DECONSTRUCTING HOPF SPACES
NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
Abstract. We characterize Hopf spaces with finitely generated cohomology *
*as
algebra over the Steenrod algebra. We "deconstruct" the original space in*
*to an H-
space Y with finite mod p cohomology and a finite number of p-torsion Eil*
*enberg-
Mac Lane spaces. One reconstructs X from Y by taking extensions by princi*
*pal
H-fibrations. We give a precise description of homotopy commutative H-spa*
*ces in
this setting and give a criterion to recognize connected covers of H-spac*
*es with finite
mod p cohomology. The key observation is that the module of indecomposab*
*les
QH*(X; Fp) lies in some stage of the Krull filtration of the category of *
*unstable
modules over the Steenrod algebra. We compare this algebraic condition w*
*ith a
topological one, namely that some iterated loop space of X is BZ=p-local.
Introduction
An H-space is a pointed space with a multiplication where the base point acts*
* as
a two side unit up to homotopy. The mere existence of a multiplication provides*
* a
very rich structure. For example the cohomology is then a Hopf algebra which is
compatible with the action of the Steenrod algebra Ap, and the space is simple *
*(that
is, its fundamental group is abelian acting trivially on the higher homotopy gr*
*oups).
Finite H-spaces - whose underlying space is of the homotopy type of a finite *
*CW -
complex - have been very well studied because they are homotopical analogues of
compact Lie groups. For some time it was conjectured that all finite H-spaces w*
*ere
compact Lie groups, S7 or products of those, but the localization and mixing te*
*ch-
niques of Zabrodsky (see e.g. [31]) allowed to construct new examples out of t*
*he
p-local structure of classical Lie groups. Together with the study of torsion t*
*his mo-
tivated to focus on mod p cohomological properties of H-spaces, one prime at a *
*time.
In particular an H-space is said to be mod p finite if it is p-complete with fi*
*nite mod
p cohomology (which we denote simply by H*(-)).
Natural examples of H-spaces arising in connection with finite ones are their*
* Post-
nikov sections and connected covers. Recall that the n-th Postnikov section X[n*
*] of a
(finite) H-space X can be reconstructed from the knowledge of the n first homot*
*opy
groups of X together with the (n - 1) first k-invariants, which are primitive c*
*ohomol-
ogy classes. We will call H-Postnikov piece an H-space which has only finitely *
*many
non-trivial homotopy groups. The n-connected cover X of X is the homotopy
fiber of the natural map X ! X[n]. In other words it is an H-space which fits a*
*s the
___________
The authors are partially supported by MCyT grant BFM2001-2035 and the third *
*author by the
program Ram'on y Cajal, MCyT, Spain.
1
2 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
total space of a fibration where the base is a (finite) H-space and the fiber *
*X[n] is
an H-Postnikov piece.
The mod p cohomology of the n-connected cover of a finite H-space is not fini*
*te
in general but finitely generated as algebra over the Steenrod algebra Ap (see *
*Sec-
tion 6 for details). Up to p-completion the known examples of H-spaces which ha*
*ve
their mod p cohomology finitely generated over Ap are all finite H-spaces, Eile*
*nberg-
Mac Lane spaces of type K(Z=pr, n) and K(Z^p, n), or more generally p-torsion H-
Postnikov pieces.
Our main result shows that those are basically the only examples. More precis*
*ely
one can recover any such H-space by taking products and certain extensions by H-
fibrations.
Theorem 6.5. Let X be a connected H-space such that H*(X) is a finitely gener-
ated algebra over the Steenrod algebra. Then X is the total space of an H-fibra*
*tion
F ____-X ____-Y
where Y is an H-space with finite mod p cohomology and F is a p-torsion H-Postn*
*ikov
piece whose homotopy groups are finite direct sums of copies of cyclic groups Z*
*=pr
and Prüfer groups Zp1.
As an application of this result we extend Hubbuck's Torus Theorem (see [15]),
which is certainly one of the most important results on finite H-spaces. It say*
*s that
a homotopy commutative finite H-space is homotopy equivalent to a finite product
of circles. We offer the following generalization:
Corollary 6.6. Let X be a connected homotopy commutative H-space with finitely
generated cohomology as algebra over the Steenrod algebra. Then, X ' (S1)n x F *
*, up
to p-completion, where F is p-torsion H-Postnikov piece.
When H*(X) is finitely generated as algebra we get back M. Slack's result [29*
*], as
well as their generalization by Lin and Williams in [20]. They proved that up t*
*o p-
completion the homotopy commutative H-spaces with finitely generated cohomology
as algebra are finite products of S1's, K(Z=pr, 1)'s, and K(Z, 2)'s.
The arguments to prove our main theorem are the following. When H*(X) is
finitely generated over Ap we show in Lemma 6.1 that the unstable module of ind*
*e-
composable elements QH*(X) is also finitely generated over Ap. The key observat*
*ion
is that such a module belongs to some stage Un of the Krull filtration in the c*
*ategory
of unstable modules. This filtration has been studied by L. Schwartz in [27] in*
* order
to prove Kuhn's non-realizability conjecture [17]. Moreover, if H*(X) is finite*
*ly gen-
erated as Ap-algebra then Lannes' T functor T H*(X) is always of finite type, a*
*nd
so is TV H*(X) for any elementary abelian p-group V . Therefore T H*(X) computes
actually the cohomology of the mapping space map (BZ=p, X), see [18].
The stage U0 of the Krull filtration is particularly interesting since it con*
*sists in all
locally finite modules (direct limits of finite modules). The condition that QH*
**(X)
is locally finite is in fact equivalent to ask that the loop space X be BZ=p-l*
*ocal, i.e.
DECONSTRUCTING HOPF SPACES 3
the space of pointed maps map *(BZ=p, X) is contractible, see [12, Prop 3.2] a*
*nd
[25, Proposition 6.4.5].
We extend this topological characterization to H-spaces with QH*(X) 2 Un.
Theorem 5.3. Let X be a connected H-space such that TV H*(X) is of finite type
for any elementary abelian p-group V . Then QH*(X) is in Un if and only if n+1X
is BZ=p-local.
We apply now P. Bousfield's results on the Postnikov-like tower associated to*
* the
BZ=p-nullification functor PBZ=p (relying on his "Key Lemma", see [4, Chapter 7*
*]).
They enable us to reconstruct those H-spaces such that n+1X is BZ=p-local from
PBZ=pX in a finite number of principal H-fibrations with p-torsion Eilenberg-Mac
Lane spaces.
Theorem 5.5. Let X be a H-space such that TV H*(X) is of finite type for any
elementary abelian p-group V . Then QH*(X) is in Un if and only if X is the tot*
*al
space of an H-fibration
F ____-X ____-PBZ=pX
where F is a p-torsion H-Postnikov piece whose homotopy groups are finite direct
sums of copies of cyclic groups Z=pr and Prüfer groups Zp1 concentrated in degr*
*ees
1 to n + 1. |
When n = 0 we recover the results of C. Broto, L. Saumell and the second named
author in [7, 10, 8] (see Corollary 5.6). We notice finally that when H*(X) is*
* a
finitely generated Ap-algebra, then the space PBZ=pX is an H-space with finite *
*mod
p cohomology. We learn indeed from Miller's solution [22, Theorems A,C] of the
Sullivan conjecture that finite H-spaces are BZ=p-local. The extension by Lann*
*es
and Schwartz in [19] of Miller's theorem characterizes algebraically the nilpot*
*ent
BZ=p-local spaces: their mod p cohomology is locally finite.
It is worthwhile to mention that working with H-spaces is crucial as illustra*
*ted by
the example of BS3, see Example 3.7. Its loop space S3 is BZ=p-local, but the f*
*iber
of the nullification map has infinitely many non-trivial homotopy groups.
The paper is organized as follows. Sections 1 and 2 contain results about Lan*
*nes'
T functor and the Krull filtration. Section 3 is dedicated to Bousfield's nulli*
*fication
functor. In Section 4 you will find a discussion on when the pointed mapping sp*
*ace
map *(BZ=p, X) is an infinite loop space. The proof of Theorems 5.3 and 5.5 are*
* given
in Section 5. Finally the main results on H-spaces with finitely generated coho*
*mology
as algebra over Ap are proven in Section 6. Proposition 6.8 in this section pro*
*vides a
criterion to recognize connected covers of H-spaces whose mod p cohomology is f*
*inite.
Notation. Throughout the paper cohomology will be understood with Fp coeffi-
cients. We say that H*(X) is of finite type if Hn(X) is a finite Fp-vector spac*
*e for
any integer n 0.
Acknowledgments. Most of this work has been done in the coffee room of the
Maths Department at the UAB. We would like to thank Alfonso Pascual for his
4 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
generosity. We warmly thank Carles Broto for his questions which regularly open*
*ed
new perspectives.
1. Lannes T functor
Lannes' T functor was designed as a tool to compute the cohomology of mapping
spaces with source BV , the classifying space of an elementary abelian p-group *
*V . It
was used also by J. Lannes to give an alternative proof of Miller's Theorem on *
*the
Sullivan's conjecture.
Let U (resp. K) be the category of unstable modules (resp. algebras) over t*
*he
Steenrod algebra. The functor TV is the left adjoint of - H*(BV ) in K where *
*V is
an elementary abelian p-group._The left adjoint of - eH*(BV ) is called the r*
*educed
T functor and denoted by TV . For each unstable module_M 2 U we have a splitting
of modules over the Steenrod algebra TV M = M TV M. We will use T to denote
TZ=pand ~Tto denote ~TZ=p.
If M = H*(X), the evaluation map BV xmap (BV, X) ! X induces by adjunction
a morphism of unstable algebras over the Steenrod algebra
~V : TV H*(X) ____-H*(map (BV, X)).
Among the results giving conditions under which ~V is an isomorphism (cf.[25]),*
* we
highlight the following proposition, since it adapts specially well to our situ*
*ation.
Proposition 1.1. [18, prop 3.4.3] Let X be a p-complete space such that H*(X) is
of finite type and let V be an elementary abelian p-group. If one of the foll*
*owing
hypotheses
a) H*(map (BV, X)) is of finite type,
b) TV H*(X) is finite type,
is verified. Then, the following three conditions are equivalent:
(1) the map TV H*(X) ____-H*(map (BV, X)) is an isomorphism in K.
(2) The space map (BV, X) is p-complete.
(3) The space map (BV, X) is p-good.
When X is an H-space, then map (BV, X) is again an H-space, and so is the
connected component map (BV, X)c of the constant map (see [32]). Moreover, if X*
* is
connected, all connected components of the mapping space have the same homotopy
type. Since an H-space is always p-good, condition (3) in Proposition 1.1 is al*
*ways
satisfied. In particular, conditions a) and b) are equivalent. Proposition 1.1 *
*can now
be reformulated in the following way when the spaces involved are H-spaces.
Proposition 1.2. Let X be a H-space such that H*(X) is of finite type. Assume
that TV H*(X) (or equivalently H*(map (BV, X))) is of finite type. Then
TV H*(X) ~=H*(map (BV, X))
as algebras over the Steenrod algebra. Moreover map (BV, ^Xp) is p-complete.
DECONSTRUCTING HOPF SPACES 5
Proof.There is weak equivalence map *(BV, X) ' map *(BV, ^Xp) for any elementary
abelian p-group V by [22, Theorem 1.5] since X is an H-space. As the evaluati*
*on
map is an H-map and has a section, it follows that there is a splitting
map (BV, X) ' X x map *(BV, X) .
Recall that X is p-good. Therefore ^Xpis p-complete and the completion map indu*
*ces
an isomorphism H*(X) ~=H*(X^p) as algebras over the Steenrod algebra. Combin-
ing these last two remarks with the fact that TV H*(X^p) ~= H*(map (BV, ^Xp)) (*
*see
Proposition 1.1), we obtain the desired isomorphism. *
* |
When working with H-spaces it is often handy to deal with the pointed mapping
space instead of the full mapping space. This is possible since the above fini*
*teness
condition on TV H*(X) can be given in terms of the pointed mapping space.
Lemma 1.3. Let X be an H-space such that H*(X) is of finite type. Then TV H*(X)
is of finite type if and only if H*(map *(BV, X)) is of finite type. Moreover, *
*if X is
such that TV H*(X) is of finite type for any elementary abelian p-group V then*
* the
same holds for map *(BW, X) for any elementary abelian p-group W .
Proof.By Proposition 1.2, if TV H*(X) is of finite type then
H*(map (BV, X)) ~=H*(X) H*(map *(BV, X))
is of finite type. In particular H*(map *(BV, X)) is of finite type. On the oth*
*er hand,
if H*(X) and H*(map *(BV, X)) are of finite type then H*(map (BV, X)) is of fin*
*ite
type, which is equivalent to TV H*(X) being of finite type by Proposition 1.2.
The last statement follows from the fact that TV xW = TV TW for any elementa*
*ry
abelian p-groups V and W . |
When X is connected, the evaluation map (BV, X) ! X is a homotopy equivalence
if TV H*(X) ~=H*(X) (for finite spaces, this is the Sullivan conjecture proved *
*by Miller
[22]). Actually spaces for which this happens can be cohomologically characteri*
*zed:
their mod p cohomology is locally finite. Recall that an unstable module is lo*
*cally
finite if it is a direct limit of finite unstable modules, which is equivalent *
*to say that
the span over Ap of any element is finite.
When one restricts the evaluation map to the connected component of the con-
stant map in the mapping space, Dwyer and Wilkerson (see also [25, 3.9.7 and 6.*
*4.5])
have shown that it is a homotopy equivalence if and only if the module of indec*
*om-
posable elements QH*(X) is locally finite. This condition can also be topologic*
*ally
characterized.
Proposition 1.4. [12, Prop 3.2],[25, Proposition 6.4.5] Let X be a connected p-
complete space such that H*(X) is of finite type. Let V be an elementary abelia*
*n p-
group and c : BV ! X a constant map. Then the following conditions are equivale*
*nt:
(1) QH*(X) is a locally finite Ap-module.
(2) map (BV, X)c ____-X is a weak homotopy equivalence.
6 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
Corollary 1.5. Let X be a connected H-space. Then QH*(X) is a locally finite
Ap-module if and only if map *(BV, X) ' * for some elementary abelian p-group *
*V .
Proof.Since X^pis a connected p-complete H-space, Proposition 1.4 applies to sh*
*ow
that QH*(X) is a locally finite Ap-module if and only if map *(BV, ^Xp) is homo*
*topi-
cally discrete for any elementary abelian p-group V .
The weak equivalence map *(BV, X) ' map *(BV, ^Xp) given by [22, Theorem 1.5]
shows that this is equivalent to map *(BV, X) being contractible, i.e. the loo*
*p space
X is BV -local. |
2. The Krull filtration of U
L. Schwartz proves in [27] the "strong realization conjecture" extending his *
*previ-
ous results from [26]. This conjecture, given by N. Kuhn in [17], states that *
*if the
cohomology of a space lies in some stage of the Krull filtration of the categor*
*y U
of unstable modules, then it must be locally finite. The Krull filtration is d*
*efined
inductively. The full subcategory of U of locally finite unstable modules is de*
*noted
by U0. Suppose now that Un is defined. One defines Un+1 as follows. In the quot*
*ient
category U=Un (see [14]) consider the smallest Serre class stable under direct *
*limits
that contains all the simple objects in U=Un. Then a module M 2 U is in Un+1 if*
* and
only if as an object of the abelian category U=Un, it is in the subcategory_(U=*
*Un)0.
The modules in Un can be characterized by means of the functor T.
Theorem 2.1. [25, Theorem 6.2.4] Let M be an unstable module. Then M 2 Un if
__n+1
and only if T M = 0.
The proof of Kuhn's conjecture by L. Schwartz shows that under the usual fini*
*teness
conditions the cohomology of a space either lies in U0 or it is not in any Un. *
*Instead
of looking at when the full cohomology of a space is in Un, we will study the m*
*odule
of the indecomposable elements QH*(X). The Krull filtration induces a filtratio*
*n of
the category of H-spaces by looking at those H-spaces X for which QH*(X) 2 Un.
There exist many spaces lying in each degree of this filtration, the most obvio*
*us ones
being Eilenberg-Mac Lane spaces.
Example 2.2. Let G be any abelian discrete group such that H*(K(G, n)) is of fi*
*nite
type. Then QH*(K(G, n)) 2 Un-1 .
Since these_Eilenberg-MacLane spaces are H-spaces, by Proposition 1.2 we can
compute T if we know the homotopy type of the mapping space. Let us denote by
Gi the abelian p-group Hn-i(BZ=p; G) and notice that
Y Y
map *(BZ=p, K(G, n)) ' K(Hn-i(BZ=p; G), i) ' K(Gi, i)
0 i n 0 i n
As T commutes_with taking indecomposable elements [25, Lemma 6.4.2], we_obtain
that T QH*(K(G,_n)) ~= 0 i n-1QH*(K(Gi, i)). Therefore by applying T repeat-
n * *
edly we have T QH (K(G, n)) = 0. For example QH (K(Z=p, n + 1)) is isomorphic
to the suspension of the free unstable module F (n) on one generator in degree *
*n. In
DECONSTRUCTING HOPF SPACES 7
__
particular, the formula T F (n) = 0 i n-1F (i) (see [25, Lemma 3.3.1]) yields *
*that
QH*(K(Z=p, n + 1)) 2 Un.
From the above exampleQit then easy to see that the filtration is not exhaust*
*ive, as
the infinite product n 1K(Z=p, n) does not belong to any stage.
The following lemma shows how QH*(X) is related to QH*(map *(BZ=p, X)) by
means of the reduced T functor.
Lemma 2.3. Let X be an H-space such that T H*(X) is of finite type. Then,
__ * *
T QH (X) ~=QH (map *(BZ=p, X)) .
Proof.Under such assumptions Proposition 1.2 applies and we know that the T fun*
*c-
tor computes the cohomology of the mapping space. Thus
* *
QT H*(X) ~=QH*(map (BZ=p, X)) ~=Q H (map *(BZ=p, X)) H (X)
Since T commutes with taking indecomposable elements [25, Lemma 6.4.2], it foll*
*ows
that_T QH*(X) ~=QH*(X) QH*(map *(BZ=p, X)). In particular, this is equivalent
to TQH*(X) ~=QH*(map *(BZ=p, X)). |
We end the section with a proposition which will allow us to perform an induc*
*tion
in the Krull filtration. Observe that N. Kuhn's strategy to move in the Krull f*
*iltration
is to consider the cofiber of the inclusion X ! map (BZ=p, X), see [17]. In our*
* context
Lemma 2.3 suggests to use the fiber of the evaluation map (BZ=p, X) ! X.
Proposition 2.4. Let X be an H-space with T H*(X) of finite type. Then, for n *
* 1,
QH*(X) is in Un if and only if QH*(map *(BZ=p, X)) is in Un-1.
Proof.By Theorem 2.1 the unstable module QH*(X) belongs to Un if and only if
__n+1 __n_ __n
T QH*(X) = 0. As T T QH*(X) = T (QH*(map *(BZ=p, X)) = 0 by Lemma 2.3,
we see that QH*(X) 2 Un if and only if QH*(map *(BZ=p, X)) 2 Un-1. |
3. Bousfield's BZ=p-nullification filtration.
The plan of this section follows step by step the preceding one, replacing th*
*e al-
gebraic filtration defined with the module of indecomposables by a topological *
*one.
Recall for example from [13] that a space X is said to be A-local if the evalua*
*tion at
the base point in A induces a weak equivalence of mapping spaces map (A, X) ' X.
When X is connected, it is sufficient to require that the pointed mapping space
map *(A, X) be contractible.
E. Dror-Farjoun and P. Bousfield have constructed a localization functor PA f*
*rom
spaces to spaces together with a natural transformation l : X ! PA(X) which is *
*an
initial map among those having a local space as target (see [13] and [2]). This*
* functor
is known as the A-nullification. It preserves H-spaces structures since it comm*
*utes
with finite products. Moreover when X is an H-space the map l is an H-map and i*
*ts
fiber is an H-space.
We recall some well-known facts about nullification functors which can be fou*
*nd in
[13, Theorem 3.A.1, Section 1.A.8, Corollary 3.D.3, Lemma 7.B.6].
8 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
Proposition 3.1. [13] Let A and X be connected spaces.
(1) P A X ' PA X. Therefore X is A-local if and only if X is A-local.
(2) If X is A-local then so is nX for any n 0.
(3) Let F ! E ! B be a fibration where B is A-local. Then PAF ! PAE ! B
is a fibration and F is A-local if and only if E is A-local.
(4) For i j, P iAP jAX ' P iAX.
P. Bousfield has determined the structure of the fiber of the nullification m*
*ap
l : X ! PAX under certain assumptions on A. We are interested in the situation *
*in
which A = nBZ=p. We give the proof for the convenience of the reader.
Theorem 3.2. [3, Theorem 7.2] Let n 1 and X be a connected H-space such that
nX is BZ=p-local. The homotopy fiber of the localization map X ! P n-1BZ=pX
is then an Eilenberg-Mac Lane space K(P, n) where P is an abelian p-torsion gro*
*up
(possibly infinite).
Proof.Let us denote by F the homotopy fiber of the nullification map l. On the *
*one
hand F is a n-1BZ=p-acyclic H-space by [13, Theorem 1.H.2]. Hence it follows as
a direct consequence of Bousfield's Key Lemma [4, Lemma 5.3] that P n-1BZ=pF is*
* a
GEM.
On the other hand X is nBZ=p-local. But since P n-1BZ=pX is n-1BZ=p-local
by definition, it must also be nBZ=p-local. The base and total space in the fi*
*bration
F ! X ! P n-1BZ=pX are nBZ=p-local spaces, hence so is F (see Proposition 3.1).
That is, F ' P nBZ=pX, which proves that F is a GEM.
Notice next that l induces an isomorphism in homology with rational coefficie*
*nts
(and in mod q homology for q 6= p as well). Thus F is a p-torsion GEM. It is
n-1BZ=p-acyclic and nBZ=p-local, so its only non-trivial homotopy group lives*
* in
degree n. |
As mentioned by Bousfield in [3, p. 848] an inductive argument allows to obta*
*in a
precise description of the fiber of the BZ=p-nullification map for H-spaces for*
* which
some iterated loop space is local.
Theorem 3.3. Let n 0 and X be a connected H-space X such that nX is BZ=p-
local. Then there is an H-fibration
F ____-X ____-PBZ=pX
where F is a p-torsion H-Postnikov piece whose homotopy groups are concentrated*
* in
degrees from 1 to n. |
We introduce a ün llification filtration" by looking at those H-spaces X such*
* that
the iterated loop space nX is BZ=p-local. The example of Eilenberg-Mac Lane
spaces shows that there are many spaces living in each stage of this filtration*
* as well,
compare with Example 2.2.
Example 3.4. Let G be an abelian discrete group with non-trivial mod p coho-
mology. Then the Eilenberg-Mac Lane space K(G, n) enjoys the property that its
DECONSTRUCTING HOPF SPACES 9
nQfold iterated loop space is BZ=p-local (it is even discrete). The infinite p*
*roduct
n 1K(Z=p, n) does not live in any stage of this topological filtration.
Another source of examples of spaces in this filtration is provided by connec*
*ted
covers of finite H-spaces.
Example 3.5. Let X be a finite connected H-space. Consider its n-connected cover
X. By definition of the n-connected cover we have a fibration
F ____-X ____-X
where F = X[n] is a Postnikov piece with homotopy concentrated in degrees n-*
*1.
Observe that n-1F is a discrete space hence BZ=p-local. As X itself is BZ=p-lo*
*cal
by Miller's theorem [22], so is n-1X, and thus n-1(X) is also BZ=p-local.
For a connected H-space X such that nX is BZ=p-local the study of the homo-
topy type of the pointed mapping space map *(BZ=p, X) is drastically simplified*
* by
Theorem 3.3 since it is equivalent to map *(BZ=p, F ) where F is a Postnikov pi*
*ece.
We prove now the topological analogue of the reduction Proposition 2.3.
Proposition 3.6. Let X be a connected H-space such that nX is BZ=p-local, then
n-1 map*(BZ=p, X) is BZ=p-local.
Proof.Under the hypothesis that nX is BZ=p-local, Theorem 3.3 tells us that we
have a fibration
F ____-X ____-PBZ=pX
where F is a p-torsion Postnikov system with homotopy concentrated in degrees f*
*rom
1 to n. Thus map *(BZ=p, X) ' map *(BZ=p, F ) because PBZ=pX is a BZ=p-local
space. Now n-1 map*(BZ=p, F ) is BZ=p-local (in fact it is a homotopically dis*
*crete
space) and thus so is n-1 map*(BZ=p, X). |
We note that working with H-spaces is crucial. Theorem 3.3 does not hold for
general spaces (not even simply connected). The example we indicate also shows *
*that
Theorem 5.5 fails for arbitrary spaces.
Example 3.7. Let us consider the space BS3. As the loop space BS3 = S3 is fini*
*te,
it is BZ=p-local by Miller's theorem. The BZ=p-nullification of classifying sp*
*aces
has been computed by B. Dwyer in [11, Theorem 1.7, Lemma 6.2]: PBZ=p(BS3) '
Z[1=p]1 (BS3). In particular it is p-torsion free and thus the fiber of the nul*
*lification
map cannot be a p-torsion Eilenberg-Mac Lane space.
4. Infinite loop spaces
In order to compare the topological with the algebraic filtration one of the *
*key
ingredients comes from the theory of infinite loop spaces. In this section we e*
*xplain
when a pointed mapping space map *(A, X) is an infinite loop space, but we are *
*of
course specially interested in the case when A is BZ=p. We make use of Segal's
techniques of -spaces and follow his notation from [28], which is better adapt*
*ed to
our needs than that of Bousfield and Friedlander, see [5]. Recall that the cat*
*egory
10 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
is the category of finite sets and a morphism ` : S ! T between two finite se*
*ts
is a partition of a subset of T into |S| disjoint subsets { `(ff) }ff2S. A -s*
*pace is a
contravariant functor from to the category of spaces with some extra conditio*
*ns.
We first construct a covariant functor Ao : ! Spaces for any pointed space A *
*by
setting An = An (so in particular A0 = *) and a morphism ` : [n] ! [m] induces *
*the
map `* : An ! Am sending (a1, . .,.an) to the element (b1, . .,.bm ) with bj = *
*ai if
and only if j 2 `(i) and bj = * otherwise.
Hence we get a contravariant functor for any pointed space X by taking the po*
*inted
mapping space map *(-, X). For this to be a -space one needs to check it is sp*
*ecial,
i.e. the n inclusions ik : [1] ! [n] sending 1 to k must induce a weak equival*
*ence
map *(An, X) ! map *(A, X)n.
Lemma 4.1. Let A and X be pointed spaces and assume that the inclusion An_A ,!
An x A induces a weak equivalence map *(An x A, X) ! map *(An _ A, X) for any
n 1. Then map *(Ao, X) is a -space.
Proof.By induction one shows that map *(An+1, X) is precisely weakly equivalent*
* to
map *(An _ A, X) ' map *(An, X) x map *(A, X) ' map *(A, X)n+1. |
Proposition 4.2. Let A be a pointed connected space and X an H-space. Assume
that the space map *(A, X) is A-local. Then map *(Ao, X) is a -space.
Proof.The cofiber sequence An _ A ! An x A ! An ^ A yields a fibration of point*
*ed
mapping spaces
map *(An ^ A, X) ____- map *(An x A, X) ____- map *(An _ A, X)
By adjunction the fiber map *(An ^ A, X) ' map *(An, map *(A, X)) is contractib*
*le
since any A-local space is also An-local (An is A-cellular or use Dwyer's versi*
*on of
Zabrodsky's Lemma in [11, Proposition 3.4]). The inclusion An_A ! AnxA induces
moreover a bijection on sets of homotopy classes [An x A, X] ! [An _ A, X] by [*
*32,
Lemma 1.3.5]. As all components of these pointed mapping spaces have the same
homotopy type we have a weak equivalence map *(An x A, X) ' map *(An _ A, X)
and conclude by the preceding proposition. |
Theorem 4.3. Let A be a pointed connected space and X be a loop space such that
map *(A, X) is A-local. Then map *(A, X) is an infinite loop space, and so is *
*the
corresponding connected component map *(A, X)c of the trivial map.
Proof.If X is a loop space, so is the mapping space map *(A, X). Therefore Sega*
*l's
result [28, Proposition 1.4] applies and shows that the -space structure expla*
*ined
above allows to deloop it further. *
* |
We specialize now to the case A = BZ=p, where we can even say more about the
intriguing infinite loop space map *(BZ=p, X)c.
Proposition 4.4. Let X be a loop space such that map *(BZ=p, X) is BZ=p-local.
Then all homotopy groups of the infinite loop space map *(BZ=p, X)c are Z=p-vec*
*tor
spaces.
DECONSTRUCTING HOPF SPACES 11
Proof.As ßn map *(BZ=p, X)c ~= [BZ=p, nX], we want to understand the maps
BZ=p ! nX. We claim all are homotopic to H-maps. Indeed by [32, Propo-
sition 1.5.1] the obstruction lives in the set [BZ=p ^ BZ=p, nX], which is tri*
*vial
since map *(BZ=p, X) is BZ=p-local. But any non-trivial H-map out of BZ=p has
order p. |
5. Structure theorems for H-spaces
The purpose of this section is to give an inductive description of the H-spac*
*es
whose module of indecomposable elements lives in some stage of the Krull filtra*
*tion.
This is achieved by comparing this algebraic filtration with the topological on*
*e and
by making use of Bousfield's result 3.3. We subdivide the proof of the main the*
*orem
into two steps.
Proposition 5.1. Let X be an H-space such that TV H*(X) is of finite type for a*
*ny el-
ementary abelian p-group V . Assume that nX is BZ=p-local. Then QH*(X) 2 Un-1.
Proof.We proceed by induction. For n = 1 assume that X is BZ=p-local, that
is, map *(BZ=p, X)c ' *. Then map *(BZ=p, X) is homotopically discrete since
map *(BZ=p, X)c is so and all components of the mapping space have the same ho-
motopy type. Hence QH*(map *(BZ=p, X)) = 0 and by Lemma 2.3, QH*(X) 2 U0.
If n > 1, let X be an H-space such that nX is BZ=p-local. We see by Proposi-
tion 3.6 that n-1 map*(BZ=p, X)c is BZ=p-local as well. Now map *(BZ=p, X)c is
an H-space such that n-1 map*(BZ=p, X)c is BZ=p-local. Moreover, by Lemma 1.3
TV H*(map *(BZ=p, X)) is of finite type for any elementary abelian p-group V . *
* By
induction hypothesis QH*(map *(BZ=p, X)c) 2 Un-2. Since all components have the
same homotopy type we obtain that QH*(map *(BZ=p, X)) 2 Un-2 and we conclude
by Corollary 2.4 that QH*(X) 2 Un-1. |
Proposition 5.2. Let X be a connected H-space such that QH*(X) 2 Un. Suppose
that TV H*(X) is of finite type for any elementary abelian p-group V . Then n+*
*1X
is BZ=p-local.
Proof.We infer the equivalence map *(BZ=p, iX) ' map *(BZ=p, i^Xp) for any i *
* 0
from [22, Lemma 1.5]. Hence iX is BZ=p-local if and only if i^Xpis so. Since X
is p-good the Ap-algebra H*(X) is isomorphic to H*(X^p) and it is enough to pro*
*ve
that n+1X^pis BZ=p-local. Without loss of generality we can thus assume that X*
* is
p-complete.
Let us proceed by induction. The case n = 0 is given by Corollary 1.5. Now
assume the result true for n - 1 and consider X such that QH*(X) 2 Un. Then by
Lemma 2.4 QH*(map *(BZ=p, X)) 2 Un-1 and the induction hypothesis ensures that
n map *(BZ=p, X)c ' map *(BZ=p, nX) is BZ=p-local. Apply now Theorem 4.3 to
deduce that the space map *(BZ=p, nX)c is an infinite loop space, with a p-tor*
*sion
fundamental group by Proposition 4.4.
These are precisely the conditions of McGibbon's main theorem in [21]: the BZ*
*=p-
nullification of connected infinite loop spaces with p-torsion fundamental grou*
*p is
12 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
trivial up to p-completion. Moreover our infinite loop space is BZ=p-local, so
n ^
(map *(BZ=p, nX)c)^p' PBZ=p(map *(BZ=p, X)c) p ' *
As we assume that X is p-complete, so are the loop space nX and the pointed ma*
*p-
ping space map *(BZ=p, nX)c. Thus we see that map *(BZ=p, nX)c must be con-
tractible. Since all components of the pointed mapping space have the same homo*
*topy
type as the component of the constant map, we infer that map *(BZ=p, nX) is ho-
motopically discrete. Looping once again, this implies that map *(BZ=p, n+1X) *
*' *,
i.e. n+1X is BZ=p-local as we wanted to prove. |
We finally sum up these two results in one single statement, which extends wi*
*dely
Dwyer and Wilkerson's Proposition 1.4 when X is assumed to be an H-space.
Theorem 5.3. Let X be a connected H-space such that TV H*(X) is of finite type
for any elementary abelian p-group V . Then QH*(X) is in Un if and only if n+1X
is BZ=p-local. |
Combining these results with Bousfield's Theorem 3.3 about the nullification *
*func-
tor PBZ=p enables us to give a topological description of the H-spaces X for wh*
*ich the
indecomposables QH*(X) live in some stage of the Krull filtration. Our main the*
*o-
rem proposes an[inductive description. Recall that the Prüfer group Zp1 is defi*
*ned
as the union Z=pn. It is a p-torsion divisible abelian group.
n 1
Theorem 5.4. Let X be a connected H-space of finite type such that TV H*(X) is *
*of
finite type for any elementary abelian p-group V . Then QH*(X) 2 Un if and only*
* if
X fits into a fibration
K(P, n + 1) ____-X ____-Y
where Y is a connected H-space such that QH*(Y ) 2 Un-1 and P is a p-torsion
abelian group which is a finite direct sum of copies of cyclic groups Z=pr and *
*Prüfer
groups Zp1.
Proof.Let F be the homotopy fiber of the nullification map X ! P nBZ=p(X). By
Theorem 3.2, F ' K(P, n + 1) where P is an abelian p-group. Moreover the equiv-
alence of mapping spaces map *( nBZ=p, K(P, n + 1)) ' map *( nBZ=p, X) shows
that the set
ßn map *(BZ=p, X) ~=ß1map *( nBZ=p, X) ~=Hom (Z=p, P )
is finite since all homotopy groups of map *(BZ=p, X) are p-torsion and its coh*
*omology
is of finite type. Thus P is isomorphic to a finite direct sum of copies of cyc*
*lic groups
Z=pr and Prüfer groups Zp1 by Lemma 5.8, which we prove at the end of the secti*
*on.
We conclude by taking Y = P nBZ=p(X). The cohomology H*(Y ) is of finite type
since H*(K(P, n + 1)) and H*(X) are of finite type, and so is H*(map *(BV, Y )).
Moreover, since nP nBZ=p(X) ' PBZ=p( nX) is BZ=p-local, Theorem 5.3 implies
that QH*(Y ) 2 Un-1. |
DECONSTRUCTING HOPF SPACES 13
Thus the example of Eilenberg-Mac Lane spaces we have seen in Example 2.2
are actually the only true new examples arising at each new stage of the filtra*
*tion.
Equivalently one can reformulate this result by describing the fiber of the BZ=*
*p-
nullification map for H-spaces such that QH*(X) belongs to Un for some n.
Theorem 5.5. Let X be a H-space such that TV H*(X) is of finite type for any
elementary abelian p-group V . Then QH*(X) is in Un if and only if X is the tot*
*al
space of an H-fibration
F ____-X ____-PBZ=pX
where F is a p-torsion H-Postnikov piece whose homotopy groups are finite direct
sums of copies of cyclic groups Z=pr and Prüfer groups Zp1 concentrated in degr*
*ees
1 to n + 1. |
In other words H-spaces such that QH*(X) 2 Un for some n are BZ=p-local H-
spaces, p-torsion Eilenberg-MacLane spaces and extensions of the previous type.
If we restrict our attention to the case n = 0 we are working with H-spaces s*
*uch
that QH*(X) is locally finite and H*(X) is of finite type. Our result reproves *
*in a
more conceptual way the theorems given by C. Broto, L. Saumell and the second
named author in [7, 10, 8].
Corollary 5.6. [8, Theorem 1.2] Let X be a connected H-space such that H*(X) is
of finite type and QH*(X) is locally finite. Then X is the total space of a pri*
*ncipal
fibration
K(P, 1) ____-X ____-Y
where Y is a BZ=p-local H-space and P is a finite direct sums of copies of cyc*
*lic
groups Z=pr and Prüfer groups Zp1
What do we learn from our study about H-spaces which do not belong to any sta*
*ge
of the filtration we have introduced in this paper? From a cohomological point *
*of view
such H-spaces have a very large module of indecomposables since it does not bel*
*ong
to any stage of the Krull filtration. It can be however easier to see if equiva*
*lently not
a single iterated loop space is BZ=p-local (remember Theorem 5.3). Let us disc*
*uss
the interesting example of the classifying space BU for reduced complex K-theor*
*y.
Example 5.7. The mod p cohomology of BU is a polynomial algebra on the Chern
classes ci in degrees 2i. From the action of the Steenrod operations on the Ch*
*ern
classes one can see that QH*(BU) is not a finitely generated unstable module. M*
*ore
precisely, QH*(BU) ~= 2H*(BS1) and we can compute explicitly the value of the T
functor on this module
T ( 2H*(BS1)) ~= 2( pH*(BS1))
since T commutes with suspensions (and BS1 = K(Z, 2) so the mapping space
map (BZ=p, BS1) is equivalent to a product (BS1)p)). This shows that QH*(BU)
does not belong to any Un.
On the other hand McGibbon's theorem in [21] tells us that the p-completion of
PBZ=pBU is contractible (BU is indeed an infinite loop space with trivial funda*
*mental
14 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
group). As BU is not a Postnikov piece (not even up to p-completion) none of i*
*ts
loop spaces can be BZ=p-local. Alternatively notice that Bott periodicity would*
* imply
that BU itself should be BZ=p-local, but this is not the case.
Therefore the Krull filtration for the indecomposables detects in BU the fact*
* that
the BZ=p-nullification Postnikov-like tower does not permit to deconstruct it i*
*nto
elementary pieces. In fact BU is K(Z=p, 2)-local by a result of Mislin (see [23*
*, The-
orem 2.2]).
We prove finally the technical lemma about abelian p-torsion groups which was
used in the proof of Theorem 5.4, and which will be needed again in the last se*
*ction.
Lemma 5.8. Let P be a p-torsion discrete group. If the set Hom (Z=p, P ) is fi*
*nite
then P is a finite direct sum of copies of cyclic groups Z=pr and Prüfer groups*
* Zp1.
Proof.By Kulikov's theorem (see [24, Theorem 10.36]) P admits a basic subgroup,
which is a direct sum of cyclic groups. It must be of bounded order since Hom (*
*Z=p, P )
is finite and a result of Prüfer (see [24, Corollary 10.41]) shows now that thi*
*s subgroup
is a direct summand. Since the quotient is divisible and Hom (Z=p, P ) is finit*
*e, P is
a finite direct sum of copies of cyclic groups Z=pr and Prüfer groups Zp1. *
* |
6. H-spaces with finitely generated algebra over Ap
We will assume in this section that H*(X) is finitely generated as algebra ov*
*er the
Steenrod algebra. Then the BZ=p-nullification of X is a mod p finite H-space up*
* to
p-completion, as we prove in Theorem 6.4.
The next lemma shows that under this finiteness condition, the H-spaces consi*
*dered
in this section satisfy the hypothesis of Theorem 5.3 (they belong to some stag*
*e of the
filtration we study in this paper). Let us recall that F (n) denotes the free u*
*nstable
module on one generator in degree n and F (n) 2 Un since T F (n) ~= i nF (i), w*
*hich
is a finitely generated module over Ap (see [25, Lemma 3.1.1]).
Lemma 6.1. Let K be a finitely generated unstable Ap-algebra. Then there exists
some integer n such that the module of indecomposables QK belongs to Un. Moreov*
*er
TV K is a finitely generated unstable Ap-algebra for any elementary abelian gro*
*up V .
Proof.First of all QK is a finitely generated module over Ap, i.e. it is a quot*
*ient of
a finite direct_sum of free modules. There exists_hence an epimorphism ki=1F (*
*ni) !
m
QK. Since T is an exact functor it follows that T (QK) = 0 where m is the larg*
*est
of the ni's and so QK 2 Um-1 .
Moreover TV commutes with taking indecomposables elements [27, Lemma 6.4.2].
Therefore Q(TV K) is a finitely generated unstable module. The above discussion
shows then that TV K is a finitely generated Ap-algebra. *
* |
Our first proposition is inspired by the situation studied by L. Smith in [30*
*].
Proposition 6.2. Let p : X ! B be a principal H-fibration classified by an H-map
' : B ! BF . Then there is an isomorphism of algebras H*(X) ~=H*(B)=='* A
where A is a subalgebra of H*(X) and H*(B)=='* is the quotient by the ideal gen*
*erated
DECONSTRUCTING HOPF SPACES 15
by the positive degree elements in Im ('*). Moreover, if H*(BF ) and H*(X) are
finitely generated Ap-algebras then so is H*(B).
Proof.Consider the Serre spectral sequence associated to the principal fibration
F ____-X ____-B
with E2-term E*,*2= H*(F ) H*(B) which converges to H*(X). Let {E0r} be the
spectral sequence associated to the universal path fibration
F ____-P BF ____-BF.
The map ' induces a morphism of spectral sequences E0r! Er which, in the E2-ter*
*m,
is the identity on the vertical axes and '* on the horizontal axes.
Since the spectral sequence for the universal path fibration converges to Fp,*
* it
follows from naturality that all elements in the image of '* are hit by some di*
*fferential.
Moreover an element on the horizontal axis is hit by a differential exactly if *
*it lies
in the ideal generated by the elements which are killed by an element on the ve*
*rtical
axis. Therefore E*,01~=H*(B)=='*.
The morphism p* induced in cohomology factors through the edge homomorphism
H*(B) ! H*(B)=='* H*(X). In particular H*(B)=='* ~=Im (p*) is a Ap-Hopf
subalgebra. Therefore by the proof of the Borel-Hopf decomposition theorem in [*
*16,
Section 2.2], there is a complement A such that H*(X) ~=H*(B)=='* A, as algebra*
*s.
If H*(X) is a finitely generated algebra over Ap, then H*(B)=='* is also a fi*
*nitely
generated algebra over Ap. On the other hand, Im('*) H*(B) is a Ap-Hopf subal-
gebra which is also finitely generated as algebra over Ap since H*(BF ) is. Thu*
*s so is
H*(B). |
The proof of the next theorem is done by induction, in which the reduction st*
*ep
relies again on mapping spaces. We need thus to control the finiteness conditio*
*ns of
such mapping spaces.
Lemma 6.3. Let F be an H-space which is a p-torsion Postnikov piece such that
H*(map *(BZ=p, F )) is a finitely generated algebra over Ap. Then so is H*(F ).
Proof.The result holds for Eilenberg-MacLane spaces since map *(BZ=p, K(P, n)) *
*is
homotopy equivalent to a product of lower dimensional Eilenberg-Mac Lane spaces
K(Pn-1, n - 1) x . .x.K(P0, 0). In particular Pn-1 ~=Hom (Z=p, P ) must be fin*
*ite
and P is a finite direct sum of cyclic and Prüfer groups by Lemma 5.8.
By induction the same holds for Postnikov pieces. Let F be a Postnikov piece *
*with
homotopy concentrated in degrees from 1 to n. There is a principal fibration
K(P, n) ____-F ____-F 0.
The highest non-trivial homotopy group of the mapping space map *(BZ=p, F ) is
isomorphic to Hom (Z=p, P ). As the mod p cohomology is of finite type, this mu*
*st be
a finite group. Hence P is a finite direct sum of copies of Z=pr by Lemma 5.8.
Applying map *(BZ=p, -) to the fibration F ! F 0! K(P, n + 1) we notice that
map *(BZ=p, F 0) has finitely generated cohomology as Ap-algebra by Proposition*
* 6.2.
16 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
By induction hypothesis, H*(F 0) enjoys the same property and therefore the same
holds for H*(F ) by Proposition 6.2 again. *
* |
We can now state our main finiteness result. It enables us to understand bet*
*ter
the BZ=p-nullification, which is the first building block in our deconstruction*
* process
Theorem 5.4. For H-spaces which have finitely generated cohomology as Ap-algebra
it must be up to p-completion a finite space.
Theorem 6.4. Let X be a connected H-space such that H*(X) is finitely generated
as algebra over the Steenrod algebra. Then PBZ=pX is an H-space with finite mod*
* p
cohomology.
Proof.By Lemma 6.1, there exists an integer n such that QH*(X) 2 Un-1, so Theo-
rem 5.3 applies and we know that nX is BZ=p-local.
We will show that if H*(X) is finitely generated as algebra over Ap and nX is
BZ=p-local then H*(PBZ=pX) is finitely generated as algebra over Ap. We proceed
by induction on n. When n = 0 the statement is clear. Assume the statement holds
for n - 1.
Since H*(X) is a finitely generated Ap-algebra, so is T H*(X) by Lemma 6.1.
Lemma 2.3 shows that H*(map *(BZ=p, X)) is a finitely generated algebra over Ap
as well. Let F be the homotopy fiber of the nullification map X ! PBZ=pX. Be-
cause of the weak equivalence map *(BZ=p, F ) ' map *(BZ=p, X), the cohomology
H*(map *(BZ=p, F )) is finitely generated as algebra over Ap and by Lemma 6.3 t*
*he
same holds for H*(F ) since F is a p-torsion Postnikov piece. By Theorem 3.2 th*
*ere
is a principal H-fibration
K(P, n) ____-X ____-P n-1BZ=pX
where P is a finite direct sum of copies of Z=pr, 1 r 1 and the nth homotopy
group of F is precisely P . In particular H*(K(P, n)) is a finitely generated A*
*p-algebra
as well as H*(K(P, n + 1)).
It follows from Lemma 6.2 that H*(P n-1BZ=pX) is finitely generated as alge-
bra over Ap. Moreover n-1P n-1BZ=pX = PBZ=p n-1X is BZ=p-local, so the in-
duction hypothesis applies. The cohomology of the BZ=p-nullification PBZ=pX '
PBZ=pP n-1BZ=pX is finitely generated as algebra over the Steenrod algebra.
Since H*(PBZ=pX) is locally finite, this implies that PBZ=pX has finite mod p*
* co-
homology. |
Combining this last result with Theorem 5.5 we obtain the folllowing.
Theorem 6.5. Let X be a connected H-space such that H*(X) is a finitely generat*
*ed
algebra over the Steenrod algebra. Then X is the total space of an H-fibration
F ____-X ____-Y
where Y is an H-space with finite mod p cohomology and F is a p-torsion H-Postn*
*ikov
piece whose homotopy groups are finite direct sums of copies of cyclic groups Z*
*=pr
and Prüfer groups Zp1. |
DECONSTRUCTING HOPF SPACES 17
As a first application of the above results we propose an extension of Hubbuc*
*k's
Torus Theorem on homotopy commutative finite H-spaces.
Corollary 6.6. Let X be a connected homotopy commutative H-space with finitely
generated cohomology as algebra over the Steenrod algebra. Then, X ' (S1)n x F *
*, up
to p-completion, where F is a connected p-torsion H-Postnikov piece.
Proof.Consider the fibration F ! X ! PBZ=pX. We know from the preceding
theorem that the fiber is a p-torsion Postnikov piece and the basis is an H-spa*
*ce
with finite mod p cohomology. Both are homotopy commutative. In particular the
mod p Torus Theorem of Hubbuck (see [15] and [1]) implies that PBZ=pX is up to
p-completion a finite product of circles (S1)n. As the fiber is p-torsion, the*
* above
fibration is split and the result follows. *
* |
When X is a mod p finite H-space, this corollary is the original Torus Theorem
due to Hubbuck and Aguad'e-Smith (which we actually use in the proof). When X is
an H-space with noetherian cohomology, QH*(X) 2 U0, the Postnikov piece F is an
Eilenberg-Mac Lane space K(P, 1) where P is a p-torsion abelian group and we get
back Slack's results [29], as well as their generalization by Lin and Williams *
*in [20].
Corollary 6.7. [20, Theorem B] Let X be a connected homotopy commutative H-
space with finitely generated cohomology as algebra. Then, up to p completion, *
*X is
the direct product of a finite number of S1's, K(Z=pr, 1)'s, and K(Z, 2)'s.
Proof.When H*(X) is a finitely generated algebra, the module of indecomposables
is finite and belongs thus to U0. Therefore the fiber F above is a K(P, 1) wher*
*e P
is a finite direct sums of copies of cyclic groups Z=pr and Prüfer groups Zp1. *
*Up to
p-completion this is equivalent to a finite product of K(Z=pr, 1)'s, and K(Z, 2*
*)'s. |
In our setting it is of course not true anymore that F is a product of Eilenb*
*erg-
Mac Lane spaces. The homotopy fiber of Sq2 : K(Z=2, 2) ! K(Z=2, 4) is indeed an
infinite loop space which satisfies the assumption of the corollary.
In our second application we offer a criterion to recognize cohomologically t*
*he n-
connected cover of a mod p finite H-space, using the Krull filtration on the mo*
*dule
of the indecomposable elements. Recall Example 3.5, where we showed that the n-
connected cover of a finite H-space X belongs to the filtration we investigate *
*since
n-1(X) is BZ=p-local. Moreover, an easy Serre spectral sequence argument wi*
*th
the covering fibration
F ____-X ____-X
shows that H*(X) is finitely generated as algebra over the Steenrod algebra.*
* The-
orem 5.3 applies and QH*(X) 2 Un-2. That is, the cohomology of X is finit*
*ely
generated as algebra over Ap, is n-connected, and QH*(X) 2 Un-2.
The following result is a converse of this fact. We prove that if the cohomol*
*ogy of
a p-complete H-space X satisfies these three conditions, then X is the n-connec*
*ted
cover of an H-space which is a mod p finite H-space up to p-completion. When n *
* 2
this does not bring anything new since the universal cover of a mod p finite H-*
*space
is again a mod p finite H-space which is even 2-connected [9, Theorem 6.10].
18 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
Proposition 6.8. Let n 2 and X be a p-complete connected H-space such that
H*(X) is an n-connected finitely generated Ap-algebra and QH*(X) is in Un-2. Th*
*en
X is the n-connected cover of an H-space with finite mod p cohomology.
Proof.Since QH*(X) is in Un-2 and H*(X) is a finitely generated algebra over Ap,
by Theorem 5.5 we know that X fits in an H-fibration
F ____-X ____-PBZ=pX
where F is a p-torsion Postnikov piece with homotopy concentrated in degrees fr*
*om
1 to n - 1. Note also that X is an n-connected space because we assume that X
is p-complete (see [6, Connectivity Lemma I.6.1]). By inspecting the homotopy l*
*ong
exact sequence for this fibration we check that X is the n-connected cover of t*
*he
H-space PBZ=pX which has finite mod p cohomology by Theorem 6.4. |
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Nat`alia Castellana and Jer^ome Scherer Juan A. Crespo
Departament de Matem`atiques, Departament de Economia i de Hist`oria
Universitat Aut`onoma de Barcelona, Econ`omica,
E-08193 Bellaterra, Spain Universitat Aut`onoma de Barcelona,
E-mail: natalia@mat.uab.es, E-08193 Bellaterra, Spain
jscherer@mat.uab.es E-mail: JuanAlfonso.Crespo@uab.es,