HSPACES 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 Hspaces
with noetherian mod two cohomology algebra. We will show that, up to 2c*
*ompletion,
they are, essentially, finite mod 2 Hspaces and their 3connected 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 Hspaces and the*
* newer
concept of pcompact group ([11]).
A finite Hspace is an Hspace whose underlying space is homotopy equivalent*
* to
a CWcomplex with a finite number of cells. In the localized version, a mod p *
*finite
Hspace stands for an Hspace which is finite up to pcompletion or equivalentl*
*y for
an Hspace which mod p cohomology ring is finite dimensional. By pcompletion *
*we
understand BousfieldKan pcompletion [5]. An Hspace, being simple, it is pgo*
*od in
the sense of BousfieldKan, so we will assume without loose of generality that *
*our mod p
Hspaces are pcomplete.
Other related spaces that play an important role are the three connected cov*
*ers of
compact Lie groups, finite Hspaces or mod p finite Hspaces. 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 pcompleted homotopy type of the origina*
*l finite
Hspace. 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 BousfieldKan pcompletion and proved that for a 1connected finite com*
*plex
X with finite ss2(X), if X denotes the nconnected cover of X, then F (X)*
* ' Xbp
for any positive integer n.
The three connected cover of a finite Hspace is not homotopy equivalent to *
*a finite
CWcomplex. 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 Hspace X is again a mo*
*d p
finite Hspace X" ' X<1>. X" is thus 2connected and its third homotopy group *
*is
torsion free ([8]). In order to construct the 3connected cover of X, we choos*
*e a map
_____________
The authors are partially supported by DGICYT grant PB940725.
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 3connected 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 Hspaces 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 Hspaces are essentially all mod p Hspaces 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 1connected mod 2 Hspace with noetherian mod 2 coho
mology algebra, then there exists a mod 2 finite Hspace F = F (X) and a princ*
*ipal
Hfibration
(1) ((CP 1)b2)n ! X ! F (X) :
Thus we easily obtain a cohomological characterization of three connected co*
*vers of
mod 2 finite Hspaces
Corollary 1.2. A mod 2 Hspace is the three connected cover of a mod 2 finite*
* Hspace_
if and only if its mod 2 cohomology ring is three connected and noetherian. *
* __
Corollary 1.3. All 1connected mod 2 Hspaces with noetherian mod 2 cohomolog*
*y ring __
are finite mod 2 Hspaces, (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 Hspace X:
(2) "X! X ! Bss1(X)
Corollary 1.4. All connected mod 2 Hspaces with noetherian mod 2 cohomology *
*ring
are finite mod 2 Hspaces, (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 1connected and pcomplete 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 Hspaces 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 Hspaces 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 Hspaces with noetherian
mod 2 cohomology, based in the corresponding result for mod 2 finite Hspaces (*
*[12],
[17]),
HSPACES WITH NOETHERIAN COHOMOLOGY 3
Corollary 1.5 (Slack [25], LinWilliams [20]).Let X be a homotopy commutative c*
*on
nected mod 2 Hspace with noetherian mod 2 cohomology. Then X is the direct pro*
*duct
of a finite number of EilenbergMacLane 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 *
*Hspace
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 1connect*
*ed mod 2
finite Hspace 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 Hmap 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 ^Z2module, has n*
*o non
trivial primitives in its 2adic 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 pcompleted 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 Hspace structure is pco*
*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
Hstructure 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 Hspaces (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 = (pi1 + 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 Hspace 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 Hspaces. Namely, Hspaces 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
n1 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 Hspace 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 ; (Sqn1 )2 1; (Sqn2 )2 2; : :;:(Sq1)2
F2
with n 0, m0 = 0, m1 = 1 and mk1 mk mk1 + 1 or
D m m mn mE
Miin= Sqn ; (Sqn1 )2 1; (Sqn2 )2 2; : :;:(Sq1)2 ; 2
F2
with n 0, m0 = 0, m1 = 1, mk1 mk mk1+ 1 and m mn, and an epimorphism
of unstable Amodules:
"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 1connected 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 Amodules Mn.
Also Theorem 1.1 can be stated in a more general form, for 1connected mod 2*
* H
spaces satisfying conditions F1,F2 and F3 (see theorem 8.4).
HSPACES 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 (Sqn1 )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 Hspace 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 Hfibrations B*
*Z=2 !
X ! E ! B2Z=2 for a given mod 2 Hspace 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 Hspace 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 DwyerWilkerson [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 Hfibrations. 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)fG'c:
The mod 2 cohomology of an Hspace 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
AHopf 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 Aalgebras R S one have a natural isomophism
~=
`(R) `(S) _____>`(R S). And therefore, the localization of an unstable AHopf*
* al
gebra is again an unstable AHopf algebra and the coaugmentation is as well a m*
*ap of
unstable AHopf algebras.
Theorem 2.2. Let X be a connected Hspace that satisfies conditions F1 and F*
*2. The
localization of H*(X; F2) gives a map of unstable AHopf algebras
X :H*(X; F2) ! H*(BV ; F2)
where V is an elementary abelian 2group 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.
HSPACES 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 AdamsWilkerson 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 Hmap.
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 Hmap.
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 Hspace 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 Hspace in s*
*uch a
way that the fibrations (7) are Hfibrations.
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 2group.
f
Lemma 2.4. Let V be any elementary abelian 2group 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 2group, 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 nullhomotop*
*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 nullhomotopic. 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 nullhomotopic. 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 Hspace that satisfies F1, F2 and*
* F3.
Assume that
f g
BV _____>X _____>E
is a principal fibration where V is an elementary abelian 2group, f is an Hm*
*ap and
H*(BV ; F2) becomes finitely generated as H*(X; F2)module induced by f*. Then,*
* E is
an Hspace and g an Hmap.
HSPACES 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 Hmap. 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 Hspace and g an Hmap. *
*__
It follows from these results that the space E of (6) is an Hspace and that*
* the
fibrations (7) are Hfibrations. 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 Hspace 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 Hfibrations
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.PGBAideals 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 Hfibration F _____>E _____>B Z=2.
Definition 3.1. Let R be an AHopf algebra. An ideal of R is called a Primiti*
*vely
Generated Borel Aideal (PGBAideal for short) if it is an Aideal 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
PGBAideals 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;
>>
>>
>>(Sqn1 )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(Sqn1 ) = (Sqn1 )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 (Sqn1 ) we obtain that
s n 2s+2n2s1 2s1 2n+2s1 2s1
Sq2 Sq = Sq Sq (Sqn1 ) = Sq Sq (Sqn1 ):
Iterating this formula we obtain
s n 2n+2s1 2n+2s2 2n+1 1
Sq2 Sq = Sq Sq : :S:q Sq (Sqns )
n+1 2
but Sq1(Sqnss) = (Sqns )2 and then Sq2 (Sqns ) ) = 0 by the Cartan formula*
* __
so Sq2 Sqn = 0 *
* __
s
Lemma 3.3. The minimal PGBAideal of H*(B2Z=2; F2) containing (Sqn )2 , n *
* 0,
s 0, is
1 2s+n 2s+1 2s 2s
J(n; s) = (Sq ) ; : :;:(Sqn1 ) ; (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+i1 ) = (Sqn+i ) andsthen we obtain that these elements are requi*
*red
in an Aideal containing (Sqn )2 .
On the other hand, we know Sq1Sqn = (Sqn1 )2 and so:
s n 2s 2s+1
Sq2 (Sq ) = (Sqn1 )
Iterating this result we obtain:
s+i 2s+i 2s+i+1
Sq2 (Sqni ) = (Sqni1 )
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 Aideal and in fact a PGBA*
*ideal. __
HSPACES WITH NOETHERIAN COHOMOLOGY 11
Proposition 3.4. The PGBAideals 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 ) ; : :;:(Sqn1 ) ; (Sqn ) ; : :;:(Sqn+r ) ; : : :;
where m0 = s, m1 = s + 1 and mk = mk1 + ffl, ffl = 0; 1.
Type iis:
2m 1 2mn 2m1 2s 2s
J = ; (Sq ) ; : :;:(Sqn1 ) ; (Sqn ) ; : :;:(Sqn+r ) ; : : :;
where m0 = s, m1 = s + 1, mk = mk1 + ffl, ffl = 0; 1 and m mn.
Observe that the trivial PGBAideal "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, (Sqn1m)2 62 J. Now our ideal might contai*
*n other
primitive generators like (Sqnk )2 withkmk s + k so it would be
1 2mn 2mk 2m1 2s 2s
J = (Sq ) ; : :;:(Sqnk ) ; : :;:(Sqn1 ) ; (Sqn ) ; : :;:(Sqn+r ) *
*; : :;:
with m1 = s + 1. However, since it should be an Aideal and as we have observed
mk+nk+1 2mk 2mk 2mk1 2mk1
before Sq2 ((Sqnk ) ) = (Sqnk+1 ) and Sq ((Sqnk+1 ) ) =
(Sqnk )mk1+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 Amodule M is said to be nilpotent of cla*
*ss l or
lnilpotent if for every homogeneous element of degree m and 0 k < l
r(mk) 2(mk) mk
Sq2 : :S:q Sq x = 0
for a large enough r.
An unstable Amodule M is nilpotent if it is 1nilpotent and it is reduced i*
*f it does
not contain a nontrivial nilpotent submodule.
For example lfold suspensions of unstable Amodules are lnilpotent.
Let J be a PGBAideal 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 Amodules
(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 ; : :;:(Sqn1 ) >F2 and
o L = 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 Amodule
(10) L^= :F2
and for J = H"*(B2Z=2; F2), L^ = :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 Amodules 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 M1 ~*
*=F2 if
J = "H*(B2Z=2; F2).
Alternatively, we can describe the unstable Amodules Mon, n 0, as the subq*
*uotients
of H*(B2Z=2; F2) described as vector spaces by
m1 2m2 1 2mn
Min= F2
with m0 = 0, m1 = 1 and mk = mk1 + ffl, ffl = 0; 1, or
m1 2m2 1 2mn 2m
Miin= F2
with m0 = 0, m1 = 1, mk = mk1 + ffl, ffl = 0; 1, and m mn.
In the example 1.7 are described the modules possible modules Minfor n = 1; *
*2.
HSPACES 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 Hfibration 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) Bmn+1 .
2. Furthermore, if the transgression d: Bn1 ! P n(A) is trivial, then d 0.
Proof. For an element x 2 Bm , we can write
X
d(x) = aibi2 En;mn+1 ~=An Bmn+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;1n+p E0;q) (E0;p En;1n+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 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) Bmn+1 En;mn+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)Bmn+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 kbkskxsk11and 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;n1 E0;mn+1, that might be written as x1 y + b y0, where b is a polyno*
*mial
HSPACES WITH NOETHERIAN COHOMOLOGY 15
on generators different of x1 and y; y0 are any elements in Bmn+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;mn+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 Hfibrations over B2Z=2
We are interested in the behavior of the Serre spectral sequence of an Hfib*
*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 Hfibration, 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 En1. If dn1 is trivial En ~=En1*
* and there
is nothing to prove. Thus, we suppose that dn1 is non trivial. According to le*
*mma 4.1
there should be a non trivial transgression which target is a primitive element*
* of An1
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 An1 is written as
H*(B2Z=2; F2) F2 [; Sq1; : :S:qn ; : :]:
An1 ~=______________ ~= ________________________________ssr1
(1; : :;:r1) (Sqm1 )2 ; : :;:(Sqmr1 )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
An1 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 En1 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 Hfibration, 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; : :;:xi1) 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.
HSPACES 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; : :;:xq1 are transgressive and satisfy (1*
*), (2) and (3).
Then we first prove:
Claim 5.3. (4) and (5) applies to the generators x1; : :;:xq1.
k1 mq12kq1
Proof. Pick a monomial y = xm121 . .x.q1 , with m1; : :;:mq1 odd integers.*
* A
k1 2kq1
differential of y is computed in terms of that of x21 ; : :x:q1. 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; :*
* :;:xq1.
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; : :;:xq1; 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; : :;:xq1. 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; : :;:xq1, so the whole bky2sk+1is a polynomial on the *
*generators
x1; : :;:xq1. Now, we check the next differential and modify x0qagain if it is*
* necessary
and so on until we have obtained "xq= xq + pq(x1; : :;:xq1) which is transgres*
*sive.
Suppose, next, that xq have non trivial transgression but there is another m*
*onomial,
y, on the previous generators, x1; : :;:xq1. 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; : :;*
*:xq1)
with the same height as xq and such that x1; : :;:xq1; "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 Hfibration 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 *
Hfibration 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 2s1x  2x *
* 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 Amodule of H*(F ; F2) generated by x1; x2; : :;:*
*xr; : :a:s
Amodule: :A
HSPACES WITH NOETHERIAN COHOMOLOGY 19
Notice that A1 should coincide with the image of p* and the kernel is ker p*
** =
(1; 2; : :;:r; : :):, a PGBAideal 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 Amodules
(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 Hfibrations over
B2Z=2
In this section we further study the transgression map in the Serre spectral*
* sequence
of an Hfibration
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 nillocalization of H*(F ; F2) (see Theorem 2.2) provi*
*des a
sequence:
0 ! ker ! A _____>A ! 0
where 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 PGBAideal I = kerp* can also be decomposed, accordin*
*g to
the sequence (9), as a maximal subAmodule, K, which is nilpotent of class 2 an*
*d the
quotient L, the suspension of a reduced Amodule,
Then we obtain a corresponding decomposition of the transgression map o (15):
ker >___>A_>>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 Hfibration where H (F ; F2) sat*
*isfies
conditions F1, F2 and F3. Then either the Hfibration 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 nullhomotopic [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 Hmap. 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
Hmap, 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 Hmap.
We have obtained a diagram of Hfibrations and Hmaps
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 Hfibrations F _____>E ____*
*_>B Z=2
according to the type of the ideal kerp*.
j p 2 *
Definition 6.2. A non trivial Hfibration F _____>E _____>B Z=2, where H (F ; *
*F2)
satisfies conditions F1, F2 and F3 is of type iif the PGBAideal 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 Hfibration with H *
*(F ; F2)
satisfying the conditions F1, F2 and F3. The transgression is determined by the*
* induced
epimorphism of unstable Amodules
o":QH*(F ; F2) ____>>Mon
for some n 1 where Mon= Minif the fibration is of type i and Mon= Miinor
M1 if the fibration is of type ii. It satisfies "o(xi) = Sqn , for xi a least*
* dimensional
HSPACES 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 M1 if n = 1.
See 3.6 for the definition of the unstable Amodules 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 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
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 Hfibrations over
B2Z=2
So far, we have obtained the structure of the E1 term of the Serre spectral*
* sequence
g h 2
of an Hfibration F _____>E _____>B Z=2:
E1 ~= A1 B1
where
H*(B2Z=2; F2)
A1 = ______________;
I
with I a PGBAideal of H*(B2Z=2; F2) and B1 is a subAHopf 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
gPA hPB
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. *
* __
HSPACES WITH NOETHERIAN COHOMOLOGY 23
7.1. Fibrations of type i. Let us now specialize to Hfibrations
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~=F2
We may suppose that
m1 1 2mn
"o(x1) = Sqn ; "o(y1) = (Sqn1 )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; : :;:Sqn1 ]
A1 ~= P [] N0; N0 ~=____________________________mnm
(Sq1)2 ; : :;:(Sqn1 )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 1conne*
*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 1connected 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 1connecte*
*d is the
Hfibration
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 = 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 1connect*
*ed
fibre; that is, Hfibrations
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~=F2 and we may *
*as
sume that
m1 1 2mn *
*2m
"o(x1) = Sqn ; "o(y1) = (Sqn1 )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; : :;:Sqn1 ]
A1 ~= N0; N0 ~=________________________________mmnm;
2 ; (Sq1)2 ; : :;:(Sqn1 )2 1
HSPACES 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 1connected 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 1connected mod 2 Hspace 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 Hmap BZ=2 ! X and form the sequence of Hfibrations
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 Hspace sati*
*sfying
conditions F1, F2 and F3 and according to Theorem 2.6 we can construct an Hmap
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 Hfibrations
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 1connected mod 2 Hspace 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 2dimensional 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 Hfibrations
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 Hfibrations
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 Ek1 ! 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 pullback diagram of Hspaces
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 1connected, so we have an exact sequence
0 ! ss2(X) ! ss2(Ek+1) ! Z=2k+1 ! 0 :
HSPACES 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 Hmap and fits in the sequence *
*of
Hfibrations
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
Hspace, 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;_:_:;:Sqn1_]____ P (y21; : :;:y2n; yn+1; : :;:y*
*s; : :]:
m m >____>N ____>>_____________________________f*
*f1ffs
(Sq1)2 n; : :;:(Sqn1 )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) ~=limkH*(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 Hspaces.
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.
HSPACES 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 Hspace, 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) ~=
lim1kss1X = 0 from [5, XI,7.4] it follows that
ss0(holimk map (BZ=2k; X)) ~=lim0kss0(map (BZ=2k; X))
and then
map (BZ=21 ; X)c ' holimkmap (BZ=2k; X)c
' holimkX
' X
Finally since X is 2complete 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 2adic 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 Hspaces 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 Hfibrations.
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
limikssimap (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 Hspace. We can easily see that f1 ; g1 ; h1 are Hma*
*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 Hspace 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 Hspace 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 Hspace
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 Hspace X1 and produce an
Hspace 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 1connected mod 2 Hspace with H*(Xn; F2) nilpotent, *
*so
map (BZ=2; Xn) ' map (BZ=2; Xn)c, and QH*(Xn; F2) locally finite as Amodule, *
*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=2local or F (Xn) ' Xn, where F is, as defined in *
*the
Introduction, the composition of the BZ=2nullification functor, LBZ=2, and Bou*
*sfield
Kan 2completion.
Theorem 8.4. Let X be a 1connected mod 2 Hspace that satisfies conditions *
*F1, F2
and F3. Then
1. There is a sequence of mod 2 Hspaces
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 Xi1 minus one and Xn = F (X) is LBZ=2local.
HSPACES WITH NOETHERIAN COHOMOLOGY 31
2. The maps
Xi _____>Xi+1
are principal Hfibrations 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
2connected finite Hopf algebra.
4. In the above sequence of mod 2 Hspaces
X = X0 _____>X1 _____>: : :____>Xn = F (X)
all Xi have noetherian mod 2 cohomology.
5. Xn = F (X) is a mod 2 finite Hspace.
6. The fibrations (17) involved in the construction of
Xi _____>Xi+1
are are of type iand Xi _____>Xi+1 is a principal Hfibrations 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 Hspace 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 Hspace. It is known ([8]) that a 1connect*
*ed mod
2finite Hspace is actually 2connected 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 2connected.
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 Hspace (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 Hspace 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 Hspace , 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 Hspace, 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 16dimensional generator. In the same way one obtains what would be *
*the
minimal possibilities for the cohomology of an Hspace with a 16dimensio*
*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 Hspace should be the 3connected cover of a mod 2 Hspace 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 Hspace. Hence these minimal cohomology Hopf algebras a*
*re
HSPACES 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 32dimensional polynomial generator. As before, one 32dimensional 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.
Email address: broto@mat.uab.es
Email address: chiqui@mat.uab.es