THE TRANSCENDENCE DEGREE OF THE MOD p
COHOMOLOGY OF FINITE POSTNIKOV SYSTEMS
JESPER GRODAL
January 7, 1997
Abstract.In this paper we examine the transcendence degree of the mod
p cohomology of a finite Postnikov system E. We prove that, under mild
assumptions on E, the transcendence degree of H*(E; Fp) is always positi*
*ve,
and give a complete classification of the Postnikov systems where the tr*
*anscen-
dence degree of H*(E; Fp) is finite. More precisely we prove that H*(E; *
*Fp) is
of finite transcendence degree iff E is Fp-equivalent to the classifying*
* space of
a p-toral group. As an application of these results we derive statements*
* about
the n-connected cover X of a finite complex X. We show for instance t*
*hat,
under suitable connectivity assumptions on X, the LS category of X is
always infinite assuming X 6= X. Finally we discuss generalizations o*
*f the
obtained results to polyGEMs.
1.Introduction
In 1953 Serre showed his celebrated result that a 1-connected finite Postnikov *
*sys-
tem E with finitely generated homotopy always has homology in infinitely many
dimensions, using his newly invented spectral sequence [28]. His methods, howev*
*er,
although revealing the asymptotic size of Betti numbers (the coefficients in the
Poincare series) of H*(E; Fp), did not in general give information about the ri*
*ng or
A-module structure of H*(E; Fp) (here A denotes the Steenrod algebra). Serre's
theorem has since then been generalized in several ways by a number of people
(Dwyer-Wilkerson [7], Lannes-Schwartz [16, 18], McGibbon-Neisendorfer [20]) all
utilizing the theory of unstable modules over the Steenrod algebra as developed*
* by
Lannes, Schwartz and others. One of the main advantages of this approach is that
it, by relating certain properties of the cohomology to questions about mapping
spaces, gives a grip on how these properties behave with respect to fibrations_*
*i.e.
it turns traditional spectral sequence questions into long exact sequence quest*
*ions.
This paper is a contribution along these lines.
____________
1991 Mathematics Subject Classification. 55S45,(55S10,55M30,55P60).
Key words and phrases. Postnikov system, Unstable module, LS category, PolyG*
*EM.
The author was partially supported by Det internationale kontor, Det naturvi*
*denskablige
fakultet, Kund Hojgaards fond and Julie Damms fond.
1
2 J. GRODAL
We offer the following two main theorems:
Theorem 1.1. Let E be a connected nilpotent finite Postnikov system with finite
ss1(E). Assume that E has finitely generated homotopy groups and that H*(E; Fp)*
* 6=
0. Then H* (E; Fp) contains an element of infinite height.
Theorem 1.2. Let E be a connected nilpotent finite Postnikov system with finite
ss1(E). Assume that E has finitely generated homotopy groups. Then H*(E; Fp)
has finite transcendence degree iff E is Fp-equivalent to a space E0 fitting in*
*to a
principal fibration sequence of the form
CP1 x . .x.CP1 ! E0! K(P; 1) ;
where P is a finite p-group.
Here we define the transcendence degree, d(K), as the maximal number of homo-
geneous algebraically independent elements in K. When K is noetherian, d(K) is
equal to the Krull dimension of K. Note that Theorem 1.1 can be reformulated as
saying that the transcendence degree of H*(E; Fp) is always positive. Theorem 1*
*.1
was previously known in the case p = 2 and E assumed to be 1-connected by work
of Lannes and Schwartz [18], but was actually rediscovered independently by the
author and formed the starting point for this work.
From now on H*(X) will denote the mod p cohomology of X for some fixed
but arbitrary prime p. In the rest of this introduction we will use some standa*
*rd
notation concerning unstable modules over the Steenrod algebra. In the next sec*
*tion
we will briefly introduce these concepts, for a general reference see e.g. Schw*
*artz
[27].
The key topological result needed in proving the results about H*(E) is the
following asymptotic growth formula in rkpV , where V is an elementary abelian
group:
logp|[BV; K(ssn(E); n)] <~logp|[BV; E]| <~C logp|[BV; K(ssn(E); n)]| ;
where n is the dimension of the highest homotopy group for which ssn(E)(p)6= 0.
The above growth formula relates to the algebra structure of H*(E) by the
following theorem of Lannes, generalizing earlier work of Miller [21]:
Theorem 1.3. [14, 15] Let X be a connected space. Suppose that X is nilpo-
tent with finite ss1(X) and H*(X) of finite type. Then the natural map f 7! f*,
[BV; X] ! Hom K(H*(X); H*(V )) is a bijection.
Here K denotes the category of unstable algebras over the Steenrod algebra.
We have the following theorem of Lannes and Schwartz:
Theorem 1.4. [26, 27] Let M be an unstable module over the Steenrod algebra.
Then the following two conditions are equivalent:
(1) M is nilpotent.
(2) Hom U(M; H*(V )) = 0 for all elementary abelian p-groups V .
THE COHOMOLOGY OF FINITE POSTNIKOV SYSTEMS 3
Here U denotes the category of unstable modules over the Steenrod algebra. The
Lannes linearization principle
*(V ))
Hom U (K; H*(V ))0' FHomKp(K;H ;
where 0denotes the (continuous) vector space dual, now immediately leads to an
analogous theorem concerning unstable algebras.
Theorem 1.5. [17] For an unstable algebra K the following conditions are equiv-
alent:
(1) Hom K(K; H*(V )) = 0 for all elementary abelian p-groups V .
(2) K is F -equivalent to the trivial unstable algebra Fp.
(3) K is nilpotent as an unstable module over the Steenrod algebra.
(4) K is a nil ideal in K, i.e. it consists of nilpotent elements.
Here the equivalence of (2), (3), and (4) are immediate consequences of the
definitions. Theorem 1.1 now follows from the growth formula and the preceding
general theorems.
It is worth noting that one in the preceding theorems has to consider element*
*ary
abelian groups V of an arbitrary size. Restriction to e.g. V = Z=p would not be
enough, which is seen by for example setting K = H*(K(Z; 3)). This makes the
property `nilpotent' a bit less well behaved that for example the property `loc*
*ally
finite'.
To prove Theorem 1.2 we look at work of Henn, Lannes and Schwartz [13] on
the structure of unstable algebras and reformulate it in terms of growth proper*
*ties.
This leads to a characterization of the transcendence degree of an unstable alg*
*ebra
K in terms of the growth of logp| Hom K(K; H*(V ))| in v = rkpV , under mild
restrictions on K. More precisely we prove:
Theorem 1.6. Let K be an unstable algebra of transcendence degree d(K) and as-
sume that Hom K(K; H*(V )) is finite for all V . If d(K) is finite*
* then
logp| Hom K(K; H*(V ))| ~ d(K)v. If d(K) is infinite then logp| Hom (K; H*(V ))|
grows faster that linearly in v.
Theorem 1.6 is a powerful tool for calculating the transcendence degree of un-
stable algebras. To demonstrate this we give a two line proof of Quillen's theo*
*rem
in the pivotal finite p-group case, by proving that the Krull dimension of H*(P*
* ) is
equal to rkpP for every finite p-group P .
The numbers logp| Hom K(K; H*(Fnp ))| (which are not necessarily integers) can
in some sense be viewed as an unstable algebra alternative to the traditional B*
*etti
numbers of an algebra. Theorem 1.6 shows that that an analog of the well known
formula relating the growth of the Betti numbers and the transcendence degree
(Krull dimension) of a noetherian algebra holds for these new numbers, now with
much weaker restrictions on the unstable algebra.
Applying Theorem 1.6 to the obtained growth formula for logp|[BV; E]|, where
E is a finite Postnikov system, and doing some work now leads to Theorem 1.2.
We also include a section where we see that the above results for example imp*
*ly
that the n-connected cover of a finite complex always has infinite LS category,
generalizing earlier partial results of McGibbon and Moller.
4 J. GRODAL
Finally we discuss and conjecture generalizations of the above results to pol*
*yGEMs,
correcting a small mistake put forth in [11].
In the proofs of the theorems we several times need a small but useful fact a*
*bout
nilpotent actions. Since we have been unable to find this fact stated in the li*
*terature
and believe that it ought to be better known we give a proof in a short appendi*
*x.
Acknowledgment: I am grateful to The Fields Institute for giving me the
opportunity to participate in the `Emphasis year in Homotopy Theory 95/96', and
especially to the organizers and participants for making the stay so enjoyable *
*and
profitable. A special thank to K. Andersen, W. Chacholski, P. Lambrechts, J.
Scherer and B. Schuster for so many hours of interesting discussions. Also than*
*ks
to R. Kane for introducing me to the beautiful theory of Lannes and Schwartz
which this paper is based on.
I would like to thank W. Chacholski for helpful suggestions and references. L*
*ike-
wise thanks to my advisor J. Moller for his support and for suggesting improvem*
*ents
in the presentation. Thanks to H. Minh Le directing my attention to the conject*
*ure
of E. Dror Farjoun. Finally thanks to C. McGibbon and J. Moller whose questions
related to Theorem 1.1 got me interested in the cohomology of finite Postnikov
systems in the first place.
2.Notation
By a space we will, for simplicity, mean an object in the pointed homotopy
category Ho* of CW-complexes. We will by [-; -] mean a free homotopy class of
maps (Theorem 1.3 is the main reason why this is convenient). We will also need
to refer to pointed homotopy classes of pointed maps which we will then denote *
*by
[-; -]pt.
We define the p-rank, rkp, of a group G as the maximal rank of an elementary
abelian subgroup V contained in G. We will at all times employ the convention
that v = rkpV , for the elementary abelian group V in question.
When talking about the asymptotic behavior of some sequence of numbers, we
will by the symbol <~mean that for all ffl > 0 there exists an N such that (lef*
*t-hand
side) (1 + ffl)(right-hand side) for all n N.
In our notation involving unstable modules over the Steenrod algebra, we will
follow the standard notation used in e.g. Schwartz [27]. We will quickly review*
* the
basic definitions:
Definition 2.1.An unstable module M is a graded module over the Steenrod al-
gebra A satisfying the following instability conditions:
o If p = 2 then Sqix = 0 for i > |x|.
o If p > 2 then fieP ix = 0 for e + 2i > |x|.
Let U denote the category whose objects are unstable modules and whose morphisms
are (degree 0) A-module maps. Note that this is an abelian category.
Definition 2.2.An unstable algebra K is an unstable module equipped with two
maps j : Fp ! K and : K K ! K making K into a commutative unital
Fp-algebra such that
o is A-linear (i.e. the Cartan formula holds).
THE COHOMOLOGY OF FINITE POSTNIKOV SYSTEMS 5
o Sq|x|x = x2 for any x 2 K if p = 2; P |x|=2x = xp for any x 2 K of even
degree if p > 2.
Let K denote the category whose objects are unstable algebras, and whose mor-
phisms are those (degree 0) Fp-algebra maps which are A-linear.
Definition 2.3.We say that an unstable module is nilpotent if the following hol*
*ds:
o If pN=-21thenNfor-every2x 2 M there exists an integer N such that
Sq2 |x|Sq2 |x|.S.q.|x|x = 0.
o If p > 2 then for every x 2 M where |x| is even there exists an integer N
such that P pN-1|x|=2P pN-2|x|=2.P.|.x|=2x = 0.
Let N ildenote the full subcategory of U whose objects are nilpotent modules.
Note that this definition extends the usual definition if I is an ideal in an*
* unstable
algebra, in the sense that we have that I lies in N iliff it is nil as an ideal*
*, that is if
all its elements are nilpotent. Beware however, for a non-noetherian algebra a *
*nil
ideal I need not be nilpotent (i.e. there exists n such that In = 0), which mak*
*es
the terminology slightly ambiguous. Note also that an unstable algebra of course
can not be nilpotent since it will contain 1.
We need a last definition:
Definition 2.4.A morphism of unstable algebras ' : K ! K0 is said to be an
F -monomorphism if the kernel of ' in the (abelian) category of unstable modules
is nilpotent as an unstable module. We define F-epimorphism and F-isomorphism
analogously.
This definition coincides with Quillen's original definition of an F-isomorph*
*ism,
and is also the same as what is sometimes referred to as a (purely) inseparable
isogeny.
Finally note that everything in this paper only depends on the space E up to *
*Fp-
equivalence (E here as everywhere connected nilpotent with finite ss1). As note*
*d by
Miller [22] map *(BV; E) ' map*(BV; ^Ep), and of course H*(E; Fp) ' H*(E^p; Fp)
- Likewise we have that [BV; E] ' [BV; ^Ep] [4, 2]. Thus we could reformulate
everything by writing: "Let E be Fp-equivalent to..." - For the sake of clarity*
* we
will refrain from doing so and instead leave reformulations like that to the re*
*ader.
3.Growth Properties
In this section we establish the growth formula for logp|[BV; E]|. To do this*
* we
first need to prove a couple of lemmas.
Lemma 3.1. Let G be a finitely generated abelian group of the form
G = Z__._._.Z-z____" Z=pr1_._._.Z=prt_-z_______"T ;
s t
where T is finite q-torsion for primes q 6= p. Furthermore let V be an elementa*
*ry
abelian p-group of rank v. Then
rkp(H1(V ; G)) ~ tv
for v ! 1 and
6 J. GRODAL
( tvk
___ ift > 0
rkp(Hk(V ; G)) ~ sk!vk-1_
(k-1)!if t = 0
for v ! 1, k 2, where we by ~ mean that the ratio tends to 1.
Proof.It is well known that H*(Z=p; Z) = Z[x]=(px) where |x| = 2. Now note
that induction on the rank v of V , using the cohomology of H*(Z=p; Z) together
with the K"unneth formula, gives us that pHk(V ; Z) = 0 for all k > 0 and all
V . By the universal coefficient theorem this also holds for Z=pr coefficients_*
*thus
Hk(V ; G) is actually a Z=p-vector space for k > 0. We have that rkp(H1(V ; G))*
* =
rkp(Hom (V; G)) = tv, so the claimed formula in the case k = 1 is clear. To show
the general case observe that
Hk(V ; Z=pr) = Hk(V ; Z) Hk+1(V ; Z)
= Hk(V ; Z=p)
for k > 0 by the universal coefficient theorem, and hence
Hk(V ; G) = (Hk(V ; Z=p))t (Hk(V ; Z))s
for k > 0. Let ak = rkpHk(V ; Z) and bk = rkpHk(V ; Z=p). We now get a recursion
formula bk = ak + ak+1 so
k-2X
ak = bk-1 - ak-1 = (-1)ibk-1-i
i=0
for k 2, since a1 = 0. It is well known that H*(V ; Z=p) has Poincare series
P (x) = __1__(1-x)v(cf. [1]) so we get
k 1 v . .(.v + k - 1)
bk = 1_k!d_dxk____(1|-xx)v=0= ______________k!
k
~ v_k!
k-1
for v ! 1 , k 1, and hence ak ~ bk-1 ~ _v___(k-1)!for v ! 1, k 2, which shows
the claimed formula in the k 2 case. __|_|
Lemma 3.2. Let X be an arbitrary space and let E be a connected finite Postnik*
*ov
system. Then Y
|[X; E]pt| |Hi(X; ssi(E))|
i>0
when E is simple.
If more generally E is nilpotent then there exists, for each i, a filtration *
*0 =
Fi;0C . .C.Fi;ti= ssi(E) such that ss1(E) acts trivially on Fi;j=Fi;j-1and
Y tiY
|[X; E]pt| |Hi(X; Fi;j=Fi;j-1)| :
i>0j=1
THE COHOMOLOGY OF FINITE POSTNIKOV SYSTEMS 7
Proof.The proof is by induction on the number of nontrivial homotopy groups.
Assume that the top homotopy group sits in dimension n. Consider first the case
where E is simple. We have a fibration sequence, which is principal since E is
simple:
K(ssn(E); n) ! E ! Pn-1(E) :
Since the fibration is principal we have an action E x K(ssn(E); n) ! E, and th*
*us
an induced action * : [X; E]ptx [X; K(ssn(E); n)]pt! [X; E]pt. This action has *
*the
property that in the exact sequence
[X; K(ssn(E); n)]pt! [X; E]pt! [X; Pn-1E]pt
we have that (f) = (g) iff there exists h 2 [X; K(ssn(E); n)] such that f * h =*
* g.
This implies that |[X;QE]pt| |[X; Pn-1E]pt||Hn(X; ssn(E))|, so by induction we
get that |[X; E]pt| i>0|Hi(X; ssi(E))| as wanted.
Now assume that E is only nilpotent. The fibration K(ssn(E); n) ! E !
Pn-1(E) might no longer be principal, but it does have a principal refinement
corresponding to a filtration 0 = Fi;0C . .C.Fi;ti= ssi(E) of ssi(E) (cf. [24]).
Using induction as before now finishes the proof in this case too. __|_|
We are now ready to prove the key growth theorem:
Theorem 3.3. Let E be a connected nilpotent finite Postnikov system with finite
ss1(E), and assume that E has finitely generated homotopy groups. Let n denote
the highest homotopy group for which ssn(E)(p)6= 0 and set k = n if ssn(E) has
p-torsion, k = n - 1 if not. Then
cvk <~logp|[BV; E]| <~Cvk ;
where c; C are positive constants.
Proof.By replacing E by the Fp-equivalent space PnE we can assume that n is the
dimension of the top non-trivial homotopy group. Since E is connected and ss1E *
*is
finite we might as well show the theorem for [BV; E]pt, which is what we will d*
*o.
We have a principal fibration sequence Pn-1E ! K(ssn(E); n) ! E so we get an
exact sequence of pointed sets
[BV; Pn-1E]pt! [BV; K(ssn(E); n)]pt! [BV; E]pt
or equivalently
(3.1) [BV; 0Pn-1E]pt! Hn(V ; ssn(E)) ! [BV; E]pt;
where [BV; 0Pn-1E]ptacts on Hn(V ; ssn(E)) as described in the previous proof
(here 0Pn-1E denotes the zero component of Pn-1E).
Now by Lemma 3.2 and 3.1 we get:
n-2X
logp|[BV; 0Pn-1E]pt| logp|Hi(V ; ssi+1(E))|
i=1
n-2X
= rkpHi(V ; ssi+1(E))
i=1
<~ Kvn-2
8 J. GRODAL
for v ! 1, since 0Pn-1E is an H-space and hence simple. Since logp|Hn(V ; ssn(E*
*))| ~
cvk, where k is as defined in the Theorem, (3.1) shows us:
logp|Hn(V ; ssn(E))|~ logp|Hn(V ; ssn(E))| - logp|[BV; 0Pn-1E]|pt
<~ log
p|[BV; E]|pt:
We want to get the other inequality by appealing to Lemma 3.2. In the simple
case it is immediate that
Xn
logp|[BV; E]pt| logp|Hi(V ; ssi(E))| <~Cvk ;
i=1
where k is defined as in the Theorem. If ssn(E) does contain p-torsion we can t*
*ake
C = c, whereas C in the case where ssn(E) does not contain p-torsion is given in
terms of ssn(E) and ssn-1(E).
In the non-simple case we need to worry a little bit about actions. We now ha*
*ve
that
Xn Xti
logp|[BV; E]pt| <~ logp|Hi(V ; Fi;j=Fi;j-1)| :
i=1j=1
It is still clear that logp|[BV; E]pt| Cvn for some C. This takes care of the *
*case
where ssn(E) does contain p-torsion. In the case where ssn(E) does not contain
p-torsion however we want the better estimate Cvn-1. This is not obvious since
the filtration quotients could have p-torsion, even though ssn(E) did not. To r*
*esolve
this problem we will have to appeal to Proposition 7.1.
We can, by replacing E by an Fp-equivalent space, assume that ss1(E) is a p-
group, since every finite nilpotent group is a product of p-groups. By Proposit*
*ion 7.1
ss1(E) acts trivially on ssn(E), so we can take the filtration of ssn(E) to be *
*the trivial
filtration, and thus there is no p-torsion introduced in dimension n. This show*
*s that
in this case we can conclude that logp|[BV; E]pt| <~Cvn-1. __|_|
Remark 3.4. It follows from the proof of the preceding theorem that one can
actually get concrete estimates for c and C. This does have some interest, espe*
*cially
in the case k = 1 where c and C actually turn out to give lower and upper bounds
on the transcendence degree of H*(E).
The first main theorem now follows from the preceding growth theorem.
Theorem 3.5. Let E be a connected nilpotent finite Postnikov system with finite
ss1(E). Assume that E has finitely generated homotopy groups and that H*(E) 6= *
*0.
Then H* (E) contains an element of infinite height.
Proof.Theorem 3.3 shows the asymptotic growth of logp|[BV; E]|_in particular
it shows that [BV; E] is nontrivial for some V . But by Theorem 1.3 and 1.5 this
now implies that H* (E) 62 N il, so H* (E) has to contain an element of infinite
height. __|_|
THE COHOMOLOGY OF FINITE POSTNIKOV SYSTEMS 9
4. Applications to n-connected covers
We now turn to investigating the consequences for n-connected covers of finite
complexes:
Corollary 4.1.Let X be a finite complex, 1-connected with finite ss2(X). Assume
furthermore that H* (PnX) 6= 0. Then H* (X) contains an element of infinite
height.
Remark 4.2. The assumption H* (PnX) 6= 0 could be formulated in a number
of alternative ways. Since X is 1-connected the assumption is equivalent to the
natural map H*(X) ! H*(X) not being an isomorphism by a Serre spectral
sequence argument. Also, since X is of finite type, H* (PnX) 6= 0 iff H* (PnX; *
*Z)
is not entirely q-torsion for primes q 6= p, which, by for example the Whitehead
theorem modulo Serre classes, is equivalent to ss*(PnX)(p)6= 0.
Proof of Corollary 4.1.We have an exact sequence
[BV; X] ! [BV; PnX] ! [BV; X] ! [BV; X] :
By Miller's theorem [BV; X] = [BV; X] = 0, so [BV; PnX] = [BV; X]. Since
H*(PnX) 6= 0 we have that ss*(PnX)(p)6= 0. Now observe that PnX sat-
isfies the assumptions of Theorem 3.3, so in particular we get that [BV; X] =
[BV; PnX] 6= 0 for some V . By Theorem 1.3 and 1.5 this is equivalent to H*(X)
containing an element of infinite height. __|_|
Remark 4.3. The relatively strong assumption on the connectivity of X cannot
be weakened, as is shown by the Hopf fibration S1 ! S3 ! S2.
Corollary 4.4.Let X be a finite complex, 1-connected with finite ss2(X). Assume
that X 6= X. Then there exists a prime q such that H* (X; Fq) contains an
element of infinite height.
Proof.If X 6= X, we have to have that H*(PnX; Fq) 6= 0 for some prime q. The
statement is now obvious from Corollary 4.1. __|_|
Corollary 4.5.Let X be a finite complex, 1-connected with finite ss2(X). Assume
that X 6= X. Then X has infinite LS category.
Proof.By Corollary 4.4 there exists a prime q such that H* (X; Fq) has infin*
*ite
cup-length, so especially X has infinite LS category. __|_|
The preceding corollaries generalize earlier partial results of Moller and McGi*
*bbon
[19]. As they point out, it can be interesting to note the radical difference b*
*etween
these results and the results obtained in the rational case. Here the rational*
* LS
category of X is always less than or equal to the rational LS category of X *
*by
the mapping theorem of Felix and Halperin [12]. But, on the other hand, in the
rational setting we have no Serre's theorem either - indeed the rational cohomo*
*logy
of K(Z; 3) does not contain an element of infinite height.
10 J. GRODAL
5. The transcendence degree of H*(E)
In this section we give a complete classification of the Postnikov systems wh*
*ose
cohomology is of finite transcendence degree. We start by using work of Henn,
Lannes and Schwartz [13] to relate the growth of logp| Hom K(K; H*(V ))| in v to
the transcendence degree of K, under mild restrictions on K. Combining these
results with our growth formula for [BV; E] = Hom K(H*(E); H*(V )) now leads to
a classification theorem for finite Postnikov systems of finite transcendence d*
*egree.
In order to state and prove our results we need to review some work of Henn,
Lannes and Schwartz. We will for the ease of the reader follow their notation. *
*We
refer the reader to [13] for more details.
Definition 5.1.For any unstable algebra K define its transcendence degree d(K)
as the maximal number of algebraically independent homogeneous elements in K.
Remark 5.2. If K is a connected graded noetherian algebra, the maximal number
of algebraically independent elements, d(K), will be finite. In this case the n*
*umber
d(K) will coincide with the Krull dimension of K, and we can furthermore choose
d(K) algebraically independent elements such that K will be finite over the alg*
*ebra
spanned by those elements. If K is an integral domain, d(K) will be the same as
the classical transcendence degree of the field of fractions of K. A nice and g*
*raded
proof of these standard facts can be found in [1].
In the following we will need to refer to elementary abelian groups of differ*
*ent
rank. Therefore we will therefore sometimes equip the elementary abelian group V
with a subscript which will then indicate the rank of V .
Definition 5.3.Let E denote the category of Fp vector spaces. Let PS denote the
category of profinite sets and let G be the category of functors E ! (PS)op. No*
*te
that we can view Gop as the category of contravariant functors E ! PS.
One should realize that objects in G contain a rich structure. This stems from
the fact that not only do we to each vector space associate a profinite set, bu*
*t we
do this in a natural way, which, loosely speaking, ties the profinite sets toge*
*ther.
By inverting all F-isomorphisms in K we obtain a quotient category of K which
will be denoted by K=N il. Let g : K ! G be given by g(K)(V ) = Hom K(K; H*(V ))
for all V . Here we will equip Hom K(K; H*(V )) with the profinite topology ind*
*uced
by writing
Hom K(K; H*(V )) = Hom K(colimffKff; H*(V )) = limffHomK(Kff; H*(V )) ;
where ff runs over the finitely generated A-subalgebras Kffof K.
By Theorem 1.4 and the Lannes linearization principle we have that a map be-
tween unstable algebras K ! K0is an F-monomorphism (resp. F-epi) iff g(K0)(V ) !
g(K)(V ) is a surjective (resp. injective) map of sets for all V . This show t*
*hat g
induces a a faithful functor K=N il! G (likewise denoted g). In [13] Henn, Lann*
*es
and Schwartz actually identify the image of g_we shall however not need this.
Definition 5.4.Let PS - End Vd denote the category whose objects are profinite
sets equipped with a continuous right action of the monoid End Vd and whose mor-
phisms are maps of profinite sets respecting the End Vd-action.
THE COHOMOLOGY OF FINITE POSTNIKOV SYSTEMS 11
To each G 2 Gop and each vector space Vd we can associate a profinite right
End Vd-set G(Vd). Namely define the right action of End Vd on G(Vd) by to each
(s; ') 2 G(Vd)xEnd Vd associating s' = G(')s. This gives us a `restriction func*
*tor'
ed : Gop ! PS - EndVd:
Likewise we can define an `induction functor'
id : PS - EndVd ! Gop : S 7! (W 7! S xEndVdHom (W; Vd)) ;
where S xEndVd Hom (W; Vd) denotes the coequalizer in the category of profinite
sets of the action of End Vd on S and Hom (W; Vd) respectively.
Now note that the canonical map G(Vd) x Hom (W; Vd) ! G(W ) induces a well
defined map on the coequalizer G(Vd) xEndVdHom (W; Vd) ! G(W ). Therefore we
can to each map S ! G(Vd) of profinite End Vd-sets and each W associate a map
S xEndVdHom (W; Vd) ! G(Vd) xEndVdHom (W; Vd) ! G(W ) :
Likewise a morphism in Gop, id(S) ! G of course induces a map S ! G(Vd)
of End Vd-sets. Since these operations are inverses of each other we get that
Hom Gop(id(S); G) = Hom PS-EndVd(S; ed(G)) for all G 2 Gop, S 2 PS - EndVd, so
id is left adjoint of ed. It is immediate that 1PS-EndVd '! edO id, given by th*
*e unit
of the adjunction, so we get an embedding of PS - EndVd as a full subcategory of
Gop. Define sd = eopdO g, where eopd: G ! (PS - EndVd)op is the opposite functor
of ed.
In [13] Henn, Lannes and Schwartz proves the following key result about the
structure of Gop.
Proposition 5.5.[13] The morphism (id O ed)(G) ! G given by the counit of the
adjunction is a monomorphism in Gop for all G 2 Gop, i.e. we have that (id O
ed)(G)(V ) ! G(V ) is an injective map of profinite sets for all V .
This gives us a filtration of G which turns out to coincide with the filtrati*
*on of
K=N ilby transcendence degree.
Proposition 5.6.[13] Let G 2 Gop. Define the transcendence degree of G as
d(G) = min{d | ((id O ed)G)(W ) ! G(W ) is bijective forWall} ;
where we take d(G) = 1 if none such d exists. For an element G 2 G we define
the transcendence degree by viewing it as lying in Gop. With these definitions *
*we
have that d(K) = d(g(K)).
We shall need some alternative ways of expressing the transcendence degree of
an object in Gop. These can be found implicit in [13]. Since they are important*
* in
their own right we find it useful to state them explicitly_we include proofs fo*
*r the
convenience of the reader. First a useful definition:
Definition 5.7.Let S be an End Vd-set, and let s 2 S. Define the rank of s 2 S
as
rk s = min{rk'| wheres = t' for somet 2 S; ' 2 End Vd} :
We say that s is regular if rks = d, i.e. if s = t' implies that ' is a regular
(invertible) matrix.
12 J. GRODAL
Proposition 5.8.We have the following formula for the transcendence of G 2 Gop:
d(G) := min {d|((id O ed)G)(W ) ! G(W ) is bijective forWall}
= max {rks|s 2 G(Vd) for somed}
= max {d|G(Vd) contains a regular element} :
Proof.We start by proving that
(5.1)
max {d|G(Vd) contains a regular}elt.= max{rks|s 2 G(Vd) for somed} :
First note that `' is obvious. To prove `' let s 2 G(W ) and suppose that rks =*
* d.
We can thus choose ss 2 End W; t 2 G(W ) such that s = tss and rkss = d. By
changing t we can assume that ss is a projection. Let ae : W ! ssW and i : ssW *
*! W
be the canonical projection and inclusion associated to ss. Now set s0 = G(i)s 2
G(ssW ) and note that G(ae)s0= G(ae)G(i)s = G(ae)G(i)G(ss)t = G(ssiae)t = s. We
claim that s0 is regular. Suppose we have ' 2 End (ssW ); u 2 G(ssW ) such that
s0= u'. Then we have
s = G(ae)s0= G(ae)G(')u = G(ae)G(')G(i)G(ae)u = G(i'ae)G(ae)u = (G(ae)u)(i'ae)
so ' has to be regular, since rks = d. This shows the wanted inequality.
We now prove that the number given by (5.1) actually coincides with the tran-
scendence degree. We will first see that it is less than or equal to the transc*
*endence
degree. Suppose therefore that we have d such that G(Vd) xEndVd Hom (W; Vd) !
G(W ) is bijective for all W , and let s 2 G(W ) be arbitrary. Choose t 2 G(Vd)
and ' 2 Hom (W; Vd) such that G(')t = s. Since the rank of ' : W ! Vd must be
less that or equal to d, we can choose a projection ss 2 End W of rank d such t*
*hat
'ss = '. But this gives us that s = G(')t = G(ss)G(')t = (G(')t)ss, which shows
that rks d, as wanted.
We finish the proof by showing d(G) max{d|G(Vd) contains a regular element}.
Let d be the maximal number such that G(Vd) contains a regular element (if there
exists regular elements in infinitely many dimension we are done). We want to p*
*rove
that G(Vd) xEndVdHom (W; Vd) ! G(W ) is surjective for all W . Let s 2 G(W ) be
arbitrary and set n = rks. Note that n d by (5.1). Choose an element ss 2 End W
of rank n and t 2 G(W ) such that s = tss. We can by changing the choice of t a*
*ssume
that ss is a projection. Letting ae : W ! ssW and i : ssW ! W denote the canoni*
*cal
projection and inclusion we get that s = G(ss)t = G(ssiae)t = G(ae)(G(ssi)t). T*
*here-
fore s is in the image of G(Vn) xEndVn Hom (W; Vn) which implies that s is in t*
*he
image of G(Vd) xEndVd Hom (W; Vd), since n d. This completes the proof, since
G(Vd) xEndVdHom (W; Vd) ! G(W ) is injective by Proposition 5.5. __|_|
Example 5.9. Let Xn be the End Vn-set consisting of two elements x1 and x0,
with the End Vn-action given as follows:
x0ff = x0 for allff 2 End Vn
ae
x1ff = x1x forff 2 AutVn
0 forff 62 AutVn :
THE COHOMOLOGY OF FINITE POSTNIKOV SYSTEMS 13
We have that x1 has rank n and x0 has rank 0. The End Vn-set Xn is thus the
smallest EndVn-set containing a regular element. It has the universal property,*
* that
any End Vn-set containing a regular element surjects onto Xn as an End Vn-set.
Just knowing the growth properties of logp|g(K)(V )| in v is actually enough *
*to
determine the transcendence degree of K, under mild restrictions on K.
Theorem 5.10. Let K be an unstable algebra and assume that Hom K(K; H*(V ))
is finite for all V . If d(K) is finite then logp|g(K)(V )| ~ d(K)v. If d(K) is*
* infinite
then logp|g(K)(V )| grows faster that linearly in v.
Remark 5.11. The assumption that Hom K(K; H*(V )) is finite for all V is a tec*
*h-
nical assumption which (by Theorem 3.3) will be satisfied for all the applicati*
*ons
we have in mind. Also it is clear that if for instance K is finitely generated *
*as an
A-algebra, then Hom K(K; H*(V )) is likewise finite. Furthermore if K is N il-c*
*losed
(cf. [5,Q13]) and of finite type then Hom K(K; H*(V )) is finite [13]. Note how*
*ever
that H*( 1i=1K(Z=p; i)) serves as an example of an unstable algebra of finite *
*type
where Hom K(K; H*(V )) is not finite.
Proof of Theorem 5.10.We want to establish the general growth formulas by estab-
lishing them for the `largest' and the smallest finite End Vn-set containing a *
*regular
element. Let Xn be the End Vn-set defined in Example 5.9. For the elements in
in(Xn)(V ) = Xn xEndVn Hom (V; Vn) we have the following relations:
(x0; ') ~ (x0; 0) for all' 2 Hom (V; Vn)
(x1; ') ~ (x0; 0) ifrk' < n
(x1; ') ~ (x1; ) ifrk' = rk = n and ker' = ker :
This shows that |in(Xn)(V )| = #( n dimensional subspaces in)V+ 1
For v > n, the number of n dimensional subspaces of V is as follows:
v - 1) . .(.pv - pn-1)
#( n dimensional subspaces in)V= (p__________________(pn - 1) . .(.pn *
*- pn-1)
~ Cpnv
for v ! 1.
Let T be a finite set and consider the the End Vn-set T x EndVn. We have that
in(T x EndVn)(V ) = T x Hom (V; Vn) so
|in(T x EndVn)(V )| = |T |pnv:
From the above we conclude that
logp|in(Xn)(V )| ~ logp|in(T x EndVn)(V )| ~ nv :
Now let K be an unstable algebra and assume that sn(K) contains a regular ele-
ment. Since sn(K) is finite we can find a surjection of End Vn-sets T x EndVn !
sn(K) for some finite set T . Also, since sn(K) contains a regular element, we *
*can
find a surjection sn(K) ! Xn of End Vn-sets. Now this means that we for all V
have surjections of sets
in(T x EndVn)(V ) ! in(sn(K))(V ) ! in(Xn)(V ) :
This shows that logp|iopn(sn(K))(V )| behaves asymptotically as nv.
14 J. GRODAL
If K has transcendence degree d < 1, then g(K) also has transcendence degree
d so logp|g(K)(V )| = logp|(iopdOeopdOg)(K)(V )| = logp|iopd(sd(K))(V )| which *
*grows
asymptotically as dv, since sd(K) contains a regular element by Proposition 5.8*
*. If
K has transcendence degree d = 1 then by Proposition 5.5
logp|g(K)(V )| logp|(iopnO eopnO g)(K)(V )| = logp|iopn(sn(K))(V )| :
Since sn(K) contains a regular element for infinitely many n by Proposition 5.8,
we get that logp|g(K)(V )| >~nv for all n as wanted. __|_|
The characteristic numbers logp|g(K)(Fnp )| can in some sense be viewed as an
unstable algebra alternative to the classical Betti numbers of a graded algebra*
* of
finite type. In the classical case we have a formula relating the growth of the*
* Betti
numbers and the transcendence degree of a noetherian graded algebra given by
d(K) = min{k 2 N0| there exists a C such thatdimFKn Cnk-1 for alln} :
Theorem 5.10 establishes a different but analogous formula for these new number*
*s,
which has among its advantages that it holds with much weaker restrictions on t*
*he
unstable algebra. Moreover, when applied to the cohomology of spaces, it is eas*
*y to
see how these numbers behave with respect to fibrations of the underlying space*
*s.
Remark 5.12. In [13] Henn, Lannes and Schwartz use Proposition 5.5 and 5.6 to
derive a far reaching generalization of Quillen's theorem about the structure o*
*f the
cohomology ring of a finite group [25]. They prove that for any unstable algebr*
*a K
of transcendence degree less that or equal to d there is an F-isomorphism
K ! limH*Vd:
sd(K)
Here we view sd(K) as a category by taking as objects the elements in sd(K),
and as morphisms the maps induced on sd(K) from endomorphisms of Vd. The
theorem however has the inherent weakness that it requires a priori knowledge of
the transcendence degree of K. Theorem 5.10 can remedy this defect, since it gi*
*ves
a very easily applicable way of calculating the transcendence of an unstable al*
*gebra
K.
To illustrate the power of this approach we will rederive Quillen's theorem i*
*n the
pivotal finite p-group case, by showing that the Krull dimension of H*(P ) is e*
*qual
to rkpP , the maximal rank of an elementary abelian group in the p-group P . It
was shown by Hopf that [BV; BP ] = Rep(V; P ) = Hom (V; P )=(conj. byp 2 P ). It
is furthermore an easy exercise to see that logp| Rep(V; P )| grows asymptotica*
*lly as
rkpP v. Theorem 5.10 now implies that d(H*(P )) = rkpP , and since we know that
H*(P ) is noetherian by the Evens-Venkov theorem (cf. [10]) we are done. Tracing
back what elements goes into this proof one sees that one of the main ingredien*
*ts is
the use of Proposition 5.8, which in some sense can be seen as being the replac*
*ement
of Serre's theorem about cohomological detection of elementary abelian groups.
Theorem 5.13. Let E be a connected nilpotent finite Postnikov system with fini*
*te
ss1(E). Assume that E has finitely generated homotopy groups. Then H*(E) has
finite transcendence degree iff E is Fp-equivalent to a space E0fitting into a *
*principal
fibration sequence of the form:
CP1 x . .x.CP1 ! E0! K(P; 1) ;
THE COHOMOLOGY OF FINITE POSTNIKOV SYSTEMS 15
where P is a finite p-group.
Proof.By Theorem 3.3 and 5.10 all spaces E0as in Theorem 5.13 have cohomology
of finite transcendence degree. To prove the converse assume that H*(E) has fin*
*ite
transcendence degree. We see from Theorem 3.3 that Hom K(H*(E); H*(V )) is
finite for all V . Theorem 3.3 and 5.10 now implies that E has to be Fp equival*
*ent
to P2E, and furthermore that ss2(E) cannot contain p-torsion. Since ss1(P2E) is
finite nilpotent, and thus a product of p-groups, we can by passing to a coveri*
*ng
replace P2E with an Fp-equivalent space whose fundamental group is a finite p-
group. These observations show that we without restriction can assume that E
has nontrivial homotopy groups only in dimension 1 and 2, ss2(E) being without
p-torsion and ss1(E) being a finite p-group. But now Proposition 7.1 tells us t*
*hat
the action of ss1(E) on ss2(E) has to be trivial. This means that the Postnikov
fibration K(ss2(E); 2) ! E ! P1E is principal (cf. [24]). We thus have a fibrat*
*ion
sequence
E ! P1E k!K(ss2(E); 3) :
Write ss2(E) = M T , where M = Z . . .Z and T is torsion. Let k0: P1E !
K(M; 3) be the map corresponding to k under the equivalence [P1E; K(ss2(E); 3)]*
* '
[P1E; K(M; 3)]. Letting E0 denote the homotopy fiber of k0we obtain a diagram
0
E0 ____//_P1E__k__//_K(M; 3)
f|| |||| |Fp'|
fflffl| || k fflffl|
E _____//P1E_____//K(ss2(E); 3)
where f is any lifting which makes the diagram commute. From this diagram we
see that f : E0 ! E has to be an Fp-equivalence as well. By construction E0 fits
into a principal fibration sequence of the form stated in the Theorem. __|_|
Remark 5.14. Remember that a p-toral group G is a group which arises as a
group extension 1 ! Tn ! G ! P ! 1, where Tn is an n-dimensional torus
and P is a finite p-group. The spaces E0 which appear in Theorem 5.13 are just
those classifying spaces of p-toral group which arise as central extensions. T*
*hey
are classified by n, P and their one extension class (= Postnikov invariant) k 2
[BP; K(Zx. .x.Z; 3)] = H3(P ; Zx. .x.Z) ' H2(P ; Tn), where in the cohomology
P acts trivially on the coefficients.
Remark 5.15. By Venkov's theorem [29], classifying spaces of p-toral groups ha*
*ve
noetherian cohomology. Theorem 5.13 thus in particular shows that for the coho-
mology of a finite Postnikov system, being of finite transcendence degree is eq*
*uiva-
lent to being noetherian. This is indeed a very striking and unusual property w*
*hich
of course does not hold for spaces in general. The cohomology of the loop space*
* of
a (1-connected, say) finite complex will for instance always be non-noetherian *
*and
have transcendence degree 0. Just knowing this intriguing property of finite Po*
*st-
nikov systems would actually be enough to rederive the above theorem using the
original Betti numbers estimates of Serre_knowing this equivalence would secure
that the wild growth of the Betti numbers could only be caused by an infinitum *
*of
polynomial generators in the ring.
16 J. GRODAL
We will end this section by showing a Proposition which precisely determines
the transcendence degree of the cohomology the spaces E0 of Theorem 5.13, and
whose proof illustrates a calculation using Theorem 5.10. The Proposition can a*
*lso
be obtained by using Quillen's theorem for compact Lie groups.
Proposition 5.16.Let E be a space which fits into a fibration of the form
CP1_x_._.x.CP1_-z_______"! E ! K(P; 1) k!K(Z x . .x.Z; 3) ;
n
where P is a finite p-group. Then
d(H*(E)) = n + max{rkpV |V ,! P; *(k) = 0} :
Especially n + rkpP d(H*(E)) n + 1, when P is non-trivial.
Proof.Consider the sequence
[BV; E] ! [BV; K(P; 1)] k*![BV; K(Z x . .x.Z; 3)] :
Let 2 [BVd; BP ] = Rep(Vd; P ) be an injection and assume that 2 ker(k*). We
have that will naturally give rise to ~ Cpdvelements in ker(k*) as v ! 1 comi*
*ng
from maps arising by precomposing with projections onto d dimensional subspaces
of V . Since Rep(VrkpP; P ) is finite we obtain that
logp| ker(k*)| ~ max{rkpV |V ,! P; *(k) = 0}v :
But the exact sequence now shows that
logp|[BV; E]| ~ nv + max{rkpV |V ,! P; *(k) = 0}v ;
so d(H*(E)) = n + max{rkpV |V ,! P; *(k) = 0} by Theorem 5.10. To get the
last part of the Proposition, note that H3(Z=p; Z) = 0. __|_|
Example 5.17. Consider the space E = Fib(K(Z=p x Z=p; 1) k!K(Z; 3)), where
k is some nonzero element in H3(Z=pxZ=p; Z) ' Z=p. From the above Proposition
it follows that d(H*(E)) = 2.
Remark 5.18. Note that the formula for the transcendence degree of H*(E) in-
volves the behavior of a certain class in the integral cohomology of P when res*
*trict-
ing to elementary abelian subgroups.
6. Generalizations to PolyGEMs
Recall that a GEM is a (possibly infinite) product of K(G; n)'s, where G is an
abelian group. A space is a polyGEM if it belongs to the smallest full subcate-
gory polyGEMs of spaces containing all GEMs and which is closed under taking
extensions by fibrations, i.e. which satisfies that if F ! E ! B is an arbitrary
fibration then B; F 2 polyGEMs ) E 2 polyGEMs . We say that a space is an
oriented polyGEM if it belongs to the smallest full subcategory polyGEMsori of
spaces containing all GEMs and which is closed under taking extensions by princ*
*i-
pal fibrations, i.e. which satisfies that if F ! E ! B is a principal fibration*
* then
F; B 2 polyGEMsori) E 2 polyGEMsori. Note that a nilpotent finite Postnikov
system is an oriented polyGEM by [24].
THE COHOMOLOGY OF FINITE POSTNIKOV SYSTEMS 17
In [11] Dror Farjoun conjectures that if X is a non-trivial p-complete polyGEM
then [BZ=p; X] is always non-trivial. As stated the conjecture is false_S^1pand
K(^Zp; 3) serve as counterexamples. The problem with ^S1pis easy_the fundamental
group is not finite. The problem with K(^Zp; 3) is deeper, it demonstrates that*
* it
in general is not enough to look at maps from just a single BZ=p (it is however
interesting to note that for any nilpotent, connected space X with finite ss1 a*
*nd
with H*(X) assumed to be noetherian, we have that [BV; X] 6= 0 ) [BZ=p; X] 6= 0
(cf. [13, Prop II 7.2])). Since the idea that polyGEMs should behave like fin*
*ite
Postnikov systems still seems very plausible, we dare to state the following mo*
*re
modest conjecture:
Conjecture 6.1. Let E be a connected nilpotent polyGEM with finite ss1(E). As-
sume that E has finitely generated homotopy groups and that H* (E) 6= 0. Then
H*(E) contains an element of infinite height.
Note that imposing the restriction on the fundamental group, we are also led *
*to
the simplification that [BV; X] = [BV; ^Xp], so the p-completion doesn't matter.
We likewise believe that Theorem 5.13 should generalize:
Conjecture 6.2. Let E be a connected nilpotent polyGEM with finite ss1(E). As-
sume that E has finitely generated homotopy groups. Then the transcendence degr*
*ee
of H*(E) is finite iff E is Fp-equivalent to a space E0 fitting into a principa*
*l fibra-
tion sequence of the form:
CP1 x . .x.CP1 ! E0! K(P; 1) ;
where P is a finite p-group.
One piece of evidence for the first conjecture is a theorem of Felix, Halperi*
*n,
Lemaire and Thomas [30] which (in the language of Dror Farjoun) states that a
1-connected oriented polyGEM E with H* (E) 6= 0 has infinite LS category.
We will give another piece of evidence for the conjecture, whose proof makes *
*use
of the Neisendorfer localization functor L = LHZ=pPBZ=p (cf. [23]). Here LHZ=p
denotes localization with respect to mod p homology (cf. [2]) and PBZ=p denotes
BZ=p-nullification (cf. [11]). First we need a definition.
Definition 6.3.Let polyGEMsft denote the smallest full subcategory of spaces
which is closed under taking extensions by fibrations and which contains all co*
*n-
nected GEMs with finite ss1.
Remark 6.4. The class polyGEMsft contains all connected finite Postnikov sys-
tems with finite solvable ss1, and is probably very close to being equal to all*
* con-
nected polyGEMs with finite solvable ss1.
Remember that an unstable module U is called locally finite if for all x 2 U *
*we
have that Ax is finite dimensional.
Proposition 6.5.Let E be a nilpotent polyGEM with H*(E) of finite type, which
belongs to polyGEMsft. Assume that H* (E) 6= 0. Then H*(E) is not locally finit*
*e.
Before giving the proof we need a lemma:
18 J. GRODAL
Lemma 6.6. We have that LE = * for all E 2 polyGEMsft, where L denotes the
Neisendorfer localization functor.
Proof of Lemma 6.6.Our main technical tool is a theorem of Dror Farjoun which
states that if Lf is any localization functor with respect to some map f, and if
F ! E ! B is any fibration sequence, then LfF = * implies that LfE '! LfB
(cf. [11]). This also applies to the Neisendorfer localization functor, since*
* we
can view L as localization with respect to just one (large) map. We first note
that the class of spaces which is acyclic with respect to Neisendorfer localiza*
*tion,
i.e. the spaces E which satisfies LE = *, is closed under taking extensions by
fibrations. It is therefore enough to prove the claim for connected GEMs with
finite ss1. Furthermore note that it is enough to prove the claim for 1-connec*
*ted
GEMs, since we can apply the theorem of Dror Farjoun to the fibration sequence
E<1> ! E ! K(ss1(E); 1), where LK(ss1(E); 1) is easily seen to be zero.
Now let E be a 1-connected GEM and write this E = E"where "Eis 2-connected.
By a theorem of Dror Farjoun [11, Prop 7.B.5] PBZ=p and PM(Z=p;1)coincide on
GEMs (where M(Z=p; 1) denotes the mod p Moore space). This gives us
PBZ=pE = PM(Z=p;1)E = PM(Z=p;2)"E;
where the last equality is by another theorem of Dror Farjoun [11, Prop 3.A.1].
Bousfield [3] has shown that
ssi(PM(Z=p;2)"E) = ssi(E") Z[1_p] ;
where we use that "Eis a 2-connected GEM. We also know the effect of the functor
LHZ=p on the homotopy groups when the spaces are nilpotent, since here it coinc*
*ides
with the Bousfield-Kan completion functor (cf. [4, 2]). If X is a nilpotent spa*
*ce
then we have a short exact sequence [4]:
0 ! Ext(Z=p1 ; ssn(X)) ! ssn(X^p) ! Hom (Z=p1 ; ssn-1(X)) ! 0 :
For a nilpotent group N we have that Ext(Z=p1 ; N) = 0 iff N is p divisible (*
*cf.
[4]) and of course Hom (Z=p1 ; N) = 0 when N is p divisible, so the above seque*
*nce
shows that the p-completion of a nilpotent space with p-divisible homotopy grou*
*ps
is zero. The above results now imply that ssi(LE) = ssi+1(LHZ=pPM(Z=p;2)"E) = 0
for all i. __|_|
Proof of Proposition 6.5.By Lemma 6.6 we have that LE = *, so especially
map*(BZ=p; E) 6= *, since map *(BZ=p; E) = * would imply that LE = ^Ep6= *.
But saying that map*(BZ=p; E) 6= * is equivalent to saying that that H*(E) is n*
*ot
locally finite (cf. [22, 17]). __|_|
Remark 6.7. Neisendorfer's theorem, stating that LX = ^Xpfor a 1-connected
finite complex X with finite ss2, follows immediately from Lemma 6.6, by applyi*
*ng
L to the fibration sequence PnX ! X ! X and using Miller's theorem to
conclude that LX = ^Xp.
THE COHOMOLOGY OF FINITE POSTNIKOV SYSTEMS 19
Remark 6.8. There is no direct relation between the LS category and the proper*
*ty
locally finite. For instance K(Z=2; 2) is an exampleQof a space with non-locally
finite cohomology but with LS category 1, whereas n2N Sn is an example of a
space of infinite LS category whose cohomology is locally finite.
7. Appendix: Nilpotent actions
In this short appendix we prove a proposition about nilpotent actions, which
we have used several times in the paper. The author thanks W. Chacholski for
pointing out a proof somewhat simpler than the author's original proof.
Proposition 7.1.Let M be an abelian group which does not contain any p-torsion
and let P be a p-group. Assume that P acts nilpotently on M. Then P acts trivia*
*lly
on M.
Proof.Suppose that P acts nilpotently on M and let 0 = F0 . . .Fn = M be
a filtration on M such that P acts trivially on the filtration quotients. We wa*
*nt
to do an induction on the length of the filtration, the induction start being t*
*rivial.
Let g 2 P and x 2 F2 be arbitrary. We can write gx = x + x1 where x1 2 F1. By
iterating this we get that gnx = x + nx1. Setting n = |g| gives us |g|x1 = 0. B*
*ut
since M does not contain p-torsion we have that multiplication by |g| is inject*
*ive
on M so x1 = 0. Since x and g were arbitrary we conclude that P acts trivially *
*on
F2. Induction on the length of the filtration now finishes the proof. __|_|
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Matematisk Institut, Universitetsparken 5, DK-2100 Kobenhavn O
E-mail address: jg@math.ku.dk