A-action on the bar resolution, one has

Sqj1+1 ? ? Sqik+l Zl kl = SqS (1 ? Sqi( *Sq 0 * 0 Sq k+l El kl) Hence, it suffices to show that

il + 1> (i2 + 1) + + (jk + 1) + (1-k)= j2 + + 3k,

for every term in the expansion of q

Recall that Vi vv 2 vij1vi So, one easily verifies that every element

V Ilk =F2 [V1, VkI is a sum of monomials v1' v k which satisfy the condition

jl >i2+ -+jk-

The lemma follows from the fact that the Dickson algebra Dk is a subalgebra of

Lemma 3.3 q is a cycle in the chain complex EB*(El-k F2), for every q C Dk This is a consequence of the following lemma, which is actually an exposition of the Adem relations

Lemma 3.4 The homomorphism

rk,p :Ak >Ak-1 -A ?A (k -I times)

v1 V1 Vp p VP1 Jp+1 Vpk k Sqjl+l 0 * 09 Sqjp+lSqjP+1+l 0 Sqik+1 vanishes onr k C Ak, for 1 < p < k

Proof Consider the diagonal b: Ak A p-A 0 A2 0 Ak-p-1 defined by

Vi O 1 1, i < p,

0 (vi) = 1 X Vj_p+l X1, p <i < p+ 1,

I lXl (Dv._P_l, p+lI < i From Proposition 2.1 of Singer [20], one gets

b(rk) c rp> oF r2 ? rk-p-l

Define the homomorphism Wt: Ft -) At by

wt (vlii *Vit ) - Sql +l1 (9 (9 Sqjt + 1 Then one has

Wk,p = (Wp-1 0 72,1 0 Wk-p-1).-

By Proposition 3.1 of Singer [20], the Adem relations yield

72,1(r2) = 0

Hence, r,k,p(rk) = 0 for 1 < p < k The lemma is proved Dl Proof of Lemma 3.3 First, we note that Sqjk+l(l-kl) = 0 for aniy ik > 0 Then,

by definition of the differential in the bar resolution, we get

k-1

dk-l(4) - Z(7rk,p 0 idx ME-

k2 ) (q 0 El1kl)

p=1

Since q C Dk C ]7k, Lemma 3.4 yields 7rk,p(q) = 0 Thus dkl(0) = 0 The lemma

For the convenience of the latter use, we define ?k,p as follows:

for 1 < p < k

Suppose as before that

q vl Vk c Dk

J=(jl, i)

For a fixed (k - s)-index (is+l ik), we define J(is+l, .k ik ) to be the set of all s- indices (j, js)'s such that (jil, .sts+ .*-,ik) occurs as a k-index in the above sum

The following lemma is a slight generalization of Lemma 3.4

Lemma 3.5 If q = j v V EDk, then

for 1 < p < s < k

Proof Let us consider the diagonal ?b2 : Ak -/\ A (8 Ak-s given by

b2 (Vi) =

i

{ 1 v_7

I

< i < Si

According to Proposition 2.1 of Singer [20], 4'(Fk) C IF, 0Fk- Since q e Dk C Fk,

it implies ZJ( +1 ik) v'.v c IF, Then, by Lemma 3.4, we have

-,S 5 Sqi' 0 .? Sqis+) -5,p VI Vsi)0

By definition of the destabilization functor D, for any left A-module M, one has

an exact sequence of chain complexes

0 -, EB*(M) E B * (M) j

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

3* : Dk (M) := Hk(DB* (M)) Hk_l(EB* (M))

Take M= ZlkF2 The following lemma deals with the connecting isomorphism

0* : Dk (ZlF2) : = Hk(DB* (ZlF2)) Hk-l(EB* (kF2))

Let [q] be the homology class of the cycle q in

Dk(El-kF2) HklI(EB*(lk F2))

Lemma 3.6 If q C Dk, then

Proof Suppose q = J ** * Vik The element Ej 1 0 Sqil?l 0 ( (? Sqjk+I (0 Z1lkl C Bk(Zl-k F2) is a lifting over jD of its class modulo EBk(El-k F2) in DBk(Zl-kF2) Let d denote the differential in B*(El-k F2), we get

d(E 1 0 Sq'l+1 ? .S ? sqik+l ? Z1-ki)

J

= 1 Sqil+l 0 0 Sqik+l ? Z1-kl

J

k-I + E 10 Wk,p( Sqj+1 0 Sqj+l) -k

+ 0 Sqil+l Sqik+ll-kl

J

By Lemma 3.4

Trk,p(ZSqil+1 0 * 0 Sqik+l) =Fk,p(q) = 0

J

On the other hand, Sqik+l(l-kl) = 0 for any jk > 0 Therefore, we obtain d(Z 1 Sqj1+1 0 0 Sqjk+1 0 E-kl) S Sqj1+' ? ? Sqik+l ? l-ki

- iE (q)

By definition of the connecting homomorphism, we have

The following theorem deals with the isomorphism aOk Dk (El-kF2) > EDk treated in Theorem 2.3

Theorem 3.7 If q C Dk, then

Ozk[q] =q

Proof We compute ak by means of the following formula

ak = A(ei(EF2) Pk-) o * o A(ei(Z3kF2) F P) o E F2)

= 6k 62 61 -

Here J, stands for A(ei(Z1-k+?sF2) ? P.-I), for brevity

Consider the short exact sequence representing e1(Z2-kF2):

0 ,> 2-kp1 i E2-kf A Zl-kF2 -*0 Then the connecting homomorphism induced by this exact sequence is nothing but

61 : Hkl,(EB* ( Flk2)) -) Hk-2(EB*(Z2k

A lifting of q = Ej Sqj1+1 0 0 Sqik+l El-kl over w is

Z Sqjll0 Sqik+l 0 Z2-kXjl1 E EB* (:2kP),

J

where we are writing P1 = F2[xk],P Span{xjI i > -1} The boundary of this element in EB*(Z2-kP) is pulled back under t to a cycle in EB*(Z2-kPl), which