NONSIMPLY CONNECTED HSPACES WITH FINITENESS
CONDITIONS
CARLES BROTO, JUAN A. CRESPO, AND LAIA SAUMELL
Abstract. This article is concerned with homotopy properties of Hspace*
*s X that
are reflected in the module of indecomposables QH*(X; Fp). It is shown *
*that mod
p Hspaces X of finite type with finite transcendence degree mod p coho*
*mology
and locally finite QH*(X; Fp) are BZ=pnull spaces, EilenbergMac Lane *
*spaces
K(^Zp; 2), K(Z=pr; 1), and extensions of those. If we restrict attentio*
*n to Hspaces
with noetherian mod p cohomology algebra, then we are left with finite *
*mod p
Hspaces and EilenbergMac Lane spaces.
1.Introduction
The mod p cohomology of an Hspace X is a Hopf algebra H*(X; Fp) that carries
a compatible action of the Steenrod algebra A. The Steenrod algebra action is*
* in
herited by the module of the indecomposables QH*(X; Fp) and we are interested *
*in
homotopy properties of X that are reflected in this Aaction, specially in the*
* non
simply connected case. In this article we will restrict our attention to the *
*class of
Hspaces X that satisfy the following finiteness conditions,
(F 1) H*(X; Fp) is of finite type.
(F 2) H*(X; Fp) has finite transcendence degree.
(F 3) QH*(X; Fp) is locally finite as module over the Steenrod algebra.
Recall that a module over the Steenrod algebra is called locally finite provid*
*ed any
Asubmodule generated by a single element is finite (cf. [11]). Since Hspace*
*s are
simple they are pgood in the sense of BousfieldKan [2] so, there will be no *
*loss of
generality if we assume all Hspaces to be pcompleted.
If we further restrict to the case where the transcendence degree of H*(X; F*
*p) is
actually zero, we end up with the class of BZ=pnull Hspaces of finite type. *
*Recall
___________
Authors are partially supported by DGES grant PB970203.
1
2 CARLES BROTO, JUAN A. CRESPO, AND LAIA SAUMELL
that BZ=pnull spaces are the local spaces for the nullification functor PBZ=p *
*of Bous
field and Dror Farjoun [6]. We will denote by F () = PBZ=p()^pthe composition*
* of
the nullification functor and pcompletion.
Proposition 1.1. A connected Hspace X of finite type is BZ=pnull if and only*
* if
it satisfies the equivalent conditions
1. The evaluation map is a homotopy equivalence, Map (BZ=p; X) ' X.
2. H*(X; Fp) is a locally finite Amodule.
3. QH*(X; Fp) is a locally finite Amodule and H*(X; Fp) has transcendence de*
*gree
zero.
Proof. Condition (1) follows by definition (see [6]). This is equivalent to co*
*ndition
(2) by [7, 6.3.1]. The fact that the transcendence degree is zero in H*(X; Fp) *
*implies
that there is just one component in Map (BZ=p; X), that of the constant map. Th*
*en
(1) and (2) are equivalent to condition (3) by [11, 3.9.7 and 6.4.5]. *
* 
Immediate examples of Hspaces that satisfy (F 1), (F 2) and (F 3) but are not
BZ=pnull are products of EilenbergMac Lane spaces K(^Zp; 2) and K(Z=pk; 1). A
product of a finite number of copies of these EilenbergMac Lane spaces is the *
*p
completed classifying space of a compact abelian Lie group and it is usually ca*
*lled an
abelian ptoral group. Our main result generalizes [3, 4] to the nonsimply con*
*nected
case and establishes the precise sense in which these examples are the only dif*
*ference
between these classes of Hspaces.
Theorem 1.2. Let X be a connected Hspace. If X satisfies (F 1), (F 2) and (F*
* 3),
then there exists a principal Hfibration
BP ____X ____F (X) ____B2P
where F (X) is a BZ=pnull Hspace of finite type and P is an abelian ptoral g*
*roup.
Condition (F 2) is the matter of discussion. We first observe that if a conne*
*cted H
space X satisfies (F 1) and (F 3), then H*(X; Fp) has transcendence degree zero*
* if and
only if it is locally finite as Amodule. Theorem 1.2 is thus interpreted as a *
*reduction
of the case of finite transcendence degree to the case of transcendence degree *
*zero.
NONSIMPLY CONNECTED HSPACES 3
On the other hand we are not aware of any example of an Hspace satisfying (F 1)
and (F 3) but not (F 2).
It is worth mentioning that Hspaces with noetherian mod p cohomology algebra
fit in the conditions of the above Theorem. Next Theorem improves the conclusion
in such cases.
Theorem 1.3. Let X be a connected Hspace. If the mod p cohomology ring
H*(X; Fp) is noetherian, then F (X) is a mod p finite Hspace.
Structure theorems for simply connected Hspaces with noetherian mod p cohomo*
*l
ogy already appeared in [3] for the 2local version and in [4] for the odd prim*
*ary case.
Notice that in the nonsimply connected situation and according to theorems 1.2*
*, 1.3
the possible extensions of mod p finite Hspaces that still have noetherian mod*
* p
cohomology can only be obtained by killing part of the three dimensional homoto*
*py.
More generally, we are allowed to kill two and three dimensional homotopy class*
*es
of BZ=pnull spaces if we want to end up with Hspaces satisfying (F 1), (F 2),*
* and
(F 3).
While dealing with nonsimply connected spaces a new interesting sort of exte*
*nsion
comes into the picture. Rather than killing homotopy classes we can enlarge the
fundamental group. The extreme case is that of Hspaces having BZ=pnull univer*
*sal
cover, or even mod p finite universal cover.
Example 1.4. The fundamental group of the compact Lie group SO(3) is Z=2. Now
for any n > 1 consider the projection Z=2n ____Z=2 and define the Hspace F by
the pullback diagram
F _____BZ=2n
__ 
 
? ?
SO(3) ___BZ=2
so that we have modified the fundamental group of SO(3) to ss1(F ) = Z=2n, while
the universal cover is still the three sphere S3. One computes
H*(F ; F2) ~=H*(S3; F2) H*(BZ=2n; F2) = E[x1; x3] P [fi(n)(x1)]
4 CARLES BROTO, JUAN A. CRESPO, AND LAIA SAUMELL
but F does not split as S3 x BZ=2n. In fact, using Miller's theorem [9] one sh*
*ows
that a section of F ____BZ=2n would factors through BZ=2n1, the homotopy fibre
of F ____SO(3). And this is not possible.
Theorem 1.5. Let X be a connected mod p Hspace of finite type, then its univ*
*ersal
cover is BZ=pnull if and only if it fits in a homotopy pullback diagram
X ________Bss1
___ 
 Bae (1)
? ?
F (X) ___Bss1(F (X))
where F (X) is a BZ=pnull Hspace and ae : ss1 ___ss1(F (X)) is a group epim*
*orphism
for some finitely generated ^Zpmodule ss1, with kernel a finite abelian pgrou*
*p Q.
Theorem 1.6. Let X be a connected Hspace. Its universal cover X" is mod p fi*
*nite
if and only if there is a diagram as 1 with F (X) mod p finite.
Example 1.7. Let p be an odd prime number. Let S3{pk} denote the homotopy
fibre of the degree pk self map of S3. It is a loop space with mod p cohomology
algebra
H*(S3{pk}; Fp) ~=[x2] E[x3]
with the relation fi(k)(x2) = x3, that is H2(S3{pk}; Z) ~=Z=pk.
ps 2 k
Consider a map S3{pk} ____B Z=p with 0 < s < k and let X be the homotopy
fibre of this map. We have an Hfibration sequence
ps 2 k
BZ=pk ____X ____S3{pk} ____B Z=p
from which it follows H1(X; Z) ~= Z=ps and H2(X; Z) ~= Z=ps. Hence we have
nontrivial classes x1 2 H1(X; Fp) and x2 2 H2(X; Fp) linked via fi(s)and also
y2 2 H2(X; Fp) and y3 = fi(s)(y2) 2 H3(X; Fp). The Serre spectral sequence with
coefficients in Z=p collapses at the E2 term, and we obtain
H*(X; Fp) ~=E[x1] [x2] P [y2] E[y3] :
Notice that in this example we combine the two facts mentioned above, on the *
*one
hand the fundamental group has been enlarged and on the other hand we have kill*
*ed
some classes in ss2(X).
NONSIMPLY CONNECTED HSPACES 5
Spaces Y with mod p cohomology P [x2] E[fi(s)(x2)] are shown in [1]. Moreover
a straightforward computation gives that H*(Y ; Fp) ~= E[y1] [fi(s)(y1)], hence
H*(X; Fp) ~= H*(Y x Y ; Fp). One may ask if the space X constructed above is
homotopy equivalent to the product Y x Y . Observe that the answer is no. On
the one hand we know that F (X) = S3{pr} which is simply connected. On the other
hand, since the functor F commutes with products F (Y x Y ) = F (Y ) x F (Y ),
but Y is already BZ=pnull, so that F (Y ) = Y and this is not simply connected.
Recall that the functor F preserves many of the structures that one likes to *
*attach
to an Hspace and therefore the above theorems apply directly to Hspaces with
additional structure. One outstanding example is that of loop structures. We ha*
*ve
therefore obtained structure theorems for mod p loop spaces under conditions (F*
* 1),
(F 2), and (F 3), or with noetherian mod p cohomology.
The paper is organized as follows. Section x2 is devoted to the general theor*
*y of
BP principal fibrations of Hspaces. In section x3 we describe the relationsh*
*ip be
tween the mod p cohomology of an Hspace of finite type and that of its univers*
*al cover
and derive the proof of theorems 1.5 and 1.6. In Section 4 we prove theorems 1.*
*2 and
1.3. All spaces will be considered pcomplete in the sense of BousfieldKan. H**
*()
will stand for mod p cohomology.
2. BP principal Hfibrations
In this section we look at the question whether or not a fibration with fibre*
* the
classifying space of an abelian ptoral group BP
f g
BP ____X ____Y
and X and Y mod p Hspaces, is a principal fibration and if this is the case, w*
*e are
also interested in whether or not this is a principal Hfibration. By this we *
*mean
h 2
that the classifying map Y ____B P is an Hmap. The results here generalize th*
*at
of [3] stated for elementary abelian pgroups.
6 CARLES BROTO, JUAN A. CRESPO, AND LAIA SAUMELL
Proposition 2.1. Let X be a connected mod p Hspace with finite type mod p coho
mology (F 1). The following conditions are equivalent,
1. The module of the indecomposables QH*(X) is locally finite as module over *
*the
Steenrod algebra (F 3).
2. For any ptoral group P and any map f :BP ____X, the evaluation
Map (BP; X)f ____X
is a homotopy equivalence.
Proof. We first prove that 1 implies 2. Since X is a connected Hspace it admit*
*s a
homotopy inverse that is inherited by the mapping space Map (BP; X), hence all *
*of
its connected components are homotopy equivalent, and it will be enough to show
the equivalence of condition 2 for the case of the constant map f = c: BP ____*
*X.
If P is elementary abelian, the result is already stated in [3] and follows f*
*rom [5] or
[11, 3.9.7]. For finite pgroups P we argue by induction on the order of P . If*
* the order
is small enough P is elementary abelian. Otherwise we can write a central exten*
*sion
Z=p ____P ____P=(Z=p) and the Zabrodsky's lemma [13, 9] applies, so that
Map (BP; X)c ' Map (B(P=(Z=p)); X)c ' X ;
this last by induction hypothesis.
Finally, if P is a ptoral group, we choose a discrete approximation; that i*
*s, a
S
sequence {Pk} of finite groups ordered by inclusion such that the inclusion k*
*Pk =
P1 ____P induces a mod p cohomology equivalence in the level of classifying s*
*paces.
That is to say hocolimk BPk _____BP is a homotopy equivalence after completion
and then
Map (BP; X) ' holimMap (BPk; X) :
k
Since every Pk is a finite pgroup, we have a homotopy equivalence, induced by *
*eval
uation
holimMap (BPk; X)c ' hocolimX ' X
k k
thus holimk Map (BPk; X)c contains just one component and
Map (BP; X)c ' holimMap (BPk; X)c ' X :
k
NONSIMPLY CONNECTED HSPACES 7
That condition 2 implies 1, follows immediately from [11, 6.4.5]. *
* 
Lemma 2.2. Let X be an Hspace and P an abelian ptoral group. For an Hmap
f :BP ____X the following conditions are equivalent
1. H*(BP ) becomes a finitely generated H*(X)module induced by f*.
2. For any other abelian ptoral group P 0and homomorphism ae: P 0_____P , t*
*he
Bae f
composition BP 0____BP ____X is nullhomotopic if and only if ae is triv*
*ial.
3. For any other abelian ptoral group P 0, the induced homomorphism
f]:[BP 0; BP ] ____[BP 0; X]
is injective.
Proof. Assume first condition 1. Let ae: P 0____P be a nontrivial homomorphi*
*sm
of abelian ptoral groups. Notice that we can restrict to the case where ae is *
*injective,
otherwise we can factor ae through its image. If ae is injective, then H*(BP 0*
*) is
finitely generated over H*(BP ) and hence it is also a finitely generated H*(X)*
*module
induced by the composition Bae* O f*. Hence f O Bae cannot be nullhomotopic.
Conversely, assume that condition 2 is satisfied. Let V be the maximal elemen*
*tary
abelian subgroup of P . If we choose an isomorphism (Z=p)r ~=V , each restrict*
*ion
BZ=p _____BP _____X is not nullhomotopic, hence H*(BZ=p) becomes finitely
generated over H*(X). Then H*(BV ) is itself finitely generated over H*(X), in
particular, the image of H*(X) ____H*(BV ) is a sub Hopf algebra of H*(BV ) of
maximal transcendence degree, and then the image of H*(X) ____H*(BP ) is also a
sub Hopf algebra of maximal transcendence degree of H*(BP ). Therefore H*(BP )
is finitely generated as H*(X)module.
We have shown that conditions 1 and 2 are equivalent. That 2 and 3 are equiva*
*lent
follows immediately from the fact that [BP 0; BP ] is a group, [BP 0; X] also i*
*nherits
from X a memorialisation and a zero element so that f] preserves multiplication*
* and
the zero element. Hence to check injectivity we only need to look at the kernel*
* of f].

Definition 2.3. An Hmap BP _____X is called mod p homotopy injective if it
satisfies the equivalent conditions of Lemma 2.2.
8 CARLES BROTO, JUAN A. CRESPO, AND LAIA SAUMELL
Remark 2.4. There is also a way to define a concept of homotopy kernel for an
f
Hmap BP _____X. If P is a discrete approximation of P , we set K = { x 2
P  B _____BP _____X is nullhomotopic}. The map BK = (BK )^p_____BP
induced by the inclusion, might be called the homotopy kernel of BP ____X.
It turns out that if X satisfies (F 1) and (F 3), then any map g :BP 0_____BP
for which the composition is nullhomotopic, factors through the homotopy kernel
BK _____BP . Also, f itself factors uniquely as BP _____B(P=K) _____X with
B(P=K) ____X mod p homotopy injective.
f
Proposition 2.5. Let P be an abelian ptoral group and BP _____X _____Y a
fibration with Y connected. Assume that X is an Hspace and f and Hmap. If f is
mod p homotopy injective, then for any abelian ptoral group P 0, Map (BP 0; X)*
*c ' X
if and only if Map (BP 0; Y )c ' Y .
f
Proof. Apply Map (BP 0; ) to the fibration BP ____X ____Y . One obtains anot*
*her
fibration:
Map (BP 0; BP ) ____ Map (BP 0; X)c ____ Map (BP 0; Y )c
where c stands for the constant map and denotes the set of maps h 2 [BP 0; BP ]
h
such that BP 0____BP ____X is nullhomotopic. Since f : BP ____X is mod p
homotopy injective then = {c}. Evaluation provides a map of fibrations
Map (BP 0; BP )c__Map (BP 0; X)c__Map (BP 0; Y )c
  
'  
? f ? ?
BP ______________X ______________Y
where the left vertical arrow is known to be a homotopy equivalence by Proposi
tion 2.1, and the result follows. *
* 
Assume now that we have a connected mod p Hspace X that satisfies (F 1) and
(F 3) and an Hmap
f :BP ____X
from the classifying space of an abelian ptoral group. According to Propositio*
*n 2.1
ev :Map (BP; X)f ____X
NONSIMPLY CONNECTED HSPACES 9
is a homotopy equivalence, thus we have obtained a different model for the spac*
*es
X that supports an action of the topological group BP . We define the homotopy
quotient as
XhBP = Map (BP; X)f xBP EBP
and we have a sequence of fibrations
f g h 2
BP ____X ____XhBP ____B P : (2)
In what follows we will show that XhBP is an Hspace and the maps g and h are
Hmaps. Furthermore, the sequence (2) is the only way to complete f :BP ____X
to a sequence of fibrations.
Proposition 2.6. Let X be a connected mod p Hspace that satisfies conditions *
*(F 1)
and (F 3). If P is an abelian ptoral group and f :BP _____X a mod p homotopy
injective Hmap, then
1. XhBP is an Hspace that satisfies (F 1) and (F 3),
2. the quotient map g :X ____XhBP is an Hmap, and
3. if Y is another Hspace satisfying (F 1) and (F 3), a map k :XhBP ____Y i*
*s an
g k
Hmap if and only if the composition X ____XhBP ____Y is an Hmap.
Proof. We use an argument similar to that of [3, 2.5]. Proposition 2.5 provide*
*s a
homotopy equivalence Map (BP x BP; XhBP )c ' XhBP . Then, the diagram
mBP
BP x BP _________BP
 
f x f f
? mX ?
X x X ___________X
 
g x g g
? mX ?
XhBP x XhBP _ _h_B_PXhBP
is completed by the map mXhBP due to Zabrodsky's lemma. Hence XhBP becomes
an Hspace and g an Hmap.
That XhBP satisfies (F 1) follows from the Serre spectral sequence. Proposit*
*ions
2.1 and 2.5 imply that XhBP satisfies condition (F 3).
10 CARLES BROTO, JUAN A. CRESPO, AND LAIA SAUMELL
Finally we prove point 3. If k is an Hmap then k O g is clearly an Hmap. Co*
*n
versely, if k O g is an Hmap again an argument using Zabrodsky's lemma shows t*
*hat
the map X xX ____Y xY ____Y factors uniquely through XhBP xXhBP . Now, the
mhBP k kxk mY
compositions XhBP xXhBP ____XhBP ____Y and XhBP xXhBP ____Y xY ____Y
are two possible factorizations, hence they are homotopic; that is, k is an Hm*
*ap. 
Proposition 2.7. Let X be a connected mod p Hspace that satisfies conditions (*
*F 1)
and (F 3), P is an abelian ptoral group and f :BP _____X a mod p homotopy
injective Hmap.
f g0
If Y is a connected space and BP ____X ____Y a fibration, there is a homoto*
*py
equivalence XhBP ' Y that completes the commutative diagram
f g
BPw ____Xw ____XhBP
ww w 
ww www '
f g0 ?
BP ____X ______Y
Furthermore if Y is an Hspace and g0an Hmap then all arrows in the above diag*
*ram
are Hmaps.
Proof. Proposition 2.1 implies that Map (BP; X)c ' X and this together with Pro*
*po
sition 2.5, that Map (BP; Y )c ' Y . So, Zabrodsky's lemma applies to the princ*
*ipal
fibration
f g
BP ____X ____XhBP
and maps with target Y . Since g0:X ____Y restricts trivially to BP , it facto*
*rs as
g k
X ____XhBP ____Y , making the above diagram homotopy commutative. It is then
clear that k is a homotopy equivalence.
Finally, if Y is an Hspace Proposition 2.6 applies and hence k is an Hmap. *
* 
Theorem 2.8. Let X be a connected mod p Hspace that satisfies conditions (F *
*1)
and (F 3). If P is an abelian ptoral group and f :BP _____X a mod p homotopy
injective Hmap, the extension
f g h 2
BP ____X ____XhBP ____B P (3)
provides an Hfibration sequence.
NONSIMPLY CONNECTED HSPACES 11
Proof.We have just to check that the map h is an Hmap. An abelian ptoral group
P is the product of a pcompleted torus T and a finite abelian pgroup Q. Thus,*
* we
can write two components hT :Y ____B2T and hQ :Y ____B2Q where Y = XhBP .
Since g is an Hmap, g O hQ ' * and the induced map ss1(X) _____ss1(Y ) is
an epimorphism, we can apply proposition 2.3.1 in [12] and conclude that hQ is *
*an
Hmap.
Let us turn our attention to hT. Notice that if B2pT is the projection from B*
*2P
to B2T then hT factors as the composition B2pT O h and this extends to a diagram
of fibrations
fQ ^g ^h 2
BQ ____Xw _____F _____B Q
 ww  __  2
BiQ  ww j B iQ
? f g ? h ?
BP ____X _____Y _____B2P
 
hT  B2pT
? ?
B2T === B2T
Since the upper row fibration fits in the conditions of propositions 2.6 and *
*2.7, F
is an Hspace, ^gis an Hmap and furthermore j is an Hmap.
Finally, j :F _____Y and hT :Y _____B2T satisfy the conditions of proposit*
*ion
2.3.1 in [12], namely the composition is nullhomotopic and by inspection of the
Serre spectral sequence, j induces in homology an isomorphism in degree one and*
* an
epimorphism in degree two, hence hT is an Hmap. 
3. Universal covers
Let X denote a connected mod p Hspace with finite type mod p cohomology. The
fundamental group will be a finitely generated ^Zpmodule
ss1(X) ~=Z=pr1Z x . .x.Z=prkZ x (^Zp)l:
p j
Let X" denote the universal cover of X. The fibration "X____X ____Bss1(X) i*
*s an
Hfibration and its Serre spectral sequence shows that j*: H1(Bss1(X)) ____H1(*
*X)
is an isomorphism and j*: H2(Bss1(X)) ____H2(X) is a monomorphism. The image
of j* is a sub Hopf algebra of H*(X) isomorphic to H*(Bss1(X))= Kerj* and then
12 CARLES BROTO, JUAN A. CRESPO, AND LAIA SAUMELL
the argument in the proof of [10, thm. 7.11] proves in our case that there is a
complementary Hopf algebra A of finite type such that
H*(Bss1(X))
H*(X) ~=____________ A
Ker j*
where A is 1connected. Up to a change of generators we can write
H*(Bss1(X)) ~=E[u1; : :;:uk+l] P [v1; : :;:vk]
with deg ui = 1, deg vi = 2, and fi(ri)(ui) = vi for i = 1; : :;:k in such a wa*
*y that
ff1 pffs
Ker j*, being 2connected, can be written as (vp1 ; : :;:vs ), where ffi2 N an*
*d s k,
so that,
P [v1; : :;:vs]
H*(Bss1(X))= Kerj* ~=E[u1; : :;:uk+l] ______________pff1pffs P [vs+1; :(:v:*
*k]4;)
(v1 ; : :;:vs )
and therefore
P [v1; : :;:vs]
H*(X) ~=E[u1; : :;:uk+l] ______________pff1pffs P [vs+1; : :v:k](5A):
(v1 ; : :;:vs )
The EilenbergMoore spectral sequence
*(Bss (X))* *
E*;*2~=TorH*;* 1 (H (X); Fp) =) H (X") :
will now make clear the relation between H*(X) and H*(X"). It is not hard to ob*
*tain
that TorP[v]*;*(_P[v]_(vpff); Fp) ~=E[z] with bidegz = (1; 2pff) and then
*
*(Bss (X))* H*(Bss (X))H (Bss1(X))
TorH*;* 1 (H (X); Fp) ~=Tor*;* 1 ____________ A; Fp ~=
Ker j*
*
~=TorH*(Bss1(X))*;*H_(Bss1(X))_; Fp A ~=E[z1; : :;:zs] A
Ker j*
where bidegzi = (1; 2pffi) and A remains in bidegrees (0; *). Notice then that*
* the
EilenbergMoore spectral sequence collapses at the E2 term and we obtain a exact
sequence of AHopf algebras
1 ____A ____H*(X") ____H*(X")==A ____1 (6)
where H*(X")==A is finitely generated by elements represented by z1; : :;:zs2 H*
**(X"),
deg zi= 2pffi 1 3, and A is the image of p*: H*(X) _____H*(X"). Since the
elements ziappear in odd degrees, if we are working at an odd prime p, zi2= 0 a*
*nd
NONSIMPLY CONNECTED HSPACES 13
then the above sequence splits to give H*(X") ~=AE[z1; : :;:zs], but this is no*
*t clear
for p = 2. In any case H*X" turns out to be generated by z1; : :;:zs as an Amo*
*dule.
Proposition 3.1. The mod p cohomology of a connected mod p Hspace of finite t*
*ype
X is described as
H*(Bss1(X); Fp)
H*(X) ~=_______________ A
Kerj*
where
1. A is a finite 1connected Hopf algebra if X" is mod p finite, or
2. A is a locally finite 1connected AHopf algebra if X" is BZ=pnull.
Proof. Follows from (6) 
Proposition 3.2. Let X be a connected mod p Hspace of finite type. Then, its
universal cover X" is also a mod p Hspace of finite type and
1. H*(X) is noetherian if and only if H*(X") is noetherian.
2. H*(X) has finite transcendence degree if and only if H*(X") has finite tra*
*nscen
dence degree.
3. QH*(X) is locally finite if and only if QH*(X") is locally finite.
Proof. (1) and (2) follow from the description of H*(X) and H*(X") in (5) and (*
*6).
The proof of (3) is same involved. Although it could be obtained in a purely al*
*gebraic
setting it seems to us more immediate a topological argument. From (5) and (6) *
*and
the fact that maps out from BV are controlled by mod p cohomology, we obtain th*
*at
the sequence
0 ____[BV; "X]__[BV; X]___[BV; Bss1(X)]
is exact for every elementary abelian pgroup V . Hence we have a fibration
Map (BV; "X)c ____ Map (BV; X)c ____ Map (BV; Bss1(X))c
Evaluation gives a homotopy equivalence Map (BV; Bss1(X))c ' Bss1(X) and then
Map (BV; "X)c ' "Xif and only if Map (BV; X)c ' X. This is the geometric reason*
* of
the statement (3) by Proposition 1.1. 
14 CARLES BROTO, JUAN A. CRESPO, AND LAIA SAUMELL
Let X be a connected BZ=pnull Hspace, then in equation (5) s = k and A is a
locally finite Amodule. Hence X" is also BZ=pnull. The universal cover migh*
*t be
BZ=pnull in more general situations. Those are determined by Theorem 1.5.
Proof of Theorem 1.5.If X fits in a pullback diagram like (1), its universal c*
*over is
homotopy equivalent to that of F (X) and this last is BZ=pnull by hypothesis.
Assume now that X" is BZ=pnull. In particular H*(X") is a locally finite Am*
*odule
and according to equation (6) A = Im{p*: H*(X) _____H*(X")} is also a locally
finite Amodule and H*(X) is described according to (5)
P [v1; : :;:vs]
H*(X) ~=E[u1; : :;:uk+l] ______________pff1pffs P [vs+1; : :v:k] A
(v1 ; : :;:vs )
with A a locally finite 1connected AHopf algebra.
Notice that the factor P [vs+1; : :v:k] is what prevents the mod p cohomology*
* of X
from being locally finite and this factor is inherited from H*(Bss1(X)), more p*
*recisely
from the cohomology of the torsion part of ss1(X).
Our objective is to eliminate such polynomial generators. Fix v to be one of *
*those.
Nillocalization provides a map of AHopf algebras
f* :H*(X) ____H*(BZ=p)
that detects precisely our polynomial generator v [4, thm. 2.10] .
By Lannes theory [8] we can realize f* as an Hmap f :BZ=p ____X that turns
out to be mod p homotopy injective. Now by Theorem 2.8 f fits in an Hfibration
sequence
f g h 2
BZ=p ____X ____E ____B Z=p
where E is the Borel construction Map (BZ=p; X)f xBZ=pEBZ=p (see [3] and [4] for
the details) that in turn extends to a pullback diagram
BZ=p === BZ=p

f  jXO f
? j ?
X"w______pXX___XBss1(X) (7)
www  __ 
w g  
? j ?
X" ______pEE____EBss1(E)
NONSIMPLY CONNECTED HSPACES 15
where jX O f is non trivial and ss1(E) ~=ss1(X)=Z=p.
Notice that jX O f is nontrivial because the generator v 2 H*(X) detected
by f was inherited from H*(Bss1(X)). Now, the same construction applies to
jX O f :BZ=p _____Bss1(X) and gives the fibration in the right column of the d*
*i
agram 7. Finally the two fibration fit together by naturality of the constructi*
*on.
Now observe that the Hspace E satisfies the same conditions as X namely, it *
*is
connected, mod p Hspace of finite type and its universal cover E"' "Xis BZ=pn*
*ull.
Hence, in case it is not yet BZ=pnull we can iterate the same construction. We
obtain, in this way, a sequence of Hspaces and Hmaps
X = E0 ____E1 ____E2 ____E3 ____ : : :
where Ek and Ek+1 are related by a pullback diagram
BZ=p ====== BZ=p

fk  jkO fk
? j ?
X"w______pkEk_____kBss1(Ek)
www  __ 
w  
? j ?
X" _____pk+1Ek+1_k+1Bss1(Ek+1)
where again ss1(Ek+1) ~=ss1(Ek)=Z=p.
Since the torsion part of ss1(X) is finite, and at each step we reduce the to*
*rsion
part of the fundamental group this process will finish at a finite stage Em tha*
*t will
be a BZ=pnull connected mod p Hspace.
Gluing together all the sequence of pullbacks we obtain
X ____Bss1(X)
__ 
 
? ?
Em ___Bss1(Em )
where Bss1(X) ____Bss1(Em ) is induced by an epimorphism ae: ss1(X) ____ss1(E*
*m )
with kernel a finite abelian pgroup.
In order to finish the proof of Theorem 1.5 we only need to show that Em is
obtained functorially from X as F (X). This follows from basic properties of t*
*he
nullification functor that can be found in [6]. Since PBZ=p annihilates BZ=p, *
*each
16 CARLES BROTO, JUAN A. CRESPO, AND LAIA SAUMELL
fk
fibration BZ=p _____Ek _____Ek1 induces a homotopy equivalence PBZ=p(Ek) '
PBZ=p(Ek+1). Hence PBZ=p(X) ' PBZ=p(Em ). But Em is already BZ=pnull, and then
PBZ=p(X) ' Em , thus also F (X) ' Em . 
Proof of Theorem 1.6.If F (X) is mod p finite, then its universal cover "F(X) i*
*s mod
p finite too. If we assume in addition that there is a pullback diagram
X ________Bss1
__ 
 Bae
? ?
F (X) ___Bss1(F (X))
then we have that F"(X) coincides with the universal cover of X.
Conversely, assume that X" is mod p finite, in particular BZ=pnull. Hence Th*
*eo
rem 1.5 applies and we obtain a diagram like the one above, thus proving that t*
*he
universal cover of F (X) is homotopy equivalent to X". Since F (X) is a BZ=pnu*
*ll H
space of finite type, Proposition 1.1 implies that H*(F (X)) has transcendence *
*degree
zero and then Proposition 3.1 together with equation 5 gives
P [v1; : :;:vs]
H*(F (X)) ~=E[u1; : :;:uk+l] ______________pff1pffs A
(v1 ; : :;:vs )
with A finite. In particular F (X) is mod p finite. *
* 
4. Proof of the Main Theorem
This section is devoted to the proof of our structural results for connected *
*Hspaces
with the prescribed finiteness conditions (F 1), (F 2), and (F 3), that is, The*
*orems 1.2
and 1.3. We first need to establish the statement that fibrewise localization [*
*6] pre
serves Hfibrations.
Lemma 4.1. If X ! E ! B is an Hfibration, then its fibrewise localization
F (X) ! E ! B is also an Hfibration. Furthermore, the induced map E ____E is
also an Hmap.
Proof. By the naturality of the fibrewise localization and by using the homotopy
equivalence F (X x Y ) ' F (X) x F (Y ). *
* 
NONSIMPLY CONNECTED HSPACES 17
Proof of Theorem 1.2.Assume X is a connected Hspace such that its mod p coho
mology satisfies conditions (F 1), (F 2) and (F 3). We can write its universal*
* cover
and its fundamental group in an Hfibration sequence
"X! X ! Bss1(X) : (8)
According to Proposition 3.2 the mod p cohomology of "Xsatisfies also the condi*
*tions
(F 1), (F 2) and (F 3) and then the results of [3, 4] apply. There is then an H*
*fibration
sequence
BP ____X" ____F (X") ____B2P (9)
where P is an abelian ptoral group, and F (X") is a simply connected BZ=pnull
Hspace.
Fibrewise localization of the fibration (8) is again an Hfibration. Together*
* with
fibration (9) this fits in a diagram
BPw ______"X______F (X")____B2Pw
ww __  ww
ww   ww
? ?
BP ______X ________X _____B2P (10)
 
 
 
? ?
Bss1(X)== Bss1(X)
Notice that the existence of the Hfibration sequence in the middle row is not *
*imme
diate but rather follows from Theorem 2.8. Indeed, X ' XhBP .
Observe also that it follows from the above diagram that F (X") is homotopy e*
*quiv
alent to the universal cover of X , and so, therefore Theorem 1.5 applies. We o*
*btain
a pullback diagram
BQ ======= BQ
 
 
? ?
X ______Bss1(X ) (11)
__ 
 Bae
? ?
F (X )___Bss1(F (X ))
where BQ is the classifying space of a finite abelian pgroup Q.
18 CARLES BROTO, JUAN A. CRESPO, AND LAIA SAUMELL
Now (10) and (11) combine to a diagram of fibrations
BPw _____W _____BQ
ww 
ww  
? ?
BP _____X ______X
  
  
 ? ?
?
*____F (X )== F (X )
where W is the homotopy fibre of the map X ! F (X ). From the fibration
BP ____W ____BQ
we deduce that W ' BP 0is the classifying space of a ptoral group and since W *
*is
the fibre of an Hmap it is itself an Hspace, and therefore P 0is an abelian p*
*toral
group.
Finally F (X) ' F (X ) and by Theorem 2.8 we obtain the required Hfibration
sequence
BP 0____X ____F (X) ____B2P 0 
Proof of Theorem 1.3.Assume now that H*(X) is noetherian. In particular X sat
isfies conditions (F 1), (F 2), (F 3), and so, in particular, the arguments in *
*the proof
of Theorem 1.2 above apply. Now, by Proposition 3.2 one has that H*(X") is also
noetherian. Hence following [3, 4], F (X") is a finite Hspace. But F (X") is h*
*omotopy
equivalent to the universal cover of X , hence Theorem 1.6 applies and we get t*
*hat
F (X ) ' F (X) is mod p finite. *
* 
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Carles Broto and Laia Saumell Juan A. Crespo
Departament de Matematiques, Centre de Recerca Matematica,
Universitat Autonoma de Barcelona, Institut d'Estudis Catalans,
E08193 Bellaterra, Spain E08913 Bellaterra, Spain
Email address: broto@mat.uab.es, Email address: chiqui@crm.es
laia@mat.uab.es