WU
APPLICATIONS DE LA K-THEORIE
A LA TOPOLOGIE
DES H-ESPACES FINIS
Forme réduite de la THESE
présentée à la faculté des sciences pour obtenir
le grade de docteur es sciences par
ALAIN JEANNERET
UNIVERSITE DE NEUCHATEL
Institut de Mathématiques
Chantemerle 20
2000 NEUCHATEL (Suisse)
IMPRIMATUR POUR LA THÈSE
.Applications..de.la....K-.théprie.à la topologie
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.des.¦H.-.espaces..finis................* (
de Monsieur...Aia jn ...J.e.a.n.ne.re.t
UNIVERSITÉ DE NEUCHATEL
FACULTÉ DES SCIENCES
La Faculté des sciences de l'Université de Neuchatel
sur le rapport des membres du jury,
.M.....les...p.r.o.tas.s.eu.r.s....U......Su.ter.r...F......S.i.grist...et...
.G.....Mls.l.i.n....(£PF.-.Zu.ric.h.)...................................................................
autorise l'impression de la présente thèse.
Neuchatel, le ........22...juin..19.9.2..........................................................
Le doyen :
A.  Robert
Réalisation topologique de certaines
algèbres associées aux algèbres de Dickson
Alain Jeanneret - Ulrich Suter
Topological realisation of certain algebras associated to the Dickson
algebras
Abstract - We discuss the topological realisation of certain Z/2-algèbras A(n) over the mod 2
*
Steenrod algebra Ci(I). If an associative H-space X(n) satisfies H (X(n);Z/2) = A(n), the mod 2
cohomology of its classifying space is isomorphic to the algebra of invariants of the canonical
GL^Z^-action on a graded polynomial algebra in n variables of degree 1.
1. Introduction. Dans cette note, on envisage de déterminer les entiers
n > 2 pour lesquels l'algèbre A(n) définie sous (1) peut être réalisée comme
algèbre de cohomologie modulo 2 d'un espace topologique. La Z/2-algèbre
graduée
(1)     A(n) = Z/2 [V1]/( v* ) <g> E(V2, V3,..., Vn-1) où degré(vk)= 2n-2n-k-l
est munie d'une unique structure de module sur l'algèbre de Steenrod Ql (2)
~n-l 19                                                                        9' 1               1
imposée par l'égalité Sqz    "1V1 = v    et les relations d'Adem Sq^"1 = Sq1
Sq2 ...Sq21"1. Par exemple, A(2) = H*(SO(3);Z/2) et A(3) = H*(G2;Z/2), où G2
est le groupe de Lie exceptionnel.
Les algèbres A(n) sont liées à certaines algèbres de Dickson D(n)
définies comme suit : le groupe linéaire GLn(Z/2) agit canoniquement sur
l'algèbre de polynômes graduée Z/2 It1,..., tn], où degre(tfc) = 1 (k = 1,..., n),
et par définition, D(n) est l'algèbre des invariants. Dickson a démontré que
D(n) = Z/2 [W1,..., Wn], où degré(wk) = 2n-2n_k. L'algèbre D(n) est un module
sur Qi (2) et la question suivante est naturellement liée au problème de
Steenrod:
Existe-t-il un CW-complexe Y(n) tel que H*(Y(n);Z/2) = D(n) ?
-1-
Les exemples connus sont : D(I) = H*(BZ/2;Z/2), D(2) = H*(BSO(3);Z/2) et
D(3) = H*(BG2;Z/2). Dans [ 6 ] L. Smith et M. Switzer montrent qu'un tel
espace n'existe pas si n > 6. Les cas où n = 4 et n = 5 sont plus délicats à
traiter.
Si Y(n) existe, on obtient facilement à l'aide de la suite spectrale
d'Eilenberg-Moore que l'espace des lacets £2Y(n) = X(n) vérifie :
(2)                                           H*(X(n);Z/2) = A(n).
La question précédente peut donc se reformuler en ces termes:
Existe-t-il un H-espace associatif X(n) vérifiant la condition (2) ?
J. Lin et F. Williams ont répondu par la négative au cas où n = 5 [4]. Leur
démonstration utilise la machinerie des opérations cohomologiques secondaires
et tertiaires. A l'aide de la K-théorie complexe, il est possible de démontrer,
sans faire appel aux opérations cohomologiques d'ordres supérieurs, le résultat
plus général suivant :
Théorème. Pour n > 5, l'algèbre A(n), définie sous (1), ne peut pas être
réalisée comme algèbre de cohomologie modulo 2 d'un espace topologique.
Notons que ce théorème exclut l'existence de X(n), pour n > 5, comme
espace topologique et non seulement comme H-espace mais qu'il n'apporte
aucune information au cas où n = 4 pour lequel il existe deux résultats
contradictoires [ 2 ], [ 5 ].
Le reste de cette note est consacré à l'esquisse de la démonstration du
théorème. Elle peut être résumée comme suit : si X(n) satisfait (2), on constate
d'abord que la cohomologie entière de QX(n) est sans 2-torsion, et on montre
ensuite, grâce aux résultats d'Ativan [ 1 ], que les opérations de Steenrod Sq* et
l'opération d'Adams 1F2 ne sont pas compatibles pour QX(n), où n > 5.
Nous n'expliciterons que le cas où n = 5, les autres se traitant de manière
identique. Dans la suite de cette note, nous supposerons que X est un CW-
complexe fini simplement connexe vérifiant :
(3)                          H*(X;Z/2) = Z/2 [x15]/( x'5 ) <8> E(x23, X27, X29)
.7-
Un tel complexe a le même type d'homotopie rationnelle que le produit de
sphères S15 x S23 x S27 x S59. Faisant appel à la technique du "mixing
homotopy types", nous supposerons également que, pour les premiers impairs
p, la cohomologie modulo p de X est isomorphe à l'algèbre extérieure
^Z/p(Z15' Z23' Z27' Z59^
Nous tenons à remercier J. Lin avec qui nous avons eu de nombreuses et
fructueuses discussions.
2. Calculs de cohomologie ordinaire.
Soit p^ : H (X;Z)-----> H (X;Z/2)   l'homomorphisme de réduction
modulo 2. A l'aide de la formule des coefficients universels on montre
l'existence d'éléments y15, y23, y27, Z59 6 H (X;2) pour lesquels p*(yk) = \
(k = 15, 23, 27) et P+(Z59) = x29-x25 ; de plus,
H*(X;Z) = P(Y15) <8> Ez/2 (y23, y27, Z59) / (y2$ <S> Z59),
où P(y, J = Z [y,,]/ ( y*    2 y2   ) = Z © Z © Z/2 © Z/2 est engendré
additivement par 1, y15, y25, y35.
Nous allons décrire la cohomologie modulo 2 de OX au moyen de la
suite spectrale d'Eilenberg-Moore. Notre référence, dans ce contexte, est le
livre de R. Kane [ 3, eh. Vu] dont nous adoptons aussi les notations. Le terme
E2 de la suite s'obtient à l'aide de la résolution bar BH (X; Z/2); dans notre
situation, on obtient
E2 = Torir(X;2/2)(Z/2,Z/2) s E(U14) ® Ru22, U26, U28U58),
où uk = sxk  (k = 14, 22, 26, 28) sont des éléments  "suspension" représentés
par [xk] et U58 est un élément "transpotent" représenté par [x   Ix  ] = [X15Ix  ].
Le degré total de tous les générateurs étant pair, on conclut que la suite
speciale est triviale,i.e. E2 s E00, et on obtient ainsi les isomorphismes additifs:
H*(ÛX;Z/2) = E°(H*(aX;Z/2)) = E(u14) <8> T(u22, U26, U28 U58)
Lemme 1.   Soit U56 l'unique élément non nul dans H56(QX;Z/2). Alors:
Sq2U56 * O.
Démonstration:         La suite spectrale d'Eilenberg-Moore est une suite de
modules sur Cl.(2) et dans Tor «,         (Z/2.Z/2) on a l'élément U56 = T2(U28) =
[X29Ix29] ainsi que les relations Sq2O29Ix29] =    £ [Sq1X29ISq-Jx29] = [x2 Ix2 ] =
i+j=2
U58 * 0.
Notons encore que la cohomologie entière de QX est sans torsion (fait résultant
des égalités H2n+1(ßX;Z/2) = 0, pour n > 0), et que l'on peut choisir des
générateurs vk e Hk(QX;Z) = Z (k = 14, 22, 26, 28, 56) tels que :
<4>                             P*(Vk ) = Uk  et  Vi24= V28'   Vi4 = VL= 2V56-
3. Calculs de K-théorie.
Une étude soigneuse de la suite spectrale d'Atiyah-Hirzebruch en K-
théorie mod 2 et entière pour l'espace X montre que, dans les deux cas, la seule
différentielle non-triviale est d3. On obtient ainsi E4 = E00 = E°(K (X)), et le
calcul explicite fournit que K (X) = E(C15, C23, C27 C59), les générateurs ÇK
satisfaisants aux conditions suivantes : dans l'ordre indiqué, les éléments ^15,
C23, C27, C59, 2C59 correspondent, dans le sous-quotient  E00 de H (X;Z), aux
éléments représentés par y15, y23 2y27, y,5-y27' z59; ^a filtrat*011 rationnelle de
ÇK est égale à k. On définit T|K e K °(SX) par
riK = G(^-1)        (k = 16, 24, 28, 60),
où o : K *(X) = K°(ZX) et on envisage de déterminer l'opération d'Adams Y2
sur ces éléments.
-4-
Pour tout CW-complexe Y et tout entier q > 0 soit F^K(Y) ç K(Y) =
K0OO le sousjgroupe des éléments de filtration > q, c'est-à-dire
F^K(Y) = Ker{K(Y)-----> K(Y<H)}. Les éléments J]K(k = 16, 24, 28, 60)
forment une base du groupe abelien libre K(ZX) / F62K(ZX).
Lemme 2. Pour un choix convenable der\2„ g K(ZX) on a dans
K(ZX)/F62K(ZX) :
(i)                                     4^(Ti16)EE 2T|28     (mod 4)
(ii)                                     ^2Ol28) - °            (mod 4)
Démonstration. Remarquons préalablement que ^2Cn) = T)2 = 0 (mod 2)
pour tout r\ e K(ZX). A homotopie près, il existe un sous-complexe
Z = S16 u e24 u e28   de ZX qui porte la cohomologie entière jusqu'à la
dimension 28. Soit i : Z------> ZX l'inclusion . La K-théorie de Z est libre à
trois générateurs, Cc16 = i (Tl16), cc24 = i (t|24) et Cc28, où 2a28 = i (Ti28). Par le
résultat d'Atiyah [ 1 ], on obtient vî/2(a16) = 28OC18 +24 a oc24 + 22 b (X28, où
a = b = 1 (mod 2), il suit que
(5)              ^2(ri 16) = 28Tj16+24 aii24+2b Ti28+ 2 CTi60.
Cette dernière égalité implique directement (i) dans le cas où c est pair; si c est
impair on remplace d'abord le générateur Ti28 par fj2g = Ti28 + Tj60.
Pour la démonstration de (ii), on note que ^111 (T]28) = m 14Tj28 + dm Ti60,
^(1W = m30T|60 et on considère l'égalité T3^(Ti28) = T2^(Ti28).
Remarque : dans le cas où n = 4 la relation 1F31F2 = Y2T3 implique que
d2 = 0 (mod 2) mais ne permet pas d'affirmer que ce coefficient est divisible
par 4.
Considérons maintenant l'application canonique e :   Z2QX-----» ZX. On
définit u.K g K(QX) (k = 14, 22, 26) par a2(|iK ) = s*(T|K+2) et on rappelle
que 2 o"2vF2(p.K ) = e*vF2(TiK+2). La cohomologie entière H*(QX;Z) étant sans
torsion, il en est de même pour K(QX) / F60K(QX). En calculant dans ce
dernier groupe on obtient du lemme 2 d'abord que \i~   = *Fz(u,14) = ]X26
o-
(mod 2) et ensuite que Ji^ = Y2^4 ) = ^2Cp76) = O (mod 2). Il existe donc un
élément co e K(QX) tel que
u*4 =2(û     (mod F60K(QX)).
La relation (5) implique que V2Ql    ) = 228 \i     (mod F60K(QX)) et on obtient
ainsi :
M^(CO) = 228 co    (mod F60K(QX)).
La suite spectrale d'Atiyah-Hirzebruch de QX en K-théorie est triviale, i.e.
H (QX; Z) = E°(K*(QX)). L'élément Ji14 représente V14, et il suit de (4) que co
représente V56 e H56(QX;Z). A l'aide du théorème d'Atiyah [ 1 ] on déduit que
Sq2U56 = O; ce qui contredit le lemme 1 et achève la démonstration du
théorème.
Références bibliographiques
[ 1] M. F. ATTYAH. Power operations in K-theory, Quart. J. Math. Oxford (2) 17 (1966), 165
- 193.
[ 2] W. G. DWYER et C. W. WILKERSON. A new finite loop space at the prime 2, Preprint.
[ 3] R. M. KANE. The homology of Hopf spaces, North Holland Math. Study # 40 (1988).
[ 4] J. P. LIN et F. WILLIAMS. On 14-connected finite H-spaces, Israel J. Math. 66 (1989),
274 - 288.
[ 5] J. P. LIN et F. WILLIAMS. On 6-connected finite H-spaces with two torsion, Topology
28 (1989), 7 - 34.
[ 6] L. SMITH et R. M. S WTTZER. Realizability and nonrealizability of Dickson Algebras as
cohomology rings, Proc. Amer. Math. Soc. 89 (1983), 303 - 313.
Institut de Mathématiques et d'Informatique,
Université de Neucliâtel, Chantemerle 20, CH-2000 Neuchâtel, Suisse.
-6-
Algebras over the Steenrod Algebra and
Finite H-Spaces
Alain Jeanneret
0 . Let p be a prime and P = Z/p [x^,..., Xn] be a graded polynomial algebra
over the mod p Steenrod algebra Ci (p). A fundamental question in algebraic
topology is to decide whether P can be realized as the mod p cohomology of a
topological space. Ideas from the Galois and invariant theories provided a
complete answer to the realization problem for the non - modular case (i.e. when
deg(xj) ^O (mod p), i = 1,..., n) see [A-W]. In contrast the modular case is still
not settled.
An interesting family of modular polynomial algebras at the prime 2 is
provided by the so called Dickson algebras defined as follows :
The canonical GL(q;Z/2) - action on (ZfIy induces a GL(q;Z/2) - action on
H*(B(Z/2)q;Z/2) = Z/2[t!,..., tq], (deg(ti) = 1, i = 1,..., n). As the Steenrod
algebra acts via the squaring map (which is linear in characteristic 2), the actions
of W = GL(q;Z/2) and Cl (2) commute. The Dickson algebra is the invariant
algebra
D(q) = H*(B(Z/2)q;Z/2)W
which is again an algebra over Cl(2). Dickson has shown that :
D(q) = Z/2[wi,..., wq], (UCg(W1) = 2q-2q_1).
The Cl (2) - action on D(q) can be described as follows : Sq^ w^ = W^+ \
(k = 1,..., q-1). A natural question is therefore :
Does there exist a topological space Y(q) such that H (Y(q);Z/2) = D(q) ?
The well known classical examples are : D(I) = H (BZ/2;Z/2), D(2) =
H*(BSO(3);Z/2) and D(3) = H*(BG2;Z/2) where Go denotes the exceptional
Lie group. L. Smith and R. Switzer have proved in [S-S] that Y(q) does not exist
for q > 6.
Let E(xj,..., xn) denote the exterior algebra on x\,..., xn and let
A(q) = Z/2 [Vl]/( Vj ) ® E(V2, V3 ,..., Vq-1)    (deg^)= 2q-2q_1-l)
be an algebra over Cl(2). The action of the Steenrod algebra is given by
oq-k-l                                             I              2
Sq          vk = vk+l  (k= 1V- q^.Sq1 Vq-1 = V1.
Standard methods show that the cohomology ring of X(q) = ^Y(q) satisfies
H*(X(q);Z/2) = A(q). For n = 5, U. Suter (unpublished), J. Lin and F. Williams
[L-W] have shown that X(5) cannot support an H - structure. This implies the
non realizability of D(5) as the cohomology algebra of a topological space. In
[J-S], U. Suter and the author proved a stronger version of the result of Lin -
Williams and Smith - Switzer, namely :
For q > 5 the algebra A(q) cannot be realized as the mod 2 cohomology
algebra of a topological space.
The purpose of this note is to give a proof of a further generalization of
the last result. Set
A1 =A(q)        (i = l,..., m)
Bj = E(Wj)     (deg(Wj) = 2nJ-l, rij > q-1, j = 1,..., n)
m                n
K(q, m, n) =   ®   A\   ® ®   Bj    (m>l,n>0).
i=l           j=l
m
We assume that  ®   Ai is an Cl (2) - subalgebra of K(q, m, n) and that Sq w- =
i = l
Sq2W-= 0 (j = 1,..., n).
Theorem. For q > 5 the 0.(2) - algebras K(q, m, n) cannot be realized as the
mod 2 cohomology of topological spaces.
Remark. The hypothesis on the Ci (2) - action on K(q, m, n) is not essential (see
the final remark at the end of this note).
This result is related to the question of connectivity of finite H - spaces in
the following way. Let us first recall a result of J. Lin :
Theorem.(see [L]) Let X be a 14 - connected finite H - space with H*(X;Z/2)
associative, then Ç(H15(X;Z/2)) * 0 where % : Hn(X;Z/2)-----> H2n(X;Z/2) is
the squaring map.
The algebras K(5, m, n) satisfy the condition Ç(K(5, m, n) ) * 0, so
they are not ruled out by Lin's theorem. Our methods show however that they
cannot even be the cohomology algebras of topological spaces.
We shall give the proof of the theorem for q = 5 only, the other cases can
be treated exactly in the same way.
I am grateful to U. Suter for his guidance and his help. I would also like
to take the opportunity to thank J. Lin for the many useful discussions I had with
him.
Notations. Throughout the paper we set
A1 = 2/2 [ x^]/((X^)4) ® E(X^, x^, x^ )      (degtx^) = k, i = 1,..., m)
Bj = E(Wj)       (deg(wj) = 2nJ-l, nj > 4, j = 1,..., n)
m                n
K =   ®   Ai   ® ®    Bj
i=l            j=l
We assume that there exists a simply connected CW - complex X such that
H (X;Z/2) = K. We shall show that the Adams operations and the action of the
Steenrod algebra are not compatible on QX. This will rule out the existence of
such a space.
1. In this section we shall describe the cohomology of QX, the loop space
on X, and give some results on the Zr>\ - cohomology of X where Zn) stands
for the ring of integers localized at the prime 2.
Our main device to obtain the mod 2 cohomology of QX is the Eilenberg-
Moore spectral sequence (E-Mss for short). Though we adopt the notation of [K]
let us recall briefly some properties of the E-Mss. It is a second quadrant spectral
sequence of Cl(2) - modules with
E2 = TorH*(X.z/2)(Z/2,Z/2) and E00 = E°(H*(QX;Z/2))
(E (H) is the graded module associated to a filtration of H). Let H denote the
algebra E(x) or Z/2 [x] / ( x ), and let H denote the augmentation ideal of H. A
straightforward computation, using the bar construction, shows that :
Tor      (Z/2,Z/2) = T(sx) and TorZ/2         4 (Z/2,Z/2) = E(sx) ® T(tx)
The element sx, called suspension element has bidegree (-1, deg(x)) and is
represented (via the bar construction) by [x] e H, whereas tx, called transpotence
2   ?
element has bidegree (-2, deg(x)) and is represented by [x lx~] or by
[xlx ] e H ® H (these two cycles differ from a boundary and therefore represent
the same element in the E2 term). As usual T(u) is the divided polynomial algebra
on u. We also mention the following property of Tor :
TorN 0 M(Z/2,Z/2) = TorN(Z/2,Z/2) ® TorM(Z/2,Z/2).
The E-Mss for X can then be computed. We obtain easily :
m                n
E9 = Tor  (Z/2,Z/2) = ®    Ci   ® ®    Dj
1= 1            j = l
where
CiSE(u®)®r(u«u«u«u«)   (i«l.....m)
Dj = T(Vj)        G = l.-.n)
with ufn = Sx^+1   (n = 7, 11, 13, 14), u^ = tx(^ and Vj = sWj.
The differentials of the E-Mss raise the total degree by one. As E2 is
concentrated in even total degree the spectral sequence is trivial, i.e. E2 = E00,
and so we get the additive isomorphisms :
m                n
H*(ßX;2/2) = E°(H*(QX;Z/2)) = ®    Ci   ® ®    Dj.
1=1           j=l
For each i = 1,..., m there is an element u,;. e H    (QX;Z/2) represented
by [x® Ix^ ] = 72^2^ in E°(H*(ßx;^/2)) = E2- As Cl(2) acts on E2 via the
2
Cartan formula, one gets (with some abuse of notations) that   Sq U55 =
?                                                        2    ^                3
Sq [X29Ix29] =   £ [Sq1X29ISqJx29] = [X15Ix".] = [X15Ix15] = u5g * 0. So we
i+j=2
have proved the following result :
Lemma 1. The element vQ e H56(QX;Z/2), represented by [x(^ lx(^ ] in
E°(H*(QX;Z/2)) (i = 1.....m) satisfies :
Sq2u(j>  * 0.
56
An immediate consequence of the fact that H~n+ (QX;Z/2) = 0 is the next
lemma. Let p* : H (X;Z/2))-----> H (X;Z/2) denote the modulo 2 reduction
homomorphism.
3fC
Lemma 2.   The algebra H (QX; Zz2O is torsion free and there are generators
s^ e Hk(QX;Z(2)) (i = 1,-, mandk = 14, 22, 26, 28, 56) with
and there exist t- € H2    \QX;Zq\) (j = 1,..., n) with p^tj) = v- .
Proof. The only point to be checked is the assertion concerning the
multiplicative structure. As the E-Mss is a specral sequence of 0.(2) - modules,
we obtain from the 0.(2) - action on K :
Therefore we can choose s, ., s\0   (i = 1,..., m) such that    (s, ^)    = s~0  and
14     Zo                                                    14             Zo
(SW)4 = (5¾2 = 0 mod 2. Using the Hopf algebra structure on H*(QX;Z(2))
and the fact that s . is primitive, one can show there exists s _, such that (s ) =
(S^)2 = 2S^. Moreover p,(s^) = u^ (i = 1,..., m).
We end this section with a description of the Zn) - cohomology of X.
Using the universal coefficient formula and the fact that Sq detects Z/2 -
summands, one shows the existence of free generators y,     e  H (XjZz2))
(i = 1,..., m; k = 15, 23, 27, 59) and z- e H2^(XjZ (2))    (j = 1,-, n) such
that:
p*(yk)} = Xk}    (k = 15, 23' 27)' p*(y59} = X29 "(x15)2 and P*(ZJ} = wj:
moreover we have the isomorphism of Zp) - modules :
h*(X;z(2))S ® {p(y(i5)®Ez/2^y(2r y&>n<y®)2®y%))
i = 1
where P(y(^ ) = Z(2)[y(^ ]/( (y(^ )4, 2 (y(^)2) = Z(2) © Z(2) 8 Z/2 © Z/2
is additively generated by 1, y , (y ^) , (y ) . Note in particular that the two
torsion of H (X;Z(2\) has only order two.
2.       This section contains the computation of the K - theory of X. Let us first
fix the notation and recall some basic facts about the K - theory with coefficients.
If Y is a finite CW - complex, K (Y) denotes the Z/2 - graded ordinary complex
K - theory [A-H]. For G = Zrn or Z/2 the K - theory with coefficients in G,
denoted by K (Y;G), is defined in [A-T]. K (Y;G) is again a Z/2 - graded
algebra. Moreover it satisfies K °(Sn;G) = 0 if n is odd and K °(Sn;G) = G if n is
even.
The Atiyah-Hirzebruch spectral sequence (A-Hss for short) is a first and
fourth quadrant spectral sequence of algebras with
E2(Y;G) = H*(Y;K*(pte;G)) and E00(YiG) = E0^(YiG)).
The modulo 2 reduction homomorphism induces a morphism of spectral
sequences     p* : Er(Y;Z(2))------> Er(Y;Z/2). Finally, for G = Z/2 the
Z/2
differential d„    is given by
df2 = Sq2Sq1 + Sq1Sq2.
Our first step towards the determination of K (X;Z/2\) is the computation
of K*(X;Z/2). Recall that in this caseE9 = E3 = H*(X;Z/2). Using the Ci (2) -
action on K ( see section 0) we infer that :
d3(x(^) =d3(x^) = d3(x^) = 0andd3(x^) =(x®)2 (i = 1,.., m)
(D
(I3(Wj) = O G = I,-, n).
As d3 is a derivation we readily get :
E4 = ®   E(x(^, x^, x^, xf7( xf5)2 ) ® ®   E(Wj)
i= 1                                              j = 1
where x^ still denotes (for simplicity of notation) the element OfE4 represented
by Xi^ e ker d3 c E3. For dimensional reasons (dim.-?,- E4(X;Z/2) = dim^
H (X;Q)) the spectral sequence collapses after E4 , i.e. E4 = E00 and hence
K*(X;Z/2) = ®   E(G^, 9^, 9^, 9(^) ® ®   E(Tj )
i= 1                                     j = 1
where %¦ and 9// are represented by the obvious classes in E (K (X;Z/2)) = E4 .
From the universal coefficient formula for K -theory and the observation that
7k                                                                       ik                                                                                                ^k
dim2/2 K (X;Z/2) = dim^ H (X;Q) we conclude that K (X; Z ^2)) is torsion
free.
We are now ready to calculate the K-theory of X with Z(2\ - coefficients.
The modulo 2 reduction homomorphism p^ : H (X;Z/2\)-----> H (X;Z/2) is an
injection on the torsion subgroup (see the end of section 1) and since image(dj.) c
torsion(Er) we obtain using (1) that
d3(y(^) = d3(y(^) = d3(yg) = 0, d3(y(^) = (y(^)2 and d3(Zj) = 0.
We then compute (2) :
m
E4KZp,) . 8   VZ(2)<.y% y« 2y« )®EZ(2)(y®, y®, y«) y®
Observe that p* : E4(XjZr)))-----> E4(X;Z/2) is again injective on the torsion
2/2
subgroup. Since d      = 0 for r > 4 we infer that dr = 0 for r > 4, and therefore
(3)                  E4(X;Z(2)) s E00(X^2)) = E°(K*(X;Z(2))).
Before giving the explicit description of K (X;ZnO we recall some relations
between the K - theory and the rational cohomology.
Let Ç e K*(Y;Z(2)) and let ch(Ç) =   Ech (C), eh (Ç) € H^(XjQ), be
q>0
its Chern character.
Definition. The rational filtration of ^ rat.fïlt.(Ç), equals p if ch (C) = 0 for
q = 0,..., p-1 and ch (C) * 0. //£, w a torsion element, we set rat.filt.(£) = «>.
For ^eK (Y;Zn\) we denote by che the first non vanishing component of
ch(Ç), /.e. che = eh (¾ w/r/i p = rat.fïlt.(Ç).
The following result is not difficult to prove :
Lemma 3. Let Y be a finite CW - complex. There exist free elements £},..., £n
providing  a basis of K (Y;Zn)) I (torsion) such that cht    .., ch*   is a
homogenous basis ofH (X;Q).
From (2), (3) and lemma 3 we obtain the main proposition of this section.
Proposition. There exist elements C^, Ql, ^2T ^59   ^ = 1^"' m^ and vi
(j = 1,.., n) such that :
K*(X;Z(2))=  ®     E(^, ^, ^, ^) ® ®   E(Vj)
i=1                                         j = 1
where rat.filt.^^) = k and rat.filt.(v-) = 2nJ-l. Moreover Z®  ^   ^  ^,
2^9 are represented by y ^5, y ^3, 2y^?) (y ^5) .y^ y ^9 whereas Vj /5
n    *
represented by z- in E (K (X; Z ^2))).
3 .       This section contains the description of the action of the Adams operations
on some generators of K(ZX;Z/2)) and the proof of the main theorem. As we are
going to use it many times, we first recall a generalisation of an important result
of Atiyah [At].
Let Y be a finite 2 - torsion free CW - complex (i.e. H (Y; Zz2)) without
2 - torsion). The submodule of elements of filtration > 2q, V2 K(Y;Z/2)) ç
K °(Y;Z(2)) = K(Y;Z(2)) is defined by T2qK(Y;Z(2)) = Ker{K(Y;Z(2))----->
rC(Y(2n-l);Z^pj   where Y(2n_1)   is the (2n-l) skeleton of Y. As Y is 2 -
torsion free the A-Hss with Z^2) - coefficients collapses and we may identify
H*(Y;Z/2) with E°(K*(Y;Z(2))) ® Z/2; if a e V2 K(Y;Z(2)) we denote the
corresponding element of            ;Z/2) by CC. We are now ready to state the
Z/2) - localized version of Atiyah's result
Theorem.(see [H]) Let Y be a finite, 2 - torsion free CW-complex and a s
F20K(YjZz2)). Then there exist elements oc- e F2Q+2jK(Y;Z/2))   (i = 0,..., q)
such that
0                q
¥ (a)=  1     2¾-1 Ct1.
i = 0
Moreover CX.- = Sq21 ä for i = 0,..., q.
In order to state a first lemma about the Adams operations we define T]   e
K(EX;Z(2)) by Tl(kl} = 0(^1)   (i = 1,..., m, k = 16, 24, 28, 60) and Cj e
K(IX;Z(2)) by Cj = a(Vj) (j = 1,..., n), where a : K 1CXjZ(^) = K °(EX;Z(2))
is the suspension isomorphism.
Lemma 4.       There exist a choice of generators Ç  -, £,*  eli. (i = 1,..., m)
such that in K(EXjZo)) I F-^K(EXjZr?)) the following relation holds :
^(71^) = 2^28     (mod 4).
m
Proof. Up to homotopy there is a subcomplex Y = x      {Su e     u e    }•
i = 1
of IX (and inclusion i : Y-----» EX) which carries the cohomology with Zr?) -
coefficients of EX up to dimension 28. The K - theory of Y is free on generators
ß16 = ^16^ ß24 = ^245 and ß28' where 2ß28 = ^285   (i = 1^"" m)-
The last equality comes from the fact that the A-Hss of Y collapses, whereas for
EX we have d3(y^) = (y^ ^)   (i = 1,..., m). Atiyah's result now implies :
*F2(ß«) = 2* ßO) +2« a® ß® + 22 b® p®       (i = 1,..., n),
where a^1' and b^1' = 1 (mod 2). So we obtain :
(4)             ^(71^) = 2871¾ +24ti®+271® (modF32K(IX;Z(2))).
It is clear that the lemma follows immediatly from (4).
The second lemma about the Adams operations concerns the K - theory of
Q.X. Let us denote by e : I QX-----> IX the suspension of the evaluation map
and define \l® e K(QX;Z(2)) W ^Jf) = e*(T1k+2} (i = 1^"' raandk= 14'
22, 26) and     A.- e K(QX;2(2)) by a2(k-) = e*(v-) (j = 1,..., n).
Lemma 5.  In K(QXjZz2)) I F^qK(QX; Zz2O the following relations hold :
(I)                 YV2^)=O (mod 2)           (i=l,...,m)
(ii)                 ¥2(h) =0   (mod 2) (for each] such that rij = 5).
Proof. Recall first that the suspension of decomposable elements is in the kernel
of £*, therefore we obtain in K(I2QX;Z(2)) /f 62K(Z2QX;Z(2)) (for i = 1,...,
m and j such that n: = 5) :
*V(ïl®)) = P1V(^)+X^ e*(Vj) + i b<j'k) e*(ng>
<Pp(e*(Vj)) = p16e*(Vj) +X c£'k) e*(il*V
^V(Tl®)) = P30e*(Tl®)
Equating the coefficient of e*(Ti% in 4/3vF2(e*(Vj)) = ¥2vF3(e*(Vj)) gives :
(5)                   c^'k) = 0 (mod 212)     (k = 1,..., m andj such that nj = 5).
The same method applied to the coefficient of £*(v-) in ¥J¥^(£*0ry ' ))
^¼¾*^)) implies :
(6)                   a^ =0 (mod2U)     (i = 1,..., m and j such that nj = 5).
Equalities (5), (6) and the coefficient of e*0i% in W3W2 (e* (T]^l)) =
OU                                          2 Q
^V(e*(T]^)) give :
(7)                 b£,k) = O (mod 25)       (i and k = 1,..., m).
In view of (5), (6) and (7) we have (for i = 1,..., m and j such that n: = 5) in
K(Z2aX;Z(2))/f62K(L2QX;Z(2)) :
(8)                1FV(Tl^)) = ^V(Vj)) EE O (mod 4).
Recall that the Adams operations do NOT commute with the Bott periodicity
[Ad], the actual relation is :
4/2(G2(a)) = 2a2vF2(a).
As K (QX]Zn)) is torsion free (by lemma 2 H (QX]Zn)) is torsion free and
thus the A-Hss is trivial), the relation (8) forces ^¾¾ = ^2Ck-) = O (mod 2)
in K(QX]Zn)) /T^qK(QX]Zq)) and the lemma is proved.
The last part of this section is devoted to the proof of the main theorem. In
K(QX;Z(2)) I F30K(nX;Z(2)) we have by lemma 4 :
(9)                           0¾2 ^ ^¾¾ = \if6    (mod 2) (i = 1,..., m).
Equality (4) can be writen more precisely as :
^V(^))=28e^Ti^) + 24e*(Ti®)+2e*(^i)8) +1   ^W-)
(10)
m     ...        ...                  _
+ 1   ^^(¾)   (m°dF62K(L2QX;Z(2))).
Therefore congruence (9), equality (10) and lemma 5 imply that :
([if/ = W2(([if4)2) = V2(V®6) = 0 (mod 2)         (i = 1,..., m)
Où
(i)
in K(QX;Z/-2")) /F^qK(QX;Z (2))- So, there exists an element
K(QX;Z(2))  (i = 1,..., m) such that 0¾4 = 2co(i) (mod F60K(QX;Z(2))).
With (4) or (10) we get :
¥2(a>(i) ) s 228 <o(i)   (modF60rC(nX;Z(2))).
As the A-Hss for QX collapses, H*(nX;;Z(2)) = E°(K*(QX;Z(2))) and thus
H1^ is represented by S1^ (i = 1,..., m), it follows from lemma 2 that ctr1^ is
represented by sii e H (QX;Zo)) (i = 1,—, m). Using one more time
Atiyah's result (for the element co'1-' ) we obtain that Sq uL = 0 for each i =
1,..., m which contradicts lemma 1 and thus proves the main theorem.
Final remark. As mentioned in the introduction, the hypothesis on the
Cl (2) - action on K is not essential. In fact every 0.(2) - action on K can always
be described in the following way :
The relations Sq15X^ = (x^)2 (i = 1,..., m),  Sq15 = Sq1Sq2Sq4Sq8 and the
(i)  7
linear independence of {(x!j ') : i = 1,..., m} allow us to change (if necessary)
the generators x!l^, xi-, xlL such that the following Steenrod connections hold :
SqS x® = X®, Sq«x® = x®, Sq2x® = xg, Sq'x« = (*f/ (i = 1,..., ra).
m
Next let h : X-----> n   K(Z/2;15) be the map such that in mod 2 cohomology
i=l
h (I1J = X1 _ (i = 1,..., m) (i     beeing the canonical generator of the i    factor
m
of   (g)   H (K(Z/2;15);Z/2)). It is not difficult to prove that h   factors through
i=l
m
®   Ai :
i = 1
m
®   H*(K(Z/2;15);Z/2)) ----------->   H*(X;Z/2)
i = l
Therefore with the above choice of generators, Aj (i = 1,..., m) becomes an
1               9
0.(2) - subalgebra of K. Moreover it is easy to show that Sq w- = Sq w- = 0 for
each j such that n- = 4, 5. As the argument to prove the theorem does not imply
filtrations greater than 58 in K (£2X;Zrr)), it is possible to prove it without any
hypothesis on the 0.(2) - action on K. This can be achieved in the following way :
Let us perform the changes mentioned above. Lemma 1 and 2 can then be
obtained similary. Next, we consider a subcomplex Y of X which carries the
Z n) - homology of X up to dimension 61 (hmology approximation). The
inclusion Y-----> X induces a mod 2 cohomology isomorphism up to dimension
62 (note that H6 (X;Z/2) = 0). One checks then that the results of the final pan
of section 1, concerning the Zn) - cohomology of X, persist up to dimension
62. The argument in K - theory is valid for Y in place of X.
REFERENCES
[Ad] J.F. ADAMS. Vector fields on spheres, Ann. of Math. 75 (1962), 603 -
632.
[A-W] J.F. ADAMS and CW. WILKERSON. Finite H - spaces and algebras
over the Steenrod algebra, Ann. of Math. 111 (1980), 95 - 143.
[A-T] S. ARAKI and H. TODA. Multiplicative structures in mod q cohomology
theories I, Osaka Math. J. 2 (1965), 71 - 115.
[At] M.F. ATIYAH. Power operations in K-theory, Quart. J. Math. Oxford (2)
17 (1966), 165 - 193.
[A-H] M.F. ATIYAH and F. HIRZEBRUCH. Vectior bundles and homogenous
spaces, Proceedings of Symposia in Pure Mathematics 3 A.M.S. (1961), 24-51.
[H] J.R. HUBBUCK. Generalized cohomology operations and H - spaces of
low rank, Trans. Amer. Math. Soc. 141 (1969), 335 - 360.
[J-S] A. JEANNERET and U. SUTER. Réalisation topologique de certaines
algèbres associées aux algèbres de Dickson, Preprint.
[K] R.M. KANE. The homology of Hopf spaces, North Holland Math. Study #
40 (1988).
[L] J.P. LIN. Steenrod connections and connectivity in H - spaces, Memoirs
Amer. Math. Soc. # 369, 1987.
[L-W] J.P. LIN and F. WELLIAMS. On 14-connected finite H - spaces, Israel J.
Math. 66 (1989) 274 - 288.
[S-S] L. SMITH and R.M. SWITZER. Realizability and nonrealizability of
Dickson Algebras as cohomology ring, Proc. Amer. Math. Soc. 89 (1983), 303 -
313.
Institut de Mathématiques et d'Informatique
Université de Neuchatel
Chantemerle 20
Ch-2000 NEUCHATEL
Switzerland
Liste des publications
1.  Réalisation topiologique de certaines algèbres associées aux algèbres de Dickson
2.  Algebras over the Steenrod algebra and finite H-spaces
Le texte complet de cette thèse a été déposé à la bibliothèque centrale de l'Univer-
sité de Neuchâtel ainsi qu'à la bibliothèque de l'institut de mathématiques de l'Uni-
versité de Neuchâtel .
Neuchâtel le 26-6-92.