H-SPACES WITH NOETHERIAN MOD TWO COHOMOLOGY

                                   ALGEBRA



                       CARLOS BROTO AND JUAN A. CRESPO



       Abstract.  The object of this paper is to analyse the structure of conne*
 *cted H-spaces
       with noetherian mod two cohomology algebra. We will show that, up to 2-c*
 *ompletion,
       they are, essentially, finite mod 2 H-spaces and their 3-connected cover*
 *s, CP 1, BZ=2r
       and certain extensions of these.



                                 1. Introduction


   A subject of great interest in algebraic topology is the understanding of th*
 *e homotopy

theoretic generalizations of the concept of compact Lie group. Among those, fin*
 *ite H-

spaces or loop spaces and the localized versions, mod p finite H-spaces and the*
 * newer

concept of p-compact group ([11]).

   A finite H-space is an H-space whose underlying space is homotopy equivalent*
 * to

a CW-complex with a finite number of cells.  In the localized version, a mod p *
 *finite

H-space stands for an H-space which is finite up to p-completion or equivalentl*
 *y for

an H-space which mod p cohomology ring is finite dimensional.  By p-completion *
 *we

understand Bousfield-Kan p-completion [5]. An H-space, being simple, it is p-go*
 *od in

the sense of Bousfield-Kan, so we will assume without loose of generality that *
 *our mod p

H-spaces are p-complete.

   Other related spaces that play an important role are the three connected cov*
 *ers of

compact Lie groups, finite H-spaces or mod p finite H-spaces. In fact, those co*
 *nnected

covers carry most of the homotopy theoretic structure of the original finite H-*
 *spaces as,

for instance, higher homotopy groups. Furthermore, recently it has been discove*
 *red that

most of the times they even recall the p-completed homotopy type of the origina*
 *l finite

H-space.  More precisely, in [9] Dror Farjoun gives a nice construction of loca*
 *lization

functors with respect to maps. Concretely the nullification functor for BZ=p, L*
 *BZ=p, was

investigated by Neisendorfer [23] who defined F as the composition of the funct*
 *or LBZ=p

and the Bousfield-Kan p-completion and proved that for a 1-connected finite com*
 *plex

X with finite ss2(X), if X<n> denotes the n-connected cover of X, then F (X<n>)*
 * ' Xbp

for any positive integer n.

   The three connected cover of a finite H-space is not homotopy equivalent to *
 *a finite

CW-complex.  In fact the mod p cohomology ring is no longer finite.  It is howe*
 *ver a

noetherian ring.  The universal cover of a mod p finite H-space X is again a mo*
 *d p

finite H-space X" ' X<1>.  X" is thus 2-connected and its third homotopy group *
 *is

torsion free ([8]).  In order to construct the 3-connected cover of X, we choos*
 *e a map


  _____________
  The authors are partially supported by DGICYT grant PB94-0725.
                                         1




2                        CARLOS BROTO AND JUAN A. CRESPO


X" ! K(^Zpm; 3) that induces an isomorphism between the three dimensional homot*
 *opy

groups and the 3-connected cover X<3> is defined as the homotopy fibre of that *
 *map,

thus it fits in a principal fibration


                             ((CP 1)bp)m ! X<3> ! "X :


A spectral sequence argument shows now that the mod p cohomology ring of X<3> is

not finite but finitely generated; that is, noetherian.  Other H-spaces with no*
 *etherian

mod p cohomology ring are (CP 1)bpand BZ=pr for any positive integer r.

   Our aim is to prove that those H-spaces are essentially all mod p H-spaces w*
 *ith

noetherian mod p cohomology ring.  In this paper, both for clarity and simplici*
 *ty we

concentrate in the case p = 2.  The necessary changes for the odd prime case wi*
 *ll be

considered in a forthcoming paper. So, at prime two, we obtain


Theorem 1.1.    Let X be a 1-connected mod 2 H-space with noetherian mod 2 coho-

mology algebra, then there exists a mod 2 finite H-space F  = F (X) and a princ*
 *ipal

H-fibration


(1)                          ((CP 1)b2)n ! X ! F (X) :


   Thus we easily obtain a cohomological characterization of three connected co*
 *vers of

mod 2 finite H-spaces


Corollary 1.2.   A mod 2 H-space is the three connected cover of a mod 2 finite*
 * H-space_

if and only if its mod 2 cohomology ring is three connected and noetherian.    *
 *      |__|


Corollary 1.3.   All 1-connected mod 2 H-spaces with noetherian mod 2 cohomolog*
 *y ring     __

are finite mod 2 H-spaces, (CP 1)b2, products of those and extensions of the fo*
 *rm (1).   |__|


   This answers a question of Lin [18, Question 2.4], at prime 2.

   The general case is reduced to the simply connected case by taking the unive*
 *rsal cover

of our H-space X:

(2)                              "X! X ! Bss1(X)


Corollary 1.4.   All connected mod 2 H-spaces with noetherian mod 2 cohomology *
 *ring

are finite mod 2 H-spaces, (CP 1)b2, BZ=2r for any positive integer r, products*
 * of those_

and extensions of the form (1) or (2).                                         *
 *   |__|


   A connected mod p loop space X is a triple (X; BX; e) where X is connected a*
 *nd p-

complete space, BX a 1-connected and p-complete space and e is a homotopy equiv*
 *alence

e: X ! BX.  A connected mod p loop space X is a mod p finite loop space or p-

compact group if H*(X; Fp) is finite dimensional ([11]). Theorem 1.1 and coroll*
 *aries 1.2,

1.3 and 1.4 remain true for mod 2 loop spaces instead of mod 2 H-spaces because

according to [9] the nullification functor LBZ=2 and hence F preserves the loop*
 * structure.

   We can also use Theorem 1.1 in order to reduce questions about H-spaces or l*
 *oop

spaces with noetherian mod 2 cohomology to finite ones. As an example we can ea*
 *sily

obtain the classification of homotopy commutative mod 2 H-spaces with noetherian

mod 2 cohomology, based in the corresponding result for mod 2 finite H-spaces (*
 *[12],

[17]),




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                    3


Corollary 1.5 (Slack [25], Lin-Williams [20]).Let X be a homotopy commutative c*
 *on-

nected mod 2 H-space with noetherian mod 2 cohomology. Then X is the direct pro*
 *duct

of a finite number of Eilenberg-MacLane spaces

                         K(^Z2; 2);   K(^Z2; 1);   K(Z=2r; 1)

for r  1.


Proof. Let "Xbe the universal cover of X. It is as well a homotopy commutative *
 *H-space

with noetherian mod 2 cohomology and now Theorem 1.1 applies. X" is the total s*
 *pace

of a principal fibration ((CP 1)b2)n ! "X! F (X") : where F (X") is a 1-connect*
 *ed mod 2

finite H-space and, using the properties of F , also homotopy commutative. Henc*
 *e the

classical torus theorem of Hubbuck ([12], [17]) implies that F (X") is contract*
 *ible and

therefore X" ' ((CP 1)b2)n.

   We have obtained a covering ((CP 1)b2)n ! X ! Bss1(X) which is a simple fibr*
 *ation

an then it should be classified by an H-map Bss1(X) ! K(^Zn2; 3). Moreover, thi*
 *s map

should represent a primitive class in the cohomology of Bss1(X) (cf. [26]).

   But ss1(X) ~=Z^r2x Z=2k1x : :x:Z=2ks, a finitely generated ^Z2-module, has n*
 *o non

trivial primitives in its 2-adic three dimensional cohomology, hence this class*
 *ifying map

is trivial and then,
                                                                 __
                X ' ((CP 1)b2)n x BZ^r2x BZ=2k1x : :x:BZ=2ks:   |__|



   Our starting point is the classification by Aguade, Broto and Notbohm [1] of*
 * the

possible p-completed homotopy types of spaces having a mod p cohomology ring li*
 *ke

that of the three connected covering of S3,

(3)                               P [x2p]  E[fix2p]

where subscripts denote degree of the generators. Also, they proved in the fina*
 *l section

that, among those spaces, the only one that admits an H-space structure is p-co*
 *mpletion

of the true three connected cover of S3: S3<3>bp.

   Later, in [2], it was considered the case of spaces with mod p cohomology is*
 *omorphic

to the algebra

(4)                           P [x2p2]  E[y2p+1; z2p2+1]

This is the cohomology of SU(3)<3> for p = 2, Sp(2)<3> for p = 3 and G2<3> if p*
 * = 5.

A homotopy uniqueness result is obtained in that case for every prime, even wit*
 *h no

H-structure assumption.

   Our observation is the relevance in the proof of the above results of the fa*
 *ct that the

considered algebras, (3) and (4), are noetherian.

   There is one further observation about the Steenrod algebra action on the al*
 *gebras (3)

and (4) and other cohomology algebras of three connected covers of H-spaces (cf*
 * [13]).

As the degree of a polynomial generator increases there are more and more nilpo*
 *tent

generators attached to that polynomial generator by means on Steenrod operation*
 *s.

For example, in (4) at prime 2 we have Sq1x8 = z9 and Sq4y5 = z9.  Actually, th*
 *is

observation goes back to [3] where it is shown that the three connected cover o*
 *f Sp(k)

has one polynomial generator of degree 2pi, for k  = (pi-1 + 1)=2, together wit*
 *h a




4                        CARLOS BROTO AND JUAN A. CRESPO


number of exterior generators. In [1] it is shown that an algebra like (3) woul*
 *d not be

the cohomology of an H-space if the degree of the polynomial generator was larg*
 *er than

2p (or rather, of any space if it was larger than 2p(p - 1)).

   Within the proof Theorem 1.1 we find an explanation for this fact, actually,*
 * for a more

general class of H-spaces. Namely, H-spaces X satisfying the finiteness conditi*
 *ons:

(F1)  H*(X; F2) is of finite type.

(F2)  H*(X; F2) has a finite number of polynomial generators.

(F3)  The module of the indecomposables QH*(X; F2) is locally finite as module *
 *over

      the Steenrod algebra.

   Recall that a module over the Steenrod algebra is called locally finite prov*
 *ided any

submodule generated by a single element is finite (cf. [24]).  We will denote b*
 *y Athe

mod two Steenrod algebra. It is generatednby the Steenrod squares Sqi, i  0, su*
 *bject

to the Adem relations.  The squares Sq2  for a system of algebra generators.  W*
 *e will

denote
                                          n-1      2  1
                            Sqn  = Sq2nSq2   : :S:q Sq


for n  0 and formally Sqr  = 0 if r < 0. With this notation we can express the *
 *mod 2

cohomology of B2Z=2 as the polynomial algebra


                     H*(B2Z=2; F2) ~=F2[; Sq1; : :;:Sqn  ; : :]:


where  2 H2(B2Z=2; F2) is the fundamental class.


Theorem 1.6.    For any mod 2 H-space X satisfying the conditions F1, F2 and F3

and any polynomial generator x 2 H*(X; F2) of degree deg x > 1, there exists a *
 *finite

subquotient of H*(B2Z=2; F2) of the form, either

                   D                m            m              mnE
              Min=  Sqn  ; (Sqn-1 )2 1; (Sqn-2 )2 2; : :;:(Sq1)2
                                                                    F2

with n  0, m0 = 0, m1 = 1 and mk-1  mk  mk-1 + 1 or

                  D                m            m              mn   mE
            Miin=  Sqn  ; (Sqn-1 )2 1; (Sqn-2 )2 2; : :;:(Sq1)2  ; 2
                                                                       F2

with n  0, m0 = 0, m1 = 1, mk-1  mk  mk-1+ 1 and m  mn, and an epimorphism

of unstable A-modules:

                               "o: QH*X    _____>>Mon


with "o(x) = Sqn   , where Mondenotes either Minor Miin. Moreover, x might be c*
 *om-

pleted to a system of generators where "o(y) = 0 for other polynomial generator*
 *s y 6= x.

   In case that H*(X; F2) is noetherian and 1-connected it can only happen M0n.


   This shows that a polynomial generator cannot happen in a large dimension un*
 *less

is linked by Steenrod operations to other nilpotent generators in a way codifie*
 *d by the

unstable A-modules Mn.

   Also Theorem 1.1 can be stated in a more general form, for 1-connected mod 2*
 * H-

spaces satisfying conditions F1,F2 and F3 (see theorem 8.4).




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                    5


Example 1.7.

   n = 1  :Mi1= {(Sq1 ); (Sq1)2}. We can represent it in a diagram

                                         Sq1
                                      O _____>o


      where O represents the class Sqn   which will correspond to the polynomia*
 *l gener-

      ator and o represent the other classes that will correspond to nilpotent *
 *generators

      in the cohomology of X. This case clearly corresponds to S3<3>.

   n = 2  :Now Mi2is either {Sq2 ; (Sq1 )2; (Sq1)2} or {Sq2 ; (Sq1 )2; (Sq1)4};

      that is, one of the two diagrams

                                     Sq1      Sq4
                                  O _____> o <____ o


                                     Sq1      Sq2
                                  O _____> o _____>o


      So, a polynomial generator in dimension 8, x8 comes always together with *
 *two

      more generators linked by Steenrod operations according to one of the abo*
 *ve two

      diagrams.

        These cases are realized by SU(3)<3> and G2<3>.

   n  1   :If n  1, Moncontains the classes of Sqn   and (Sqn-1 )2 = Sq1(Sqn  ):

                                   Sq1
                                O _____> o : : :o : : :


      thus every polynomial generator in degree bigger than or equal to two has*
 * degree a

      power of two and non trivial Sq1. This has appeared with some restriction*
 *s in [19].


   Our final corollary was suggested to us by R. Kane.


Corollary 1.8.   Let X be a connected H-space of finite integral type.  If H*(X*
 *; F2) is

noetherian, then al rational polynomial generators appear in degree two.


Proof. Being X of finite integral type, the Bockstein spectral sequence applies*
 *. It starts

with H*(X; F2) and, according to Theorem 1.6, all polynomial generators have non

trivial Sq1, unless possibly some generators in degree two. Thus the spectral s*
 *equence

converges to a finitely generated Hopf algebra with all polynomial generators i*
 *n degree_

two. Thus the same should happen rationally.                                   *
 *  |__|


   The paper is organized as follows.  In section 2 we construct H-fibrations B*
 *Z=2 !

X ! E ! B2Z=2 for a given mod 2 H-space satisfying F1, F2 and F3 where BZ=2 ! X

detects a prescribed polynomial generator. Then we study the Serre spectral seq*
 *uence for

X ! E ! B2Z=2. Section 3 contains information about the structure of H*(B2Z=2; *
 *F2)

and section 4 about differential Hopf algebras. The spectral sequence itself is*
 * analysed

in sections 5, 6 and 7. Finally in section 8 we iterate the construction of sec*
 *tion 2 and

obtain the proofs of Theorems 1.1 and 1.6.

   The content of this paper was first presented in the Homotopy Theory Confere*
 *nce at

Palazzo Feltrinelli, Gargnano (Italy) in June 1995. The first named author is g*
 *rateful

to the organizers for their kind invitation.




6                        CARLOS BROTO AND JUAN A. CRESPO


        2. Detecting polynomial generators with central elements


   Let X denote a connected mod 2 H-space that satisfies the conditions F1, F2 *
 *and

F3. according to conditions F1 and F2 and the Borel classification of finite ty*
 *pe Hopf

algebras, there is an algebra isomorphism

                                                 P [y1; : :;:ys; : :]:
(5)                H*(X; F2) ~=P [x1; : :;:xr]  ___________________2ff1ffs
                                                (y1  ; : :;:y2s ; : :):

We call r the depth of X.

   The objective of this section is to show the existence of central elements i*
 *n X detecting

the polynomial generators; that is, maps


                                   f :BZ=2 ! X


for which a certain polynomial generator in H*(X; F2) has non trivial restricti*
 *on to

H*(BZ=2; F2) and such that map (BZ=2; X)f ' X. The main tool here is Lannes the*
 *ory

on elementary abelian groups [15]. The proof of that last homotopy equivalence *
 *will be

based on work of Dwyer-Wilkerson [10] and uses strongly the condition F3.  Fina*
 *lly,

we take the homotopy quotient by the central element thus obtaining a sequence *
 *of

fibrations

                             BZ=2 ! X ! E ! B2Z=2

that turn out to be H-fibrations.  We use Zabrodsky's Lemma in order to prove t*
 *his

fact. Let us recall it here


Lemma 2.1 ([27, 22]).   Let G be a topological group and G ! E  ! B a principal

fibration. If, for a space X, map (G; X)c ' X, where c is the constant map, then


                           map (B; X) ' map (E; X)f|G'c:


   The mod 2 cohomology of an H-space is both, an unstable algebra over the Ste*
 *enrod

algebra and a Hopf algebra in a compatible way.  We will say that it is an unst*
 *able

A-Hopf algebra.

   Let ` denote the localization functor of unstable modules or algebras over t*
 *he Steen-

rod algebra away from nilpotence (cf. [24],[6]).  From the proof of proposition*
 * 1.23

in [7] or applying proposition 8.2.1 in [16] to the construction of `, it follo*
 *ws that

for a tensor product of unstable A-algebras R  S one have a natural isomophism
             ~=
`(R)  `(S) _____>`(R  S). And therefore, the localization of an unstable A-Hopf*
 * al-

gebra is again an unstable A-Hopf algebra and the coaugmentation is as well a m*
 *ap of

unstable A-Hopf algebras.


Theorem 2.2.    Let X be a connected H-space that satisfies conditions F1 and F*
 *2. The

localization of H*(X; F2) gives a map of unstable A-Hopf algebras


                           X :H*(X; F2) -! H*(BV ; F2)


where V is an elementary abelian 2-group of rank the depth of X.

   Furthermore, if H*(X; F2) is described as in (5), then there is a basis u1; *
 *: :;:ur of
                                 fii
H1(BV ; F2) such that X (xi) = u2i , for fii 0 and i = 1; : :;:r.




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                    7


Proof. since X satisfies conditions F1 and F2 we can describe its mod 2 cohomol*
 *ogy

algebra as in (5).  Let X :H*(X; F2) ! ` H*(X; F2)  be the coaugmentation of the

localization of H*(X; F2).  The kernel of X  consists of the maximal nilpotent *
 *ideal

of H*(X; F2).  It should be, therefore, the ideal generated by the nilpotent ge*
 *nerators

y1; : :;:ys; : :.: Thus, we obtain that the image of x is isomorphic to the pol*
 *ynomial

algebra P [x1; : :;:xr] and then, ` H*(X; F2)  ~= ` P [x1; : :;:xr] .  This las*
 *t is clearly

given by the Adams-Wilkerson embedding into the mod 2 cohomology of an elementa*
 *ry

abelian group of rank r, the depth of X, hence we obtain the desired map X .

   Finally, image of the localization should be sub Hopf algebra of H*(BV; F2) *
 *and then

the precise description of X is a consequence of the Borel classification of Ho*
 *pf algebras_

(see [4]).                                                                     *
 *   |__|

   What makes this result interesting is that using results of Lannes [15] we c*
 *an realize

the algebraic map of the theorem by a geometric map

                                    f :BV ! X

with f* = X .  Moreover, since X  is a map of Hopf algebras, it commutes with t*
 *he

diagonal and geometrically this means that f is an H-map.

   Assume that V 0is any other elementary abelian group and f0: BV 0! X a map. *
 *By

universality of the coaugmentation X , the induced map f0*:H*(X; F2) ! H*(BV 0;*
 * F2)
                                    X    *                *    0               0
factors as a composition H*(X; F2) ____>H (BV ; F2) ____>H (BV ; F2) and then f*
 * itself
                          f                                  0             0
factors as BV 0_____>BV _____>X for a certain homomorphism V  ! V  and so f  is*
 * as

well an H-map.


Lemma 2.3.     In the above conditions, map (BV 0; X)f0' X, provided X satisfie*
 *s as well

the condition F3.


Proof. We have a map

                                g :BV 0x X _____>X

obtained as the composition of f0x id: BV x X ! X x X and the multiplication of*
 * X.

   The map g induces in cohomology a map g*:H*(X; F2) ! H*(BV 0; F2)  H*(X; F2)

and this in turn induces an adjoint map TV 0(H*(X; F2); f*) ! H*(X; F2), where *
 *TV 0is

the Lannes' T functor ([15]).

   The  computation  of  TV 0(H*(X; F2); f*)  follows  from  [10,  3.2,4.5]  an*
 *d  in  fact

TV 0(H*(X; F2); f*) ! H*(X; F2) becomes an isomorphism if and only if condition*
 * F3 is __

satisfied. Then, the lemma follows from [15, 3.3.2].                           *
 *      |__|

   Thus, we have proved that if we fix a connected mod 2 H-space X satisfying c*
 *on-

ditions F1, F2 and F3, any BV 0! X is central.  Next, we observe that BV 0acts *
 *on

map  (BV 0; X)f ' X and define

(6)                        E = map (BV 0; X)f xBV 0EBV 0

so obtaining a sequence of fibrations

                                f       g       h    2  0
(7)                       BV 0_____>X _____>E _____>B V  :


   It remains to prove that the constructed space E is actually an H-space in s*
 *uch a

way that the fibrations (7) are H-fibrations.




8                        CARLOS BROTO AND JUAN A. CRESPO


   The argument is a variation of an argument in [1] in which we use the fibrat*
 *ion (7)

itself in order to compute the mapping spaces map (BW; E)c, for W  any elementa*
 *ry

abelian 2-group.


                                                                f
Lemma 2.4.     Let V be any elementary abelian 2-group and BV _____>X _____>Y a*
 * fibra-

tion. Assume that H*(BV ; F2) becomes finitely generated as H*(X; F2)-module in*
 *duced

by f*. Then, for any W elementary abelian 2-group, map (BW; X)c ' X if and only*
 * if

map  (BW; Y )c ' Y .



Proof. The proof follows from the diagram of fibrations



                                              ev
                             map (BW; BV )S  ______>BV

                                    |               ||
                                    |               |
                                    |               |
                                    |               |
                                    _|       ev     _|
                              map (BW; X)c  _______>X

                                    |               ||
                                    |               |
                                    |               |
                                    |               |
                                    _|        ev    _|
                              map (BW; Y )c ________>Y



where S is the set of components of maps from BW to BV that become null-homotop*
 *ic

when composed with f.

   Now, suppose that g :BW ! BV  represents one of the components of S.  That i*
 *s,

f O g :BW ! X is null-homotopic. Maps out of classifying spaces of elementary a*
 *belian

groups are controlled by cohomology ([15]) and then the condition that H*(BV ; *
 *F2)

is finitely generated as H*(X; F2)-module induced by f* implies that f O g is n*
 *ull-

homotopic if and only if g itself is null-homotopic.  This proves that in our c*
 *ase S

consists of just one component, that of the constant map, and therefore the eva*
 *luation
                  ev
map  (BW; BV )S _____>BV  is a homotopy equivalence and the lemma follows from *
 *the
                                                                               *
 *  __
diagram.                                                                       *
 * |__|



Proposition 2.5.   Let X be a connected mod 2 H-space that satisfies F1, F2 and*
 * F3.

Assume that


                                     f       g
                                BV _____>X _____>E



is a principal fibration where V  is an elementary abelian 2-group, f is an H-m*
 *ap and

H*(BV ; F2) becomes finitely generated as H*(X; F2)-module induced by f*. Then,*
 * E is

an H-space and g an H-map.




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                    9


Proof. The argument here is the same used in [1, 9.17]. Look at the diagram

                                           mBV
                                BV x BV   _______>BV
                                    |              |
                                    |              |
                               f x f||             ||f
                                    |              |
                                    _|    mX      _|
                                 X x X  _________>X
                                    |              |
                                    |              |
                               g x g||             ||g
                                    |              |
                                    _|         mE _|
                                 E x E   _ _ _ _ _E_ _>

where the columns are principal fibrations and the top square is homotopy commu*
 *tative

because f is an H-map. By 2.4 map (BV xBV; E)c ' E and then the lemma 2.1 impli*
 *es

the existence of mE making the bottom square homotopy commutative.

   A similar argument shows that this multiplication admits a two sided unit el*
 *ement_

up to homotopy and therefore E becomes an H-space and g an H-map.             |*
 *__|


   It follows from these results that the space E of (6) is an H-space and that*
 * the

fibrations (7) are H-fibrations.  Actually, we will restrict our detection resu*
 *lt to one

single polynomial generator, e.g. x1 in which case we have obtained


Theorem 2.6.    Let X be a connected mod 2 H-space satisfying conditions F1, F2*
 * and

F3 and let x a polynomial generator of H*(X; F2). Then, there exists a map


                                  f :BZ=2 _____>X

               n+1                                      *
with f*(x) = u2   , u the one dimensional generator of H (BZ=2; F2) and a seque*
 *nce

of H-fibrations

                                f       g       h    2
(8)                      BZ=2 _____>X _____>E _____>B Z=2 :


Proof. The map f is obtained as the composition BZ=2 ! BV  ! X for a suitable  *
 *  __

chosen inclusion Z=2  V .                                                      *
 * |__|



                        3.PGBA-ideals of H*(B2Z=2; F2).


   The motivation for this section is the study of certain ideals in the mod 2 *
 *cohomology

of B2Z=2. Those among which we will find the possible kernels in mod 2 cohomolo*
 *gy
                                              j       p    2
of the projection map p of an H-fibration F _____>E _____>B Z=2.


Definition 3.1.  Let R be an A-Hopf algebra.  An ideal of R is called a Primiti*
 *vely

Generated Borel A-ideal (PGBA-ideal for short) if it is an A-ideal generated by*
 * a regular

sequence of primitive elements.


   The examples of interest to us will appear as kerp* where p is a projection *
 *as above.

The trivial ones correspond to the fibrations * ! B2Z=2 ! B2Z=2 where kerp* = {*
 *0}

and the universal principal bundle BZ=2 ! * ! B2Z=2 where kerp* is the ideal of*
 * all




10                       CARLOS BROTO AND JUAN A. CRESPO


positively graded elements of H*(B2Z=2; F2). Our aim is the classification of a*
 *ll possible

PGBA-ideals of H*(B2Z=2; F2).

   Recall that

                     H*(B2Z=2; F2) ~=F2[; Sq1; : :;:Sqn  ; : :]:

issa primitively generated polynomialsalgebra.  Thus, the primitives are the el*
 *ements

2  of degree 2s+1 and (Sqn  )2  of degree 2s(2n + 1) for all s  0 and n  0.  It*
 * will

be important in next sections the observation that in each degree there is at m*
 *ost one

primitive element.


Lemma 3.2.     For n  0 and s  0,
                                  8
                                  >>0           s > n + 1;
                                  >>
                                  ><Sqn+1       s = n + 1;
                         s  n
                      Sq2 Sq    =   0           0 < s  n;
                                  >>
                                  >>(Sqn-1 )2   s = 0; n 6= 0;
                                  >:
                                    0           s = 0; n = 0:


Proof. The case s > n + 1 follows by unstability and the case s = n + 1 is just*
 * the

recursive definitionnof the Sqn  .  Now, the case s = 0, also by unstability, w*
 *e have

Sq1Sqn   = Sq2 +1(Sqn-1 ) = (Sqn-1 )2 if n  1 or Sq1Sq1 = 0 for n = 0.

   Itsremains toslooknat the case 0 < s  n.  From the Adem relations applied to

Sq2 Sqn   = Sq2 Sq2 (Sqn-1 ) we obtain that
            s  n       2s+2n-2s-1 2s-1             2n+2s-1  2s-1
         Sq2 Sq    = Sq         Sq    (Sqn-1 ) = Sq       Sq    (Sqn-1 ):

Iterating this formula we obtain
                   s  n       2n+2s-1  2n+2s-2      2n+1  1
                Sq2 Sq    = Sq       Sq       : :S:q    Sq (Sqn-s )
                                        n+1        2
but Sq1(Sqn-ss) = (Sqn-s )2 and then Sq2   (Sqn-s ) ) = 0 by the Cartan formula*
 *  __

so Sq2 Sqn   = 0                                                               *
 * |__|

                                                                          s
Lemma 3.3.     The minimal PGBA-ideal of H*(B2Z=2; F2) containing (Sqn  )2 , n *
 * 0,

s  0, is
                      1 2s+n               2s+1         2s               2s
        J(n; s) =  (Sq )    ; : :;:(Sqn-1 )    ; (Sqn  )  ; : :;:(Sqn+r )  ; : *
 *:::


Proof. We have seen in lemma 3.2 how the Steenrod algebra operates on Sqn  . Mo*
 *re-
                                                   s+r  2r           2s  2r
over,  from  the  Cartan  formula,  we  know:   Sq2   (x  )   =   (Sq  x)  .   *
 * So,
   n+i+s         2s          2s
Sq2    (Sqn+i-1 )  = (Sqn+i )  andsthen we obtain that these elements are requi*
 *red

in an A-ideal containing (Sqn  )2 .

   On the other hand, we know Sq1Sqn   = (Sqn-1 )2 and so:
                               s   n   2s          2s+1
                            Sq2 (Sq   )  = (Sqn-1 )

Iterating this result we obtain:
                           s+i       2s+i            2s+i+1
                        Sq2  (Sqn-i )    = (Sqn-i-1 )

Then, all these elements must appear in our ideal, too. Finally one just check *
 *that the  __

ideal generated by all those elements is already an A-ideal and in fact a PGBA-*
 *ideal.   |__|




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                   11


Proposition 3.4.   The PGBA-ideals of H*(B2Z=2; F2) are either 0 or one of the *
 *fol-

lowing types for n  0, s  0,

   Type is:
                  1 2mn               2m1         2s               2s
          J =  (Sq )   ; : :;:(Sqn-1 )   ; (Sqn  )  ; : :;:(Sqn+r )  ; : : :;


      where m0 = s, m1 = s + 1 and mk = mk-1 + ffl, ffl = 0; 1.


   Type iis:
             2m     1 2mn               2m1         2s               2s
        J =    ; (Sq )   ; : :;:(Sqn-1 )   ; (Sqn  )  ; : :;:(Sqn+r )  ; : : :;


      where m0 = s, m1 = s + 1, mk = mk-1 + ffl, ffl = 0; 1 and m  mn.


   Observe that the trivial PGBA-ideal "H*(B2Z=2; F2) = (; Sq1; : :):is of type*
 * ii0 with

n = 0, m = m0 = 0.


Proof. Being the ideal J non trivial, it contains a primitive of H*(B2Z=2; F2) *
 *and there-

fore one of the minimal ideals described in lemma 3.3, but it might be bigger; *
 *that is,

it might contain generators that are roots of the generators in the minimal ide*
 *al.

   Assume that J contains the minimal ideal J(n;ss) and also that this is the l*
 *argest

J(n; s) that it contains; that is, (Sqn-1m)2  62 J.  Now our ideal might contai*
 *n other

primitive generators like (Sqn-k )2   withkmk  s + k so it would be
          1 2mn               2mk               2m1         2s               2s
  J =  (Sq )   ; : :;:(Sqn-k )   ; : :;:(Sqn-1 )   ; (Sqn  )  ; : :;:(Sqn+r )  *
 *; : :;:


with m1 = s + 1.  However, since it should be an A-ideal and as we have observed
          mk+n-k+1         2mk              2mk       2mk-1           2mk-1
before Sq2        ((Sqn-k )   ) = (Sqn-k+1 )    and Sq     ((Sqn-k+1 )     ) =

(Sqn-k )mk-1+1 for k = 1; : :;:n with m0 = s, it follows that the integers mk s*
 *hould

satisfy the condition specified in the Proposition.m  m          m

   In case J contains a power of , the equality Sq2 (2 ) = (Sq1)2  implies that*
 * m has_

to be an integer larger than or equal to mn.                                   *
 *    |__|


   Notice that those ideals are determined by a sequence of integers: m, mn, mn*
 *-1,...,

m1, s that determines the powers of the indecomposable primitives contained in *
 *the

ideal.

   We will now derive some properties of the systems of generators for the PGBA*
 *-ideals

of H*(B2Z=2; F2), and will define some important quotients.


Definition 3.5 (cf. [24]).An unstable A-module M is said to be nilpotent of cla*
 *ss l or

l-nilpotent if for every homogeneous element of degree m and 0  k < l
                             r(m-k)      2(m-k)  m-k
                          Sq2      : :S:q      Sq    x = 0


for a large enough r.

   An unstable A-module M is nilpotent if it is 1-nilpotent and it is reduced i*
 *f it does

not contain a non-trivial nilpotent submodule.


   For example l-fold suspensions of unstable A-modules are l-nilpotent.

   Let J be a PGBA-ideal of H*(B2Z=2; F2). The quotient


                      F2 H*(B2Z=2;F2)J ~=J=J . "H*(B2Z=2; F2)




12                       CARLOS BROTO AND JUAN A. CRESPO


is generated as a vector space by the regular sequence that generates J as an i*
 *deal.

Using the formulae in Lemma 3.2, we observe that the module J=J . "H*(B2Z=2; F2*
 *) is

nilpotent of class one. It is, indeed, a suspension. However it is not always 2*
 *-nilpotent.

In fact, it is nilpotent of class two if and only if s > 0.  Let N2 be the maxi*
 *mal 2-

nilpotent submodule of J=J . "H*(B2Z=2; F2) and L the quotient, so we have a sh*
 *ort

exact sequence of unstable A-modules


(9)                  0 ! N2 ! J=J . "H*(B2Z=2; F2) ! L ! 0;



where, for s = 0 and J 6= "H*(B2Z=2; F2) = (; Sq1; : :)::
                  mn               2
    o N2 = <(Sq1)2  ; : :;:(Sqn-1 ) >F2 and

    o L = <Sqn+r  | r  0>F2 is the suspension of a reduced module, L,

while, in case s > 0:

    o N2 = J=J . "H*(B2Z=2; F2) and

    o L = 0

and for J = "H*(B2Z=2; F2):

    o N2 = 0 and

    o L = <; Sq1; : :>:F2~=QH*(B2Z=2; F2).

   For J of type either i0 or ii0, as described in Proposition 3.4, and J 6= "H*
 **(B2Z=2; F2),

we define the sub A-module


(10)                     L^= <Sqn+1 ; : :;:Sqn+s ; : :>:F2



and for J = H"*(B2Z=2; F2), L^ = <Sq1; Sq1 ; : :>:F2, what amounts to set n = -1

in (10).

   The corresponding quotients will be important in understanding the transgres*
 *sion

map (15) in section 6.


Definition 3.6.  We define the unstable A-modules Mon= J=L^, where J are PGBA-

ideals of H*(B2Z=2; F2) of type either i0 or ii0. We will write Minif J was of *
 *type i0

and Miinif J was of type ii0 with J 6= "H*(B2Z=2; F2). We will simply set M-1 ~*
 *=F2 if

J = "H*(B2Z=2; F2).

   Alternatively, we can describe the unstable A-modules Mon, n  0, as the subq*
 *uotients

of H*(B2Z=2; F2) described as vector spaces by

                                    m1          2m2          1 2mn
              Min= <Sqn  ; (Sqn-1 )2  ; (Sqn-2 )   ; : :;:(Sq )   >F2


with m0 = 0, m1 = 1 and mk = mk-1 + ffl, ffl = 0; 1, or

                                   m1          2m2          1 2mn  2m
            Miin= <Sqn  ; (Sqn-1 )2  ; (Sqn-2 )   ; : :;:(Sq )   ;   >F2


with m0 = 0, m1 = 1, mk = mk-1 + ffl, ffl = 0; 1, and m  mn.


   In the example 1.7 are described the modules possible modules Minfor n = 1; *
 *2.




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                   13


                        4. Differential Hopf algebras.


   In this section we study the structure of some differential Hopf algebras re*
 *lated to the

Serre spectral sequence of an H-fibration F ! E ! B2Z=2.

   The model for a page of our spectral sequences is a bigraded Hopf algebra E *
 *= s;tEs;t,

0  s; t, which is isomorphic to a tensor product of two connected graded Hopf a*
 *lgebras

A and B:

                                   Es;t~=As  Bt

and equipped with a differential d of degree (n; 1 - n). We will identify A wit*
 *h E*;0and

B with E0;*.

   Then we have the following variant of the DHA lemma (see [14]):


Lemma 4.1.     Let (E ~=A  B; d) be a bigraded differential Hopf algebra as abo*
 *ve, then:

   1. d(Bm )  P n(A)  Bm-n+1 .

   2. Furthermore, if the transgression d: Bn-1 ! P n(A) is trivial, then d  0.


Proof. For an element x 2 Bm , we can write
                            X
                     d(x) =    aibi2 En;m-n+1 ~=An  Bm-n+1

                             i

with {bi} linearly independent.

   The diagonal applied to this element can be written in terms of the diagonal*
 * of ai

and bi:

                X
(11)   (d(x)) =    (ai)(bi) =

                 i
                 X  i               X          ji              X          j
               =      ai 1 + 1  ai+    a0ij a00ijbi 1 + 1  bi+    b0ik b00ik

                  i                  j                          k


where deg(a0ij); deg(a00ij) < n and deg(b0ij); deg(b00ij) < m - n + 1.

   On the other hand, counting degrees, one obtains that
                              M
           (d(x)) = d((x)) 2      (En;1-n+p E0;q)  (E0;p En;1-n+j)

                             p+q=m


and this implies that many homogeneous summands in equation (11) must vanish. In

particular:
                                 X
                                     a0ijbi a00ij= 0

                                  i;j

in E<n;m-n+1  E<n;0, and since {bi} are linearly independent,

                                  X
                                      a0ij a00ij= 0

                                   j


for all i, that is, the elements ai 2 An are primitives. This proves part (1). *
 *(2) follows_

like in [14] since P (E) = P (A) + P (B).                                      *
 *     |__|




14                       CARLOS BROTO AND JUAN A. CRESPO


   Next we assume that B is of finite type and therefore isomorphic, as algebra*
 *, to a

tensor product of monogenic Hopf algebras:

                                    P [x1; : :;:xr; : :]:
                              B ~=___________________2ff1ffr;
                                  (x1  ; : :;:x2r ; : :):

where ffi 2 N [ {1}, and ffi = 1 means that there is no relation at all and xi *
 *is

a polynomial generator.  With this notation we want to obtain a precise hold of*
 * the

homology H(A  B; d), for a non trivial d.  According to the previous lemma, the

differential d is trivial unless the transgression d: Bn-1 ! P n(A) is non triv*
 *ial.  And

this means that it has a non trivial value in one of the generators of B. We ca*
 *n assume,

without loose of generality, that d(x1) = a 2 P n(A) is the non trivial transgr*
 *ession and

also that x1 is of minimal height among elements x with d(x) = a. Then we have


Lemma 4.2.     Assume as above that (E  ~=A  B; d) is a bigraded differential H*
 *opf
                                          P [x1; : :;:xr; : :]:
algebra with B of finite type, thus B ~= ___________________2ff1ffras algebras.*
 *  Assume also
                                         (x1  ; : :;:x2r ; : :):
that there is a non trivial transgression d(x1) = a 2 P n(A), with x1 of minima*
 *l possible

height in d-1(a) and

  (i) P n(A) is one dimensional.

 (ii) a is not a zero divisor in A.

Then, there is a new system of generators x1; "x2; : :":xr; : :,:such that
                           P     2s +1
   1. for j > 1, "xj= xj +   kbkx1 k  and d("xj) = 0, where each bk is a polyno*
 *mial on

      the generators xi, different of x1 and with degrees smaller than m = deg(*
 *xj),
         ffj                                            P [x1; "x2; : :;:"xr; :*
 * :]:
   2. "x2j= 0, so that B is equally expressed as B ~=________________________2f*
 *f12ff2ffrand
                                                     (x1  ; "x2; : :;:"x2r; : :*
 *):
                  P [x12; "x2; : :;:"xr; : :]:       A
   3. H(E; d) ~=________________________2ff12ff2ffr ___, as algebras.
                (x1  ; "x2; : :;:"x2r; : :):        (a)


Proof. We need to study how the differential acts on the generators xi. We have*
 * assumed

that d(x1) = a 6= 0. Let xj the next generator of minimal degree such that d(xj*
 *) 6= 0.

We shall see that we can modify xj and obtain a different generator "xjwith tri*
 *vial

differential.

   By lemma 4.1 the degree of xj, m, is bigger than or equal to deg(x1) = n - 1*
 * and

d(xj) 2 P n(A)  Bm-n+1   En;m-n+1. Actually, we can prove:

                       P     2s
Claim 4.3.   d(xj) = a   kbkx1 k, where each bk is a polynomial on the generato*
 *rs xi,

different of x1 and with degrees smaller than m = deg(xj).


Proof.PAccording to lemma 4.1 d(xj) 2 P n(A)Bm-n+1 , so it can be written as d(*
 *xj) =

a   kbkxsk1, where bk are polynomials on generatorsPxi different than x1 and of*
 * degree

less than m = deg (xj).  Then 0 = d2(xj) = a2   kbkskxsk-11and bk should be zer*
 *o    __

whenever sk is odd. That is d(xj) might be written as above.                   *
 *    |__|


Now, (1) is an easy computation. In order to prove the relation (2) we use the *
 *diagonal

map: (xj) = xj  1 + 1  xj + : :.:We are particularly interested in the component

in E0;n-1 E0;m-n+1, that might be written as x1  y + b  y0, where b is a polyno*
 *mial




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                   15


on generators different of x1 and y; y0 are any elements in Bm-n+1 . Thus we ca*
 *n write:

        (xj) = xj  1 + 1  xj + x1  y + b  y0+ terms in different degrees.

Now we compute d((xj)) and look particularly at the component in En;0 E0;m-n+1:

         d((xj)) = d(xj)  1 + 1  d(xj) + a  y + terms in different degrees.

On the other hand,
                X               X                   X
   (d(xj)) = (a    bkx2sk1) = a    bkx2sk1 1 + 1  a    bkx2sk1+

                 k               k                   k
                                           X
                                      + a     bkx2sk1+ terms in different degre*
 *es.

                                            k
                                              P     2s
Hence, the equation d(xj) = d(xj) implies y =   kbkx1 kand then
                               X
    (xj) = xj  1 + 1  xj + x1     bkx2sk1+ b  y0+ terms in different degrees.

                                k
                                                  ffj                 P      2s*
 *  2ffj
From this equation it follows that the relation x2j = 0 implies that     kbkx1 *
 *k    = 0,
                      ffj     P      2s +1 2ffj       2ffj
and therefore that "x2j= xj+    k bkx1 k      = 0 or x1  =0, but this second op*
 *tion

gives us to the previous one.

   Finally, we prove (3).  We can use inductively the result of this claim and *
 *obtain a

new system of generators x1; "x2; : :":xr; : :w:ith

                       P [x1; : :;:xr; : :]: P [x1; "x2; : :;:"xr; : :]:
                 B ~=___________________2ff1ffr~=__________________2ff12ff2ffr
                     (x1  ; : :;:x2r ; : :):(x1  ; "x2; : :;:"x2r; : :):

as algebras, and such that d(x1) = a and d("xi) = 0 for all i > 1. Hence, we ca*
 *n split E

as differential algebra
                          i P [x ]      j   P ["x; : :;:"x; : :]:
                      E ~=  ____1_2ff1 A   ______2________r___2ff2ffr
                            (x1  )         ("x2 ; : :;:"x2r; : :):

where the differential on the right term is trivial. The homology of the left t*
 *erm is_easily_

computed using that d(x1) = a is not a zero divisor and then the lemma follows.*
 *     |__|


          5. Serre Spectral sequence for H-fibrations over B2Z=2


   We are interested in the behavior of the Serre spectral sequence of an H-fib*
 *ration

F ! E ! B2Z=2:

(12)              E*;*2~=H*(B2Z=2; F2)  H*(F ; F2) =) H*(E; F2)


Proposition 5.1.   Let F  ! E  ! B2Z=2 be an H-fibration, where H*(F ; F2) is of

finite type.  Then each stage of the corresponding Serre spectral sequence is a*
 * bigraded

differential Hopf algebra of the form:

                                           H*(B2Z=2; F2)
(13)                   En ~=An  Bn;   An = ______________;
                                             (1; : :;:r)

where 1; : :;:r is a regular sequence of primitive elements of H*(B2Z=2; F2) an*
 *d Bn is

a sub Hopf algebra of H*(F ; F2).




16                       CARLOS BROTO AND JUAN A. CRESPO


   Moreover, the elements i are the targets of the transgression homomorphisms *
 *of the

previous stages of the spectral sequence.


Proof. We will proof by induction on n that each term En of the spectral sequen*
 *ce has

the form (13) and the elements i are targets of previous transgressions.

   For n = 2, the E2 term of the spectral sequence is as described in (12) and *
 *it clearly

satisfies the above conditions. See section 3 for a description of H*(B2Z=2; F2*
 *).

   Assume by induction that this is true for En-1. If dn-1 is trivial En ~=En-1*
 * and there

is nothing to prove. Thus, we suppose that dn-1 is non trivial. According to le*
 *mma 4.1

there should be a non trivial transgression which target is a primitive element*
 * of An-1

in degree n - 1.

   Recall that the primitiveselements of H*(B2Z=2; F2) ~= F2[; Sq1; : :S:qn  ; *
 *: :]:are

of the form i= (Sqmi  )2 , heince by the induction hypothesis An-1 is written as


                    H*(B2Z=2; F2)         F2 [; Sq1; : :S:qn  ; : :]:
             An-1 ~=______________ ~= ________________________________ssr-1
                     (1; : :;:r-1)     (Sqm1  )2  ; : :;:(Sqmr-1 )2    1

                                  si              s   m
where the primitives i = (Sqmi  )2  have degrees 2 i(2 i + 1) < n - 1 because t*
 *hey

appear as images of previous transgressions. It follows that the remainingsprim*
 *itives in

An-1 in degrees bigger than or equal to n - 1 are the classes of (Sqm  )2 , wit*
 *h degree

2s(2m + 1)  n - 1 and m 6= mi for 1  i  r - 1.  Such elements are still non zero

divisors in An and there is at most one in each degree.

   Hence the transgression in En-1 should hit one of those primitives and we ca*
 *n apply

Lemma 4.2(3) in order to compute En and check that it has the form (13). This f*
 *inishes_

the induction step and then the proof of the Proposition.                      *
 *     |__|


Proposition 5.2.   Let F ! E ! B2Z=2 be an H-fibration, with H*(F ; F2) of fini*
 *te

type, thus we can write

                                         P [x1; : :;:xr; : :]:
(14)                      H*(F ; F2) ~=___________________2ff1ffr:
                                       (x1  ; : :;:x2r ; : :):

where the generators are ordered by degree and by height in case of coincidence*
 * of degree.

Then, there exists a system of transgressive generators for H*(F ; F2), "x1; : *
 *:;:"xr; : : :

such that

   1. for each i, "xi= xi+ pi where pi= pi(x1; : :;:xi-1) is a polynomial on th*
 *e previous

      generators,

   2. "xihas the same height as xi, so that

                                         P ["x1; : :;:"xr; : :]:
                          H*(F ; F2) ~=___________________2ff1ffr:
                                       ("x1 ; : :;:"x2r; : :):

   3. In each degree, there is at most one monomial on the "xi's with non trivi*
 *al trans-

      gression.

   4. If y 2 H*(F ; F2) is a monomial on the "xi's with non trivial transgressi*
 *on, then
                                                  k
      there exists i and k  0 such that y = ("xi)2 .
                                                              k
   5. If, for a given i, "xiis a nilpotent generator, then "x2ihas trivial tran*
 *sgression for

      all k  1.




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                   17


Proof. We proceed by induction. For this, we assume that we have a system of or*
 *dered

generators x1; : :;:xr; : :s:uch x1; : :;:xq-1 are transgressive and satisfy (1*
 *), (2) and (3).

Then we first prove:


Claim 5.3.   (4) and (5) applies to the generators x1; : :;:xq-1.

                               k1    mq-12kq-1
Proof. Pick a monomial y = xm121 . .x.q-1    , with m1; : :;:mq-1 odd integers.*
 *  A
                                                    k1      2kq-1
differential of y is computed in terms of that of x21 ; : :x:q-1. Not all of th*
 *at elements

can transgress trivially for if they did, y would transgress trivially as well.*
 *  Assume
       ki
that x2i is an element of minimal degree in the decomposition of y that transgr*
 *esses

non trivially. Then, this same differential kills y and then y would not be tra*
 *nsgressive
                       ki
unless it is exactly x2i . This proves that (4) applies to the generators x1; :*
 * :;:xq-1.

   Now, we look at (5). In case xi is nilpotent, it must restrict to zero along*
 * the induced

map BZ=2 ! F and therefore its transgression is a decomposable primitive, or ju*
 *st

trivial. Since x2i= Sqdeg(xi)xi, xi is transgressive, too. If thestransgression*
 * of xi is zero,

then so is that of x2i. Suppose that xi transgresses to (Sqm  )2 , with s  1. I*
 *n this case
                                                                             s
the degree of xi had to be odd, and then x2itransgresses to Sqdeg(xi)(Sqm  )2  *
 *= 0, by

the Cartan formula. Hence, in any case x2itransgresses to zero; that is, it is *
 *a permanent
                                                          k                    *
 *       __
cycle in the spectral sequence, so the same is true for x2i, k  1.             *
 *      |__|


   Now we show the induction step. Let xq the next generator (that might be the*
 * first

one!).  The generators x1; : :;:xq-1; xq may fail to satisfy the same condition*
 *s for two

different reasons:

    o xq is not transgressive, or

    o there is a monomial y on the previous generators such that both y and xq *
 *have non

      trivial transgression.

   Suppose that xq is not transgressive, that is, there is n with n  deg(xq) an*
 *d dn(xq) 6=

0.

   Since dn is non trivial, by lemma 4.1, there is a transgressive element y wi*
 *th dn(y) 6= 0.

y is, by degree reasons, a polynomial onPthe generators x1; : :;:xq-1.  Accordi*
 *ng to

lemma 4.2, we can modify xq to x0q= xq+   bky2sk+1withPdn(x0q) = 0 and bk polyn*
 *omials

on the generators x1; : :;:xq-1, so the whole   bky2sk+1is a polynomial on the *
 *generators

x1; : :;:xq-1. Now, we check the next differential and modify x0qagain if it is*
 * necessary

and so on until we have obtained "xq= xq + pq(x1; : :;:xq-1) which is transgres*
 *sive.

   Suppose, next, that xq have non trivial transgression but there is another m*
 *onomial,

y, on the previous generators, x1; : :;:xq-1. According to the claim, y is eith*
 *er one of

the generators or a power of a polynomial generator.

   If y is just one of the previous generators its height would be smaller than*
 * or equal

to the height of xq by our assumptions about the arrangement of the generators.*
 * Then,

Lemma 4.2 applies again.  Actually, we just choose "xq= xq + y as the new gener*
 *ator

that substitutes xq, having the same height and trivial transgression.
                                                         k
   In case y is a power of a polynomial generator: y = x2i, the transgression o*
 *f both,

y and xq, being non trivial, should be an odd dimensional primitive; that is, a*
 *n inde-

composable primitive. But this means that xq restricts non trivially along the *
 *induced

map BZ=2 ! F  and therefore that xq has infinite height.  So again we just choo*
 *se




18                       CARLOS BROTO AND JUAN A. CRESPO


x"q = xq + y as the new generator instead of xq and having infinite height as w*
 *ell but

trivial transgression.

   We have finally obtained, in any case, a new generator "xq= xq + pq(x1; : :;*
 *:xq-1)

with the same height as xq and such that x1; : :;:xq-1; "xqare transgressive an*
 *d satisfy

(1), (2) and (3), and also (4) and (5) again by the claim.  We have therefore f*
 *inished_

with the induction step and proved the Proposition.                            *
 *   |__|


   The above Proposition suggests the following definition


Definition 5.4.  Let F ! E ! B2Z=2 be an H-fibration with H*(F ; F2) of finite *
 *type.

A system of algebra generators for H*(F ; F2), x1; : :x:r; : :,:is a good syste*
 *m of trans-

gressive generators if

                     P [x1; : :;:xr; : :]:
   1. H*(F ; F2) ~=___________________2ff1ffr.
                   (x1  ; : :;:x2r ; : :):
   2. Each xi is transgressive.

   3. In each degree, there is at most one monomial on the generators xi with n*
 *on trivial

      transgression.


   Proposition 5.2, thus, proves the existence of good systems of trangressive *
 *generators

and also, according to the claim, that any good system of transgressive generat*
 *ors

satisfies as well conditions (4) and (5) of Proposition 5.2.

   Propositions 5.1 and 5.2 determine the behavior of the Serre spectral sequen*
 *ce of an
                j       p    2           *
H-fibration F _____>E _____>B Z=2 where H (F ; F2) is of finite type. Each stag*
 *e of such

a spectral sequence is a bigraded differential Hopf algebra En ~= An  Bn of the*
 * sort

considered in section 4 hence the differentials are always determined by transg*
 *ression.

   It should finally converge to

                                  E1 ~= A1  B1

where
                                            H*(B2Z=2; F2)
                         A1 ~= lim-!nAn ~=__________________
                                          (1; 2; : :;:r; : :):

with 1; 2; : :;:r; : :a:regular sequence of primitives and
                                   "
                              B1 =    Bn  H*(F ; F2)

                                    n

is a sub Hopf algebra of B2 = H*(F ; F2) that might be described using a good s*
 *ystem

of transgressive generators for H*(F ; F2): x1; : :;:xr; : :.:

   What remains to do is describing the link between those generators of H*(F ;*
 * F2)

and the regular sequence 1; : :;:r; : :o:f H*(B2Z=2; F2). This link should clea*
 *rly be

the transgression homomorphism and here is where Steenrod operations come into *
 *the

picture. In fact it is well known that the transgression homomorphism commutes *
 *with

primary operations.

   In terms of our good system of transgressive generators any transgressive el*
 *ement with
                                                       s     2s-1|x |     2|x |*
 * |x |
non trivial transgression is either xi for some i or x2i = Sq      i: :S:q   iS*
 *q  ixi, for

some s  1. Hence we can choose as source of the transgression homomorphim, with*
 *out

loose of information, the sub A-module of H*(F ; F2) generated by x1; x2; : :;:*
 *xr; : :a:s

A-module: <x1; x2; : :;:xr; : :>:A




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                   19


   Notice that A1  should coincide with the image of p* and the kernel is ker p*
 ** =

(1; 2; : :;:r; : :):, a PGBA-ideal of H*(B2Z=2; F2).  So this ideal can be chos*
 *en as

target of the transgression. But there is an indeterminacy given by possible mu*
 *ltiples

of elements hit by previous differentials.  That indeterminacy is therefore con*
 *tained

in kerp* . "H*(B2Z=2; F2) and then the trangression is determined by a well def*
 *ined

morphism of unstable A-modules

(15)           o :<"x1; "x2; : :;:"xr>A ____>>ker p*= kerp* . "H*(B2Z=2; F2)

where we have finally written the suspension of <"x1; "x2; : :;:"xr>A as source*
 * in order to

make o a degree zero homomorphism.

   Moreover, o is an epimorphism because kerp*= kerp*.H"*(B2Z=2; F2) is a vecto*
 *r space

generated by the classes of 1; 2; : :;:r; : :,:which are obtained precisely as *
 *targets of

the transgression homomorphims according to Proposition 5.1.


 6. Transgression in the Serre Spectral sequence for H-fibrations over

                                        B2Z=2


   In this section we further study the transgression map in the Serre spectral*
 * sequence

of an H-fibration
                                   j       p    2
                               F _____>E _____>B Z=2

in the case in which F satisfies the finiteness conditions F1, F2 and F3.

   Assume that x1; : :;:xr; y1; : :;:ys; : :i:s a good system of transgressive *
 *generators

for H*(F ; F2), where we distinguish between polynomial generators xi and nilpo*
 *tent

generators yi, thus we write:

                                                    P [y1; : :;:ys]
                     H*(F ; F2) ~=P [x1; : :;:xr]  _______________2ff1ffs
                                                   (y1  ; : :;:y2s )

with 0 < ffi< 1 for each i = 1; : :;:s.

   The suspension of the nil-localization of H*(F ; F2) (see Theorem 2.2) provi*
 *des a

sequence:


           0 !  ker ! <x1; : :;:xr; y1; : :;:ys>A _____><x1; : :;:xr>A ! 0


where <x1; : :;:xr>A  P [x1; : :;:xr] >___>H*(BV ; F2) is reduced and ker  (y1;*
 * : :;:yr)

is nilpotent, hence  ker is nilpotent of class two.

   On the other hand, the PGBA-ideal I = kerp* can also be decomposed, accordin*
 *g to

the sequence (9), as a maximal subA-module, K, which is nilpotent of class 2 an*
 *d the

quotient L, the suspension of a reduced A-module,

   Then we obtain a corresponding decomposition of the transgression map o (15):

                  ker   >___><x1; : :;:xr; y1; : :;:ys>A_>><x1; : :;:xr>A
                    |                                          |
                    |                   |                      |
(16)         o| ker ||                  ||o                    ||o0
                   __|                  __|                    __|

                   K >________>I=I . H*(B2Z=2; F2)__________>>L

where o0 should also be seen as the restriction of o to the polynomial generato*
 *rs, since

those appear in degrees a power of two (see Theorem 2.2) and then by degree rea*
 *sons




20                       CARLOS BROTO AND JUAN A. CRESPO


can only map by o to elements of L. This proves in turn that o| ker  in the dia*
 *gram

is an epimorphism.

                       j       p    2                             *
Lemma 6.1.     Let F _____>E _____>B Z=2 be an H-fibration where H (F ; F2) sat*
 *isfies

conditions F1, F2 and F3.  Then either the H-fibration is trivial or the PGBA i*
 *deal

I = kerp* is of type i0 or ii0.


Proof. According to Proposition 3.4 the PGBA ideals of H*(B2Z=2; F2) are either*
 * 0 or

of type isor iis. Assume that I = kerp* is a PGBA ideal with s > 0 or it is jus*
 *t 0. In

that cases L = 0 in the diagram (16) and all of the polynomial generators of H**
 *(F ; F2)
                                                                 f
transgress trivially and in turn they map trivially along BZ=2 _____>F . Hence *
 *this map

is trivial in cohomology and therefore null-homotopic [15].

   We will now apply Zabrodsky's lemma (see Lemma 2.3) to the principal fibrati*
 *on
      f'*      g
BZ=2 _____>F _____>E in order to extend the identity of F to a section of the f*
 *ibration

s: E ! F . This turns out to be an H-map. In fact, look now at the fibration BZ*
 *=2 x

BZ=2 ! F x F ! E x E. The multiplication of F , m: F x F ! F extends to E x E
                                      sxs          m
in two different ways, namely, E x E _____>F x F _____>F and, since g :F ! E is*
 * an

                       m       s
H-map, also as E x E _____>E _____>F . Applying again Zabrodsky's lemma, these *
 *two

factorizations should be homotopic, hence the section s: E ! F is an H-map.

   We have obtained a diagram of H-fibrations and H-maps

                                g               h    2
                          Fw___________>E _________>B Z=2
                          ww                         ww
                          ww            ||           ww
                          ww            |(s; h)      ww
                          w             _||          w

                          F _______>F x B2Z=2 ____>B2Z=2


that commutes up to homotopy, thus E ' F x B2Z=2 and our original fibration is *
 *    __

trivial.                                                                       *
 *   |__|

                                                                     j       p *
 *   2
   It will be useful to distinguish the non trivial H-fibrations F _____>E ____*
 *_>B Z=2

according to the type of the ideal kerp*.

                                               j       p    2            *
Definition 6.2.  A non trivial H-fibration F _____>E _____>B Z=2, where H (F ; *
 *F2)

satisfies conditions F1, F2 and F3 is of type iif the PGBA-ideal kerp* is of ty*
 *pe i0 and

it is of type iiif kerp* is of type ii0.

                          j      p    2                                       *
Proposition 6.3.   Let F ____>E ____>B Z=2 be a non trivial H-fibration with H *
 *(F ; F2)

satisfying the conditions F1, F2 and F3. The transgression is determined by the*
 * induced

epimorphism of unstable A-modules


                             o":QH*(F ; F2) ____>>Mon


for some n  -1 where Mon= Minif the fibration is of type i and Mon= Miinor

M-1 if the fibration is of type ii. It satisfies "o(xi) = Sqn  , for xi a least*
 * dimensional




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                   21


polynomial generator of H*(F ; F2) that maps non trivially along BZ=2 ! F for n*
 *  0

or "o(xi) =  2 M-1 if n = -1.


   See 3.6 for the definition of the unstable A-modules Mon. Notice also, that *
 *by abuse

of language we denote equally by xi the element of QH*(F ; F2) represented by t*
 *he

generator xi2 H*(F ; F2).


Proof. Look at the decomposition of the transgression map o (15), given in diag*
 *ram (16).

There should be a polynomial generator that transgresses to the least dimension*
 *al ele-

ment in L. And it has to be one in least possible dimension that restricts non *
 *trivially

to H*(BZ=2; F2) along BZ=2 ! F . Thus, it appears in dimension one and "o(xi) =*
 *  or

deg  xi = 2n+1, n  0, and "o(xi) = Sqn  . It follows that the decomposables con*
 *tained

in <x1; : :;:xr>A maps into L^ I=I . H*(B2Z=2; F2).

   Now, look at the elements in ker. According to Proposition 5.2, (4), (5), am*
 *ong

those elements, only the generators y1; : :;:ys can have non trivial transgress*
 *ions. Hence,

all the decomposables contained in ker map to zero in I=I . H*(B2Z=2; F2).

   We have obtained that in the composition


             <x1; : :;:xr; y1; : :;:ys>A ____>>I=I . H*(B2Z=2; F2) ____>>Mon


all the decomposable elements remain in the kernel, hence this factors as

                                             "o   o
                               QH*(F ; F2) ____>>Mn

                                                                               *
 *     __
and this finishes the proof.                                                   *
 *    |__|



    7. Outcome of the Serre Spectral sequence for H-fibrations over

                                        B2Z=2


   So far, we have obtained the structure of the E1  term of the Serre spectral*
 * sequence
                      g       h    2
of an H-fibration F _____>E _____>B Z=2:


                                  E1 ~= A1  B1


where

                                    H*(B2Z=2; F2)
                               A1 = ______________;
                                           I

with I a PGBA-ideal of H*(B2Z=2; F2) and B1  is a sub-A-Hopf algebra of H*(F ; *
 *F2).

It follows, that g* and h* factor as edge homomorphisms inducing Im h* ~= A1  a*
 *nd

Im  g* ~=B1 .

   We thus have an associated graded ring to H*(E; F2). In this section que stu*
 *dy the

extension problems in order to get information about H*(E; F2) itself.

   Using the filtration degree of the elements of H*(E; F2) it is defined an ad*
 *ditive

isomorphism

                                ss :H*(E; F2) ! E1




22                       CARLOS BROTO AND JUAN A. CRESPO


that fits in the diagram

                          Im h*>____>H*(E; F2) ____>>Img*
                            |
                            |             |            |
                            |~          ~ |            |~
                            |=          = |ss          |=
                            |             |            |
                            _|           _|           _|
                           A1  >________>E1 ________>>B1

The map ss is not in general an algebra map.  It is true however that given ele*
 *ments

x; y 2 H*(E; F2) such that ss(x)ss(y) 6= 0 in E1 , we have ss(xy) = ss(x)ss(y).


Lemma 7.1.     If a1; : :;:am ; : :i:s a system of algebra generators for Im h**
 * ~=A1  and

b1; : :;:bn; : :f:or Im g* ~=B1 , then a1; : :;:am ; : :;:b01; : :;:b0n; : :i:s*
 * a system of algebra

generators for H*(E; F2), where b0iis any element for which g*(b0i) = bi.

   Moreover, the sequence

                        1 ! Im h* ! H*(E; F2) ! Im g* ! 1

is an exact sequence of Hopf algebras in the sense that Im g* ~=H*(E; F2)== Im *
 *h*.


Proof. Since we have E1 ~= A1 B1 for any two elements x 2 Im h* and y 2 H*(E; F*
 *2)

with g*(y) 6= 0 2 Im g* we have ss(x) = x  1 and ss(y) = 1  g*(y) + decomposabl*
 *es, so

that 0 6= ss(x)ss(y) = ss(xy).

   Consequently, we can obtain a system of algebra generators for H*(E; F2) fro*
 *m a

system of algebra generators for Im h* and one for Im g*, as indicated in the l*
 *emma.

   Also ss maps the ideal (Im h*+ ) of H*(E; F2) onto the ideal (A1 + ) of E1 ,*
 * and_then

H*(E; F2)== Im h* ~=Im g*.                                                     *
 *  |__|


Lemma 7.2.     Let A, B and C be connected graded algebras and let A ! B ! C ~=

B==A be an exact sequence of algebras. Let A, B and C denote the respective quo*
 *tients by

the ideals of the nilpotent elements. Then A >____>B  is an injection while B =*
 *=A  ____>>C

is an epimorphism with nilpotent kernel.
                                                                               *
 *   __
Proof. Straightforward.                                                        *
 *  |__|


Lemma 7.3.     Assume that A = PA  NA , B = PB  NB , C = PC  NC are connected

algebras and we have a diagram

                                 g|PA         h|PB
                           PA >_________>PB ________>>PC
                            _|           _|           _|
                            |             |            |
                            |             |            |
                            |             |            |
                            _|     g     _|    h      _|
                            A >__________>B_________>>C~= B==A

Then, there is a diagram
                                    g        h
                               NA _____>NB ____>>NC

where

    o NB ==g(NA ) ~=NC and

    o kerg consists of the elements of NA represented in the ideal (PB+) of B.
                                                                               *
 *   __
Proof. Straightforward.                                                        *
 *  |__|




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                   23


7.1. Fibrations of type i. Let us now specialize to H-fibrations

                                   g       h    2
                               F _____>E _____>B  Z=2


where F satisfies conditions F1, F2 and F3, which are of type i.

   So there is a good system of transgressive generators with


                                                  P [y1; : :;:ys; : :]:
                   H*(F ; F2) ~=P [x1; : :;:xr]  ___________________2ff1ffs
                                                 (y1  ; : :;:y2s ; : :):

and transgression
                                               m1          2m2          1 2mn
    o :QH*(F ; F2) ____>>Min~=<Sqn  ; (Sqn-1 )2  ; (Sqn-2 )   ; : :;:(Sq )   >F2

   We may suppose that
                                                m1                   1 2mn
              "o(x1) = Sqn  ; "o(y1) = (Sqn-1 )2  ; : :;:"o(yn) = (Sq )

and "otrivial elsewhere. These formulae determine the transgression hence the s*
 *pectral

sequence. That is, using inductively the computation of Lemma 4.2 (3) we get th*
 *at the

E1  term of the Serre spectral sequence for the fibration is E1 ~= A1  B1  with

                                            P [Sq1; : :;:Sqn-1 ]
               A1 ~= P []  N0;     N0 ~=____________________________mnm
                                         (Sq1)2  ; : :;:(Sqn-1 )2      1

and
                                               P [y12; : :;:yn2; yn+1; : :;:ys;*
 * : :]:
         B1 ~= P [x2; : :;:xr]  N00;     N00~= ______________________________2f*
 *f1ffs
                                                     (y1  ; : :;:y2s ; : :):

With this notation we obtain


Proposition 7.4.   For a fibration of type ias above, and provided F is 1-conne*
 *cted

   1. H*(E; F2) ~=P [x01; x02; : :;:x0r]  N where x01= h*() and g*(x0i) = xi if*
 * i  2.

   2. N is a nilpotent Hopf algebra of finite type that fits in an exact sequen*
 *ce of Hopf

      algebras
                                    h*      g*   00
                           1 ! N0 _____>N _____>N  ! 1 :


Proof. By Lemma 7.1 H*(E; F2) has in each dimension at most a finite number of

generators so it is of finite type and then there is an isomophism of algebras

                                H*(E; F2) ~=P  N

where P is a polynomial algebra and N is a nilpotent Hopf algebra of finite typ*
 *e.

   It also follows from lemma 7.1 the existence of an exact sequence of Hopf al*
 *gebras

                        1 ! Im h* ! H*(E; F2) ! Im g* ! 1 :

By lemma 7.2 there is an injection

                                     P [] ,! P :

As E is 1-connected we can choose generators for P , z1; z2; z3; : :i:n such a *
 *way that

 7! z.  Hence P==P [] ~= P [z2; z3; : :]:contains no nilpotent elements and the*
 *refore

Lemma 7.2 implies

                       P [z2; z3; : :]:~=P==P [] ~=P [x2; : :;:xr] :




24                       CARLOS BROTO AND JUAN A. CRESPO


   This means that we can choose x01= h*(), x02; : :;:x0rwith g*(x0i) = xi for *
 *i  2, and

an isomorphism of algebras

                         H*(E; F2) ~=P [x01; x02; : :;:x0r]  N :

   In order to prove (2) we first observe thatPN0\ (x01; : :;:x0r) = 0. In fact*
 *, assume that

n 2 N0 ,! H*(E; F2) can be written as n =   ikix0iin H*(E; F2). Observe that we*
 * could

have chosen x01; : :;:x0rin such a way that they are represented in E1  by the *
 *regular

sequenceP  1; 1  x2; : :;:1  xr so each ss(ki)ss(x0i) 6= 0 as soon as ki 6= 0, *
 *and then

ss(n) =   iss(ki)ss(x0i) 2 (  1; 1  x2; : :;:1  xr). But this ideal of E1 does *
 *not_contain_

ss(n) = n  1 unless n = 0. Hence by Lemma 7.3 N0 injects in N and N==N0 ~=N00. *
 *  |__|


Remark 7.5.    A counterexemple to this proposition in case F is not 1-connecte*
 *d is the

H-fibration

                             BZ=4 ! BZ=2 ! B2Z=2 :


Remark 7.6.    Notice that in the case of fibrations of type iwith n = 0; that *
 *is,

                                   g       h    2
                               F _____>E _____>B Z=2


with
                                                  P [y1; : :;:ys; : :]:
                   H*(F ; F2) ~=P [x1; : :;:xr]  ___________________2ff1ffs
                                                 (y1  ; : :;:y2s ; : :):

where x1; : :;:xr; y1; : :;:ys; : :i:s a good system of transgressive generator*
 *s and

                         "o:QH*(F ; F2) ! M0 = <Sq1>F2;

the result of Proposition 7.4 has no extension problems and if we assume that j*
 *ust x1

transgresses to Sq1, we have

                                                   P [y01; : :;:y0s; : :]:
                  H*(E; F2) ~=P [x01; : :;:x0r]  ____________________0ff10ffs
                                                 (y12 ; : :;:ys2 ; : :):

with h*() = x01, g*(x0i) = xi for i = 2; : :;:r and g*(y0i) = yi, for i = 1; : *
 *:;:s; : : :


7.2. Fibrations of type ii. We consider now fibrations of type iiwith 1-connect*
 *ed

fibre; that is, H-fibrations
                                   g       h    2
                               F _____>E _____>B  Z=2

with
                                     P [y1; : :;:ys; : :]:
   1. H*(F ; F2) ~=P [x1; : :;:xr]  ___________________2ff1ffs, where x1; : :;:*
 *xr; y1; : :;:ys; : : :
                                    (y1  ; : :;:y2s ; : :):
      is a good system of transgressive generators and

   2. the transgression is determined by "o:QH*(F ; F2) ____>>Miin, n  0.
                              m1          2m2          1 2mn  2m
   Now Miin~=<Sqn  ; (Sqn-1 )2  ; (Sqn-2 )   ; : :;:(Sq )   ;   >F2 and we may *
 *as-

sume that
                                        m1                   1 2mn             *
 *2m
      "o(x1) = Sqn  ; "o(y1) = (Sqn-1 )2  ; : :;:"o(yn) = (Sq )   ; "o(yn+1) = *
 *   ;

and "otrivial elsewhere. So, with the same arguments as above we obtain E1 ~= A*
 *1 B1

with
                                       P [; Sq1; : :;:Sqn-1 ]
               A1 ~= N0;     N0 ~=________________________________mmnm;
                                   2 ; (Sq1)2  ; : :;:(Sqn-1 )2       1




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                   25


                                              P [y12; : :;:yn+12; yn+2; : :;:ys*
 *; : :]:
        B1 ~= P [x2; : :;:xr]  N00;     N00~= ________________________________2*
 *ff1ffs
                                                     (y1  ; : :;:y2s ; : :):

and


Proposition 7.7.   For a fibration of type iiwith 1-connected fibre as above

   1. H*(E; F2) ~=P [x02; : :;:x0r]  N where g*(x0i) = xi if i  2.

   2. N is a nilpotent Hopf algebra of finite type that fits in an exact sequen*
 *ce of Hopf

      algebras
                                    h*      g*   00
                           1 ! N0 _____>N _____>N  ! 1 :

                                                                               *
 *     __
Proof. Like that of Proposition 7.4.                                           *
 *    |__|



                                8. The iteration


   Let X be a 1-connected mod 2 H-space that satisfies conditions F1, F2 and F3.

   It follows from section 2 that we can choose a polynomial generator of H*(X;*
 * F2),

detect it by an H-map BZ=2 ! X and form the sequence of H-fibrations


                            BZ=2 ! X ! E ! B2Z=2 :


Sections 5, 6 and 7 are concerned with the computation of H*(E; F2).  Now, we w*
 *ill

iterate this construction with E and the subsequent quotients and will obtain t*
 *he proofs

of Theorems 1.1 and 1.6.

   Given a polynomial generator x of H*(X; F2), where X is a mod 2 H-space sati*
 *sfying

conditions F1, F2 and F3 and according to Theorem 2.6 we can construct an H-map

f :BZ=2 ! X such that x restricts non trivially to H*(BZ=2; F2) and by construc*
 *tion

we can complete x to a system of generators where any other generator restricts*
 * trivially

to H*(BZ=2; F2). Moreover, this map fits in a sequence of H-fibrations

                                f       g       h    2
                         BZ=2 _____>X _____>E _____>B Z=2


   According to Proposition 5.2 our system of generators can be modified to a g*
 *ood

system of transgressive generators.  Actually, we can keep x itself in the new *
 *system.

For this, we should have a look at the proof of Proposition 5.2.  Since x is al*
 *ready

known to be transgressive and any other polynomial generators of degree less th*
 *an the

degree of x transgresses trivially, the only reason for changing x would be to *
 *have a

nilpotent generator y in its same degree which transgresses non trivially as we*
 *ll, but

this is impossible by diagram (16).

   The results of section 7 show that, essentially, we substitute the old polyn*
 *omial gen-

erator, x, by a new generator in dimension 2, x0, which is either polynomial in*
 * case the

fibration X ! E ! B2Z=2 was of type ior nilpotent if it was of type ii.

   Notice that E is again a 1-connected mod 2 H-space that satisfies conditions*
 * F1 and

F2 and it also satisfies condition F3, by Lemma 2.4 and [10, Theorem 3.2].

   So, therefore , we can repeat the operation with E1 = E and the subsequent q*
 *uotients

Ek using each time the new polynomial generator x(k)of degree two and, of cours*
 *e, we

stop if, eventually, our polynomial generator degenerates to a 2-dimensional ni*
 *lpotent

generator.




26                       CARLOS BROTO AND JUAN A. CRESPO


   Thus we obtain a sequence (either finite or infinite)


(17)                  X = E0 ! E1 ! : :!:Ek ! Ek+1 ! : : :


of principal fibrations

                           BZ=2 ! Ek ! Ek+1 ! B2Z=2


where BZ=2 ! Ek detects x(k) while Ek+1 ! B2Z=2 classifies x(k+1).


Proposition 8.1.     (i)The compositions in (17) are principal H-fibrations

                                fk      gk       hk   2   k
                        BZ=2k _____>X _____>Ek _____>B Z=2


 (ii) The evaluation map map (BZ=2k; X)fk ' X is a homotopy equivalence and Ek

      coincides with the Borel construction


                       Ek ' map (BZ=2k; X)fkxBZ=2kEBZ=2k


Proof. (i) For k = 1 this sequence is just the construction of the first step o*
 *f the se-

quence 17. For k > 1, assume by induction that we have H-fibrations


                            BZ=2j ! X ! Ej ! B2Z=2j


for j  k, where the fundamental class of H*(B2Z=2; F2) restricts to x(j)2 H*(Ej*
 *; F2)

which class is therefore the mod 2 reduction of the class in H*(Ej; Z=2j) class*
 *ified by

Ej ! B2Z=2j.

   In case the fibration Ek-1 ! Ek ! B2Z=2 was of type iiwe would have finished*
 * the

iteration and therefore the proof of (i). Thus, we assume that this is still a *
 *fibration of

type i and then there is a next step BZ=2 ! Ek ! Ek+1 ! B2Z=2 and we can form

the pull-back diagram of H-spaces


                   BZ=2kw ________>F ________>BZ=2 ______>B2Z=2kw

                     ww            ||__|        |           ww
                     ww            |            |           w
                     ww            ||           ||          www

                                  _|           _|
(18)               BZ=2k  ________>X __________>Ek_______>B2Z=2k
                                   |
                                   |            |
                                   |            |
                                   |            |
                                   |            |
                                  _|           _|
                                 Ek+1======   Ek+1


Then F ' K(A; 1) where A is a group that fits in an extension classified by the*
 * com-

position BZ=2 ! Ek ! B2Z=2k. We need to check the effect of this composition in*
 * co-

homology. Since the fundamental class of H*(B2Z=2k; F2) restricts to x(k)2 H*(E*
 *k; F2)

which is in turn detected by BZ=2 ! Ek the composition BZ=2 ! Ek ! B2Z=2k is

non trivial and therefore A ~=Z=2k+1.

   Both, X and Ek+1 are 1-connected, so we have an exact sequence


                       0 ! ss2(X) ! ss2(Ek+1) ! Z=2k+1 ! 0 :




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                   27


And by the Hurewicz theorem the second homomorphism represents a cohomology cla*
 *ss

classified by a map Ek+1 ! B2Z=2k+1 which is an H-map and fits in the sequence *
 *of

H-fibrations

                        BZ=2k+1 ! X ! Ek+1 ! B2Z=2k+1 :

   This finishes the induction and therefore the proof of (i). (ii) Since X is *
 *a connected

H-space, all the components in map (BZ=2k; X) are homotopy equivalent. So, in o*
 *rder

to prove that the evaluation map map (BZ=2k; X)fk ! X is a homotopy equivalence

it suffices to show the same statement for the component of the constant map.  *
 *And

this follows by induction.  We know the case k = 1 from Lemma 2.3.  And then we

apply Zabrodsky's lemma to the principal fibration BZ=2 ! BZ=2k+1 ! BZ=2k. Since

map  (BZ=2; X)c ' X, it follows that map (BZ=2k; X)c ' map (BZ=2k+1; X)c and by*
 * the

induction hypothesis map (BZ=2k+1; X)c ' X.

   Notice that the same is true for Ek. So, in the diagram

                                 BZ=2 =================   BZ=2

                                    |                       |
                                    |                       |
                                    |                       |
                                    |                       |
                                   _|                ev     _|

                            map (BZ=2k; X)fk ______________>X
                                    |                '      |
                                    |                       ||
                                    |                       |
                                    |                       |
                                   _|                       _|

                    map (BZ=2k; X)fkxBZ=2kEBZ=2k      _ _ _E_>k

                                    |                       |
                                    |                       |
                                    |                       |
                                    |                       |
                                   _|                       _|

                                B2Z=2k                   B2Z=2k

the dashed arrow might be obtained applying again Zabrodsky's lemma.  And this_*
 *is_

the required homotopy equivalence map (BZ=2k; X)fkxBZ=2kEBZ=2k ' Ek.         |_*
 *_|


   Assume now that the sequence (17) is infinite; that is, all the fibrations E*
 *k ! Ek+1 !

B2Z=2 are of type i. In this case we define

                                 E1 = hocolimkEk

Let us first study the cohomology of E1 . Suppose that

                                                 P [y1; : :;:ys; : :]:
                   H*(X; F2) ~=P [x1; : :;:xr]  ___________________2ff1ffs
                                                (y1  ; : :;:y2s ; : :):

for a good system of trangressive generators and x1 is the class detected by BZ*
 *=2 ! X.

According to Proposition 7.4

                         H*(E1; F2) ~=P [x01; x02; : :;:x0r]  N

where

       ____P_[Sq1;_:_:;:Sqn-1_]____              P (y21; : :;:y2n; yn+1; : :;:y*
 *s; : :]:
              m                 m  >____>N ____>>_____________________________f*
 *f1ffs
        (Sq1)2 n; : :;:(Sqn-1 )2 1                    (y21 ; : :;:y2s ; : :):




28                       CARLOS BROTO AND JUAN A. CRESPO


In the following steps we just detect the two dimensional class x01and produce *
 *a new x001

and so on, hence according to the remark 7.6


                       H*(Ek; F2) ~=P [x(k)1; x(k)2; : :;:x(k)r]  N


each map g :Ek ! Ek+1 maps x(k)1to zero and the other generators to the same in*
 *dexed

ones up to a polynomial in the previous generators, thus inducing an isomorphism

                                          ~=   *             k+1)
                     H*(Ek; F2)=(x(k)1) <____ H (Ek+1; F2)=(x1   )


So, using the Milnor exact sequence to compute the cohomology of a telescope we*
 * obtain


                  H*(E1 ; F2) ~=lim-kH*(Ek; F2) ~=P ["x2; : :;:"xr]  N


Observe that the polynomial class x1, finally disappeared and instead we keep, *
 *in par-

ticular, a three dimensional class in N that appeared after the first step, res*
 *tricted from

Sq1 2 H*(B2Z=2; F2). This is the one that we could classify in order the recove*
 *r the

original X. Let us make this statement precise.

   The sequences of Proposition 8.1 (i) combine in a direct system


                     : :_:_____>BZ=2k  _____>BZ=2k+1 _______>: : :

                                   |            |
                                   |            |
                                   |            |
                                   |            |
                                  _|           _|
                     : :=:======= X =========  X =========  : : :
                                   |            |
                                   |            |
                                   |            |
                                   |            |
                                   |            |
                                  _|           _|
                     : :_:_______>Ek _________>Ek+1_________>: : :

                                   |            |
                                   |            |
                                   |            |
                                   |            |
                                  _|           _|

                     : :_:_____>B2Z=2k ____>B2Z=2k+1 ______>: : :

   Hence we obtain fibrations:

                                f1      g1        h1   2   1
                       BZ=21  _____>X _____>E1  _____>B Z=2


and the mod 2 completion,

                               f^      ^g        ^h   2 1
                         BS^12____>X _____>E^1 _____>B S^2:



It remains to show that E1 , ^E1 are H-spaces.


Lemma 8.2.     For A = Z=21  or ^S12,

  (i) the evaluation map induces


                         map (BA; X)f ' map (BA; X)c ' X


      for any f :BA ! X.




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                   29


 (ii) Also induced by evaluation:


                              map (BZ=2; E1 )c ' E1

                              map (BZ=21 ; E1 )c ' E1

                              map (BS^12; ^E1)c ' ^E1


Proof. We know form Proposition 8.1 (ii) and because X is an H-space, that


                      map (BZ=2k; X)fk ' map (BZ=2k; X)c ' X


Now map (BZ=21 ; X) ' holim kmap (BZ=2k; X) and since lim1kss1map  (BZ=2k; X) ~=

lim-1kss1X = 0 from [5, XI,7.4] it follows that


                ss0(holimk map (BZ=2k; X)) ~=lim-0kss0(map (BZ=2k; X))


and then
                    map (BZ=21 ; X)c  ' holimkmap (BZ=2k; X)c

                                      ' holimkX

                                      ' X

Finally since X is 2-complete we have as well


                       map (BS^12; X)c ' map (BZ=21 ; X)c ' X


   (ii) Is a consequence of the diagrams of principal fibrations like

          map (BZ=21 ; BZ=21 )c ___>map(BZ=21 ; X)c ___>map (BZ=21 ; E1 )c

                    |                      |                     |

                  ' ||ev                 ' ||ev                  ||ev
                    |                      |                     |
                    _|                     _|                   _|
                 BZ=21   __________________>X__________________>E1

where the fibre of map (BZ=21 ; X)c ! map (BZ=21 ; E1 )c consists of those comp*
 *onents

of map (BZ=21 ; BZ=21 ) containing maps ': BZ=21 ! BZ=21  such that f1 O ' is h*
 *o-

motopy to a constant map. But this is detectable by cohomology with 2-adic coef*
 *ficients

and the only possibility is ' ' constant.

   It then follows that ev :map (BZ=21 ; E1 )c ! E1  is a homotopy equivalence.*
 *  The_

other statements are proved in the same way.                                   *
 *   |__|


Proposition 8.3.   The spaces E1 , E^1 are H-spaces and the fibrations

                                f1      g1        h1   2   1
                       BZ=21  _____>X _____>E1  _____>B Z=2


and
                                f^      ^g        ^h   2 0
                          BS^02____>X _____>E^1 _____>B S^2


are H-fibrations.


Proof. We explain two different proofs.

   First, observe that for a direct system indexed by N we have that


                  hocolimk Ek x hocolimkEk <____ hocolim kEk x Ek




30                       CARLOS BROTO AND JUAN A. CRESPO


is a homotopy equivalence and then the multiplications k: Ek x Ek ! Ek induce a

multiplication

                              1 :E1 x E1  _____>E1

It is not clear however that 1  has a homotopy neutral element. We can clearly *
 *guess

what the neutral element should be but then we need to show that the composition

                                  j1             1
                            E1  _____>E1 x E1  _____>E1


is homotopic to the identity.

   We know that the restriction to each ik: Ek ,! E1  is homotopic to the ident*
 *ity. So

the obstructions for 1 O ji to be homotopic to the identity lie in


                           lim-ikssimap (Ek; E1 )ik;   i  1:


   The Zabrodsky's lemma applied to the principal fibration BZ=2 ! Ek ! Ek+1

together with the fact that map (BZ=2; E1 )c ' E1  (see Lemma 8.2 ii) implies t*
 *hat

map  (Ek+1; E1 )ik+1' map (Ek; E1 )ik so that ssimap (Ek; E1 )ik are constant f*
 *unctors

and the higher limit functors vanish.

   We have therefore proved that 1 :E1 x E1 ! E1  has a homotopy neutral element

and then E1  becomes and H-space. We can easily see that f1 ; g1 ; h1  are H-ma*
 *ps.

   A different point of view consists in adapting the argument of Proposition 2*
 *.5 using_

be results of Lemma 8.2.                                                       *
 *  |__|


   Thus, our method produce a new H-space X1 out of X = X0 with one less polyno*
 *mial

generator and X1 still satisfies conditions F1, F2 and F3. Either, the iteratio*
 *n stops at

a finite place and the new H-space inherits a nilpotent two dimensional generat*
 *or or

the iteration does not stop and the new inherited classes start at dimension th*
 *ree.

   Observe that in any case the two dimensional classes of X are still in the n*
 *ew H-space

X1, unless x1 if it had dimension two (see Propositions 7.4 and 7.7).

   And now we can repeat our construction with the new H-space X1 and produce an

H-space X2 with one less polynomial generator. And so continue up to a final st*
 *ep Xn,

where Xn has no polynomial generator.

   We have obtained Xn, a 1-connected mod 2 H-space with H*(Xn; F2) nilpotent, *
 *so

map  (BZ=2; Xn) ' map (BZ=2; Xn)c, and QH*(Xn; F2) locally finite as A-module, *
 *hence

map  (BZ=2; Xn) ' map (BZ=2; Xn)c ' Xn. That is Xn satisfies the Sullivan conje*
 *cture,

or in other words Xn is LBZ=2-local or F (Xn) ' Xn, where F  is, as defined in *
 *the

Introduction, the composition of the BZ=2-nullification functor, LBZ=2, and Bou*
 *sfield-

Kan 2-completion.


Theorem 8.4.    Let X be a 1-connected mod 2 H-space that satisfies conditions *
 *F1, F2

and F3. Then

   1. There is a sequence of mod 2 H-spaces


                     X = X0 _____>X1 _____>: : :____>Xn = F (X)


      where all Xi satisfy as well conditions F1, F2 and F3, the depth of Xi is*
 * the depth

      of Xi-1 minus one and Xn = F (X) is LBZ=2-local.




                     H-SPACES WITH NOETHERIAN COHOMOLOGY                   31


   2. The maps

                                   Xi _____>Xi+1

      are principal H-fibrations with fibre either (CP 1)b2or BZ=2k for some k *
 * 1.

   3. The composition X _____>F (X) is as well a principal fibration with fibre*
 * the product

      of the fibres of the maps Xi _____>Xi+1.

Assume furthermore that H*(X; F2) is actually noetherian, then

   3. As algebras, H*(X; F2) ~= P  N, where P  is a polynomial algebra and N is*
 * a

      2-connected finite Hopf algebra.

   4. In the above sequence of mod 2 H-spaces

                     X = X0 _____>X1 _____>: : :____>Xn = F (X)

      all Xi have noetherian mod 2 cohomology.

   5. Xn = F (X) is a mod 2 finite H-space.

   6. The fibrations (17) involved in the construction of

                                   Xi _____>Xi+1

      are are of type iand Xi _____>Xi+1 is a principal H-fibrations with fibre*
 * (CP 1)b2.

   7. The composition X _____>F (X) is as well a principal fibration:
                               1    n
                           (CP  )b2  _____>X _____>F (X) :


Proof. 1 and 2 follow from the previous constructions. 3 and argument similar t*
 *o that

of Proposition 8.1 (i).

   Assume now that H*(X; F2) is noetherian. Using the propositions 7.4 and 7.7,*
 * we see

that at each step of our construction the obtained H-space has as well noetheri*
 *an mod

2 cohomology H*(Xi; F2) ~=Pi Ni where Pi is a polynomial algebra and Ni is a fi*
 *nite

algebra. In particular the mod 2 cohomology of Xn = F (X) is finite: H*((; F2)X*
 *n) ~=

Nn. So Xn = F (X) is a mod 2 finite H-space. It is known ([8]) that a 1-connect*
 *ed mod

2-finite H-space is actually 2-connected so its mod 2 cohomology, Nn cannot con*
 *tain

two dimensional classes. But, according to propositions 7.4 and 7.7 a two dimen*
 *sional

class in any Ni would be inherited by Nn. Hence, all Ni should be 2-connected.

   Observe as well that if one of the fibrations (17) involved in the construct*
 *ion was of

type iithen it would produce a nilpotent two dimensional class in the cohomolog*
 *y of

the constructed H-space (Proposition 7.7). Again this class would survive to Nn*
 *, which

is a contradiction, hence all the fibrations involved are of type i, the sequen*
 *ce (17) is

infinite

                Xi= E0 ! E1 ! : :!:Ek ! : :!:E1 = hocolimkEk

and the fibre of the principal fibration Xi! Xi+1= (E1 )b2is (CP 1)b2. We have *
 *proved __

3, 4, 5, 6. 7 follows as before.                                               *
 *      |__|


Proof of Theorem 1.1. It follows from Theorem 8.4 for the case in which H*(X; F*
 *2) is_

noetherian.                                                                    *
 *  |__|


Proof of Theorem 1.6. If X is a mod 2 H-space that satisfies the conditions F1,*
 * F2 and

F3 and x 2 H*(X; F2) is a polynomial generator, we can construct a fibration

                                  BZ=2 ! X ! E




32                       CARLOS BROTO AND JUAN A. CRESPO


and with the arguments of the beginning of this section, we can complete x to a*
 * good

system of transgressive generators such that x is the only one in this system t*
 *hat

transgresses non trivially.  The theorem then follows from Proposition 6.3 and *
 *Theo-_

rem 8.4(6).                                                                    *
 *  |__|


Example 8.5.    Observe that our method allows us to guess what the cohomology *
 *of

F (X) should be. Let us have a look at some examples with just one polynomial g*
 *ener-

ator.

   1. One four dimensional polynomial generator. We have already seen in exampl*
 *e 1.7

      that a four dimensional polynomial generator always appears together with*
 * its Sq1.

      So the minimal possible cohomology of an H-space , X, with a four dimensi*
 *onal

      polynomial generator is P [x4]  E(Sq1x4).

        After the first step we have H*(E1; F2) ~=P [x2]  E(x3) and then H*(X1;*
 * F2) ~=

      E(x3), the cohomology of S3.

   2. One eight dimensional polynomial generator. Also from example 1.7 we know*
 * that

      there are two minimal possibilities for the cohomology of an H-space, X, *
 *with an

      eight dimensional polynomial generator are. Namely

        -H*(X; F2) = P [x8]  E(x9; x11) which would be the three connected cove*
 *r of

         X1 with H*(X1; F2) = P [x3]=(x43)  E(x5).

        -H*(X; F2) = P [x8]  E(x5; x9) which in this case would be the three co*
 *nnected

         cover of X1 with H*(X1; F2) = E(x3; x5).

      In both cases with Sq2x3 = x5.  Those cohomology algebras correspond to G*
 *2,

      SU(3) and its three connective coverings.

   3. One 16-dimensional generator.  In the same way one obtains what would be *
 *the

      minimal possibilities for the cohomology of an H-space with a 16-dimensio*
 *nal poly-

      nomial generator. The possible Mi3are

                                Sq1      Sq8     Sq4
                             O _____> o <____ o <____ o ,

                                Sq1      Sq8     Sq2
                             O _____> o <____ o _____> o ,
                                 Sq1      Sq2      Sq8
                              O _____> o _____> o <____ o

      and
                                Sq1      Sq2      Sq4
                             O _____> o _____> o _____> o .

      Such an H-space should be the 3-connected cover of a mod 2 H-space with c*
 *oho-

      mology either

        -E(x3; x5; x9),
          P (x3)
        - ______4 E(x5; x9),
           (x3)
          P (x3; x5)
        - _________44 E(x9) or
           (x3; x5)
          P (x3; x5)
        - _________84 E(x9),
           (x3; x5)
      respectively, with Sq2x3 = x5, Sq4x5 = x9. Observe that those Hopf algebr*
 *as are

      primitively generated and by classical results (cf. [14]) they cannot app*
 *ear as the

      cohomology of an H-space.  Hence these minimal cohomology Hopf algebras a*
 *re




                    H-SPACES WITH NOETHERIAN COHOMOLOGY                   33


     not realizable. Notice that the first of the possibilities embeds in the c*
 *ohomology

     of SU(5), so SU(5)<3> realizes one on the minimal examples but with an add*
 *itional

     class in dimension seven.

  4. One 32-dimensional polynomial generator. As before, one 32-dimensional gen*
 *erator

     together with just the minimal amount of extra generators implied by Theor*
 *em 1.6

     cannot exist. However some of those minimal possibilities embed in the coh*
 *omolo-

     gies of the three connected covers of the Lie groups SU(9), E6, E7 and E8,*
 * where

     the corresponding modules Mi4are:

                           Sq1      Sq16    Sq8     Sq4
                        O _____> o <____ o <____ o <____ o ,

                           Sq1      Sq16    Sq8     Sq2
                        O _____> o <____ o <____ o _____> o ,
                            Sq1      Sq2      Sq16    Sq8
                         O _____> o _____> o <____ o <____ o

     and
                           Sq1      Sq2      Sq4      Sq8
                        O _____> o _____> o _____> o _____> o ,

     respectively (see [13]).


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   Departament de Matematiques
   Universitat Autonoma de Barcelona
   08193 Bellaterra, Spain.


   E-mail address: broto@mat.uab.es

   E-mail address: chiqui@mat.uab.es