HOMOLOGY EXPONENTS FOR H-SPACES
ALAIN CL'EMENT AND J'ER^OME SCHERER
Abstract.We say that a space X admits a homology exponent if there exist*
*s an exponent for
the torsion subgroup of H*(X; Z). Our main result states if an H-space o*
*f finite type admits a
homology exponent, then either it is, up to 2-completion, a product of s*
*paces of the form BZ=2r,
S1, CP1 , and K(Z, 3), or it has infinitely many non-trivial homotopy gr*
*oups and k-invariants.
We then show with the same methods that simply connected H-spaces whose *
*mod 2 cohomology
is finitely generated as an algebra over the Steenrod algebra do not hav*
*e homology exponents,
except products of mod 2 finite H-spaces with copies of CP1 and K(Z, 3).
Introduction
The study of the torsion in the homotopy groups and the integral homology gro*
*ups of a space
motivated the Moore conjecture, see [Sel88], and the Serre conjecture, [Ser53].*
* Serre proved that a
simply connected space with finite dimensional (and non-trivial) mod p (co)homo*
*logy H*(X; Fp)
must have infinitely many non-trivial homotopy groups. He conjectured that ther*
*e should in fact
exist infinitely many homotopy groups of X containing p-torsion, which was prov*
*ed eventually by
McGibbon and Neisendorfer [MN84 ], relying on Miller's solution [Mil84] of the *
*Sullivan conjecture.
This was then refined further by Lannes and Schwartz in [LS86]. Their criterion*
* is that H*(X; Fp) is
locally finite, as a module over the Steenrod algebra. Dwyer and Wilkerson went*
* one step further,
[DW90 ], looking only at the module QH*(X; Fp) of indecomposable elements. F'e*
*lix, Halperin,
Lemaire, and Thomas provided yet another criterion involving the depth of H*( X*
*; Fp), [FHLT89 ].
In their subsequent paper [FHT92 ] they focused on the size of the torsion part*
* in the "loop space
homology" H*( X; Z). They proved in fact a homological version of the Moore con*
*jecture, namely
that the p-torsion part of the integral homology of the loop space of a Z(p)-el*
*liptic space always has
an exponent.
In this article we are interested in understanding when the torsion subgroup *
*of the integral
homology of a large class of loop spaces, and more generally H-spaces, can have*
* an exponent.
In the spirit of Serre's theorem, we first classify those H-spaces having a hom*
*ology exponent at
the prime 2 which are Postnikov pieces (they have only a finite number of non-t*
*rivial homotopy
groups). Thus we will say henceforth that a space admits a homology exponent if*
* there exists an
integer k such that 2k. T2H*(X; Z) = 0, where T2 stands for the 2-torsion subgr*
*oup. We work with
connected H-spaces of finite type.
Theorem 5.2 Let X be an H-space of finite type which admits a homology exponent*
*. Then either
X is, up to 2-completion, a product of spaces of the form BZ=2r, S1, CP 1 and K*
*(Z, 3), or X
admits infinitely many non-trivial k-invariants and homotopy groups.
The methods we develop predict in fact explicit degrees in which to find homo*
*logy classes of
order 2r for arbitrarily large r when the space has no homology exponent, quite*
* in the spirit of
Browder's "infinite implications", [Bro61]. This builds on previous work by the*
* first author, who
analyzed the case of a Postnikov piece with at most two non-trivial homotopy gr*
*oups in [Cl'e06].
____________
2000 Mathematics Subject Classification. Primary 57T25, 55S45 ; Secondary 55P*
*20, 55S10, 55T10, 55T20.
The second author is supported by the program Ram'on y Cajal, MEC, Spain, and*
* FEDER/MEC grant MTM2004-
06686. This research was partially supported by the Swiss National Science Foun*
*dation grant FN 200020-105383.
1
2 ALAIN CL'EMENT AND J'ER^OME SCHERER
There is a class of H-spaces which is very close to the Postnikov pieces we h*
*ave been dealing
with up to now, namely those H-spaces for which the mod 2 cohomology is finitel*
*y generated as
an algebra over the Steenrod algebra. They are obtained indeed as extensions by*
* H-fibrations of
an H-space with finite mod 2 cohomology by a Postnikov piece, [CCSa ].
Theorem 7.5 Let X be a simply connected H-space of finite type such that H*(X; *
*F2) is finitely
generated as an algebra over the Steenrod algebra. Assume that X admits a homol*
*ogy exponent.
Then X is, up to 2-completion, the product of a mod 2 finite H-space Y with cop*
*ies of K(Z, 2)
and K(Z, 3).
This contrasts with the homological version of the Moore conjecture obtained *
*by F'elix, Halperin,
and Thomas in [FHT92 ]. Of course the mod 2 cohomology of the loop space on a f*
*inite complex is
very rarely finitely generated as an algebra over the Steenrod algebra.
Acknowledgements. We would like to thank Richard Kane for providing a simple*
* proof of
Lemma 3.3, and Juan A. Crespo and Wolfgang Pitsch for helpful comments. The se*
*cond au-
thor would like to thank Kathryn Hess and the IGAT, EPFL, for the invitation wh*
*ich made this
collaboration possible.
1.Reduction to simply connected spaces
In this section we explain how to reduce the study of arbitrary connected H-s*
*paces to simply
connected ones for which the second homotopy group is torsion. These are then t*
*he spaces we study
in the rest of the article. Let us start with basic terminology and notation.
Notation 1.1. A space X is a Postnikov piece if it has only finitely many non-t*
*rivial homotopy
groups. It is an H-Postnikov piece if it is moreover an H-space. The n-th Pos*
*tnikov section
in : X ! X[n] is determined, up to homotopy, by the property that it induces is*
*omorphisms on
homotopy groups ssi, for i n, and ssiX[n] = 0 for i > n. The homotopy fiber X*
* of in is the
n-connected cover of X. When X is simple (for example when X is simply connecte*
*d or when X is
an H-space), there exist k-invariants kn 2 Hn+1(X[n - 1]; ssnX) such that X[n] *
*can be recovered
as the homotopy fiber of a map kn : X[n - 1] ! K(ssnX, n + 1) representing the *
*k-invariant. When
X is an H-space, all k-invariants are primitive elements.
Let X be a space. By {B*r, dr} we denote its mod-2 cohomology Bockstein spect*
*ral sequence:
B*1~=H*(X; F2) =) (H*(X; Z)=torsion) F2. Recall that the first differential d*
*1 = Sq1 is the
Bockstein and a pair of elements x and y which survive to the page Br and such *
*that dr(x) = y
detect a copy of Z=2r in H*(X; Z) in degree |y| = |x| + 1.
We collect now a result about "small" Postnikov pieces. These will turn out *
*to be the only
H-Postnikov pieces having an exponent.
Proposition 1.2. Let X be a connected H-space of finite type such that ss2(X) a*
*nd ss3(X) are
torsion free. Then X[3] is a product of spaces of the form BZ=pr, S1, CP 1 and *
*K(Z, 3). Moreover,
X[3] admits a homology exponent.
Proof.It is well-known that a copy of the integers in ss1X corresponds to a cop*
*y of S1 splitting of
X (because of the existence of a section S1 ! X). One readily verifies that H3(*
*K(Z=pr, 1); Z) = 0,
which shows that the first k-invariant must be trivial. Thus X[2] splits as a p*
*roduct of copies of
BZ=pr's, S1's, and CP 1's. Next, the only elements in H4(K(Z=pr, 1); Z), for a*
*ny prime p and
any integer r, and H4(K(Z, 1); Z) are multiples of the square of the generator *
*in degree 2. Such
elements are not primitive (unless they are trivial), and hence cannot be the k*
*-invariants of an
H-space. Therefore, the second k-invariant of X[3] is trivial as well and the *
*space splits as a
product.
HOMOLOGY EXPONENTS FOR H-SPACES *
* 3
It remains to prove the assertion about the homology exponent. Recall that pr*
*.He*(BZ=pr; Z) = 0
(by a transfer argument), H*(S1; Z), H*(CP 1; Z) are torsion free and 2 . T2(H**
*(K(Z, 3); Z)) = 0
(as a consequence of Serre [Ser53] or Cartan's computations [Car55]).
Lemma 1.3. Let r 1 and (S1)r ! Y ! X be a H-fibration. If X admits a homology*
* exponent,
then so will Y .
Proof.Let us remark that the fibration is orientable [Spa81, p. 476]. We obtain*
* the result for r = 1
by inspecting the associated Gysin cohomology exact sequence and conclude by in*
*duction on r.
We conclude this section with the promised reduction. The existence of a homo*
*logy exponent
for an arbitrary H-space is detected in the homology of a certain covering spac*
*e.
Proposition 1.4. Let X be a connected H-space of finite type. Then there exists*
* a simply connected
H-space of finite type Y such that ss2Y is a torsion abelian group and Y fits i*
*nto the following H-
fibration:
Y ____//_X___//_Bss1X x K(Zk, 2),
for some k 0. Moreover, if X admits a homology exponent, then so will Y .
Proof.Let us first deal with the copies of Z in ss1X. They correspond to a toru*
*s (S1)r splitting
off X. There exists hence X1 such that X ' X1 x (S1)r and ss1X1 ~= aZ=2sa A, *
*where A
is a 20-torsion abelian group. For any a there is a map BZ=2sa ! K(Z, 2) corres*
*ponding to the
Bockstein operation of order sa. Let us define X2 to be the homotopy fiber of t*
*he composite map
Y
X1 ! Bss1X1 ! K(Z, 2) x K(A, 1).
a
Q
As in the proof of [LS86, Proposition 0.7], X2 splits as a product eXx a S1, w*
*here eXdenotes the
universal cover of X. By the previous lemma, X2, and thus eX, admit an exponent*
* if X does.
Finally let us write ss2X ~= Zk A0 where A0 is a finite torsion group and d*
*efine Y to be
the homotopy fiber of the map Xe ! K(Zk, 2) so that ss2Y ~=A0. The previous l*
*emma yields
the statement about the exponent. The description of the base of the H-fibrati*
*on comes from
Proposition 1.2.
2.A splitting principle
Let X be a Postnikov piece, which highest non-trivial homotopy group is ssnX *
*= H. In this
section we show that, if H is torsion free, the (n - 2)-connected cover X(n - 2*
*) splits as a product
K(H, n)xK(G, n-1). Loosely speaking, the n-th k-invariant attaches K(H, n) dire*
*ctly to X[n-2]
and cannot tie the last two homotopy groups together.
We will first need some basic results on Eilenberg-Mac Lane spaces. We follow*
* the terminology
and notation of [Sch94, Chapter 1]. For a unified treatment of the spaces K(Z=2*
*s, n), with s 1,
and K(Z, n), it is convenient to introduce a notation for the higher Bockstein *
*operations. Let un
(respectively Sq1sun) denote the generator of the 1-dimensional F2-vector space*
* Hn(K(Z=2s, n); F2)
(respectively Hn+1(K(Z=2s, n); F2)). For an admissible sequence I = (i1, . .,.*
*im ), we will write
SqIsun instead of Sq(i1,...,im-1)Sq1sun if im = 1 and instead of SqIun if im 6=*
* 1. We denote also by
un the generator of Hn(K(Z, n); F2).
Serre computed the mod-2 cohomology of Eilenberg-Mac Lane spaces.
Theorem 2.1. (Serre, [Ser53]) Let n 1 and s 1.
(1) The F2-algebra H*(K(Z=2s, n); F2) is isomorphic to the polynomial F2-alg*
*ebra on genera-
tors SqIsun, where I covers all the admissible sequences of excess e(I) *
*< n.
4 ALAIN CL'EMENT AND J'ER^OME SCHERER
(2) The F2-algebra H*(K(Z, n); F2) is isomorphic to the polynomial F2-algebr*
*a on generators
SqIun, where I covers all the admissible sequences of the form (i1, . .,*
*.in) where in 6= 1
and of excess e(I) < n.
Our first lemma relies on Serre's computations in low degrees.
Lemma 2.2. Let G be a finitely generated abelian group, H be free abelian, and *
*n 3. Then
P n+1H*(K(G, n - 1); H) = 0.
Proof.Since H*(K(G, n - 1); H) ~=H*(K(G, n - 1); Z) H, it is enough to consid*
*er the case
when H = Z. When n 4, the only elements in Hn+1(K(G, n - 1); F2) are sums of *
*elements of
the form Sq2un-1. These elements all have non-trivial Bockstein and we see the*
*refore from the
Bockstein spectral sequence that Hn+1(K(G, n - 1); Z) is 2-torsion free. As the*
*re is obviously no
odd primary torsion in this degree (from Cartan's description of H*(K(G, n - 1)*
*; Fp), [Car55]), we
have that Hn+1(K(G, n - 1); Z) = 0. When n = 3, write G ~=Zk A, where A is to*
*rsion. Then
H4(K(G, 2); Z) ~=H4(K(Zk, 2); Z). There are no primitive elements in this degre*
*e.
Proposition 2.3. Let X be a simply connected H-space of finite type such that s*
*s2X is a torsion
group. Let n 3, consider the Postnikov section X[n] and assume that ssnX is t*
*orsion free. Then
we have the following H-fibration:
X[n] ____//_X[n - 2]__//_K(ssn-1X, n) x K(ssnX, n + 1).
Proof.Set G = ssn-1X. We prove that the k-invariant kn 2 Hn+1(X[n - 1]; H) fact*
*ors through
Hn+1(X[n - 2]; H). Let us consider the fibration
K(G, n - 1) ! X[n - 1] ! X[n - 2]
and the cofibration K(G, n - 1) ! X[n - 1] ! C. From Lemma 2.2 we deduce that t*
*he H-map
K(G, n - 1) ! X[n - 1] kn-!K(H, n + 1) is null-homotopic.
Therefore kn factors through a map C ! K(H, n + 1). By Ganea's result [Gan65 *
*] the fiber of
the map C ! X[n - 2] is the join K(G, n - 1) * (X[n - 2]). This is an n-connec*
*ted space and
ssn+1(K(G, n - 1) * (X[n - 2])) ~=G ss2X, a torsion group because ss2X is so*
*. Thus K(H, n + 1)
is a K(G, n - 1) * (X[n - 2])-local space, as we assume that H is torsion free*
*. From Dwyer's
version of Zabrodsky lemma [Dwy96 , Proposition 3.5] we deduce that kn factors *
*through a map
k : X[n - 2] ! K(H, n + 1).
If kn-1 2 Hn(X[n-2]; G) denotes the previous k-invariant, this means that X[n*
*] is the homotopy
fiber of the product map X[n - 2] kn-1xk-----!K(G, n) x K(H, n + 1).
3.Gaps in the primitives
This section contains the key cohomological result which makes the analysis o*
*f the Serre spectral
sequence possible. We notice first that there are gaps in the mod 2 cohomology*
* of Eilenberg-
Mac Lane spaces and show then that these gaps propagate in the cohomology of an*
*y Postnikov
piece.
Definition 3.1. Let n 1. We set An = {a 2 N, a odd| 2(a) n + 1} where 2(a*
*) is the 2-adic
length of the integer a.
We will show that there are no indecomposable elements in the cohomology of a*
*n n-stage Post-
nikov piece in degrees a 2 An. To deduce that there are no primitive elements e*
*ither, we make
use of the relationship provided by the Milnor-Moore theorem [MM65 , Propositi*
*on 4.21]: For a
connected, associative, and commutative Hopf algebra over F2, there is an exact*
* sequence of graded
modules
0 ____//_P (,H)__//_P H___//_QH,
HOMOLOGY EXPONENTS FOR H-SPACES *
* 5
where ,H is the image of the Frobenius map , : x 7! x2, QH is the module of ind*
*ecomposable
elements and P H is the module of primitive elements of H.
Lemma 3.2. Let H be a finitely generated abelian group and n 2. Then
QaH*(K(H, n); F2) = 0 = P aH*(K(H, n); F2)
for all a 2 An.
Proof.When it is not trivial, the F2-algebra structure of H*(K(H, n); F2) is gi*
*ven by a polynomial
algebra on generators of the form SqIsun where I runs over admissible sequences*
* with excess e(I) <
n, as we have recalled in Theorem 2.1. Careful calculations show that these ge*
*nerators lie in
degrees 1 + 2h1+ . .+.2hn-1 where h1 . . .hn-1 0 (see [Ser53, Th'eor`eme 1,*
* p. 212 and
Th'eor`eme 2, p. 213]). The 2-adic length of these degrees is bounded by n. Thi*
*s shows that there
are no indecomposable elements in the degrees we claimed. These degrees being *
*odd, there are
no primitives either, because the kernel of the map P H*(K(H, n); F2) ! QH*(K(H*
*, n); F2) is
concentrated in even degrees.
The proof of the following lemma has been kindly communicated to us by Richar*
*d Kane, [Kan ].
Lemma 3.3. Let B be a connected, associative, and commutative Hopf algebra of f*
*inite type over
F2 and A a sub-Hopf algebra of B. Then the morphism QA ! QB is injective in odd*
* degrees.
Proof.We work in degree 2n + 1. Consider the Hopf subalgebra C of A, and hence *
*of B, generated
by the elements in A of degree 2n. Then one has an inclusion of quotient Hopf*
* algebras A==C ,!
B==C by [Kan88 , Corollary p.9]. Let x be an indecomposable element in QA of de*
*gree 2n + 1. It
determines a non-zero primitive element in P (A==C), hence in P (B==C). As the *
*map P (B==C) !
Q(B==C) is injective in odd degrees, we see that the composite QA ! Q(B==C) is *
*injective in
degree 2n + 1. Therefore QA ! QB must be injective in degree 2n + 1 as well.
Remark 3.4. The preceding lemma has a nice interpretation in terms of Andr'e-Qu*
*illen homology,
the derived functor of Q(-). It is proved in [CCSb , Proposition 1.3] that one *
*has, in the setting of
the lemma, an exact sequence HQ1(B==A) ! QA ! QB ! Q(B==A) ! 0, a result dual t*
*o that of
Bousfield, [Bou70, Theorem 3.6]. Moreover the graded F2-vector space HQ1(B==A) *
*is concentrated
in even degrees.
We are now ready to prove that the gaps also appear in the cohomology of any *
*Postnikov piece.
Proposition 3.5. Let n 2 and X be a simply connected n-stage H-Postnikov piec*
*e of finite type.
Then QaH*(X; F2) = 0 = P aH*(X; F2) for all a 2 An.
Proof.The proof goes by induction on n. We have the following H-fibration given*
* by the Postnikov
tower of X:
K(ssn(X), n)___//_X_p_//_X[n - 1].
We rely on the analysis of the Eilenberg-Moore spectral sequence done by Smith *
*[Smi70, Propo-
sition 3.2]. The quotient Hopf algebra R = H*(X[n - 1]; F2)== kerp* can be ide*
*ntified via p*
with a sub-Hopf algebra of H*(X; F2). The corresponding quotient S = H*(X; F2)*
*==R is iso-
morphic to a sub-Hopf algebra (and a sub-A2-algebra) of H = H*(K(ssn(X), n); F2*
*). There is
a section S ! H*(X; F2), which is a map of algebras, so that the module of inde*
*composables
QH*(X; F2) is isomorphic to QR QS, as graded F2-vector spaces. We have to pro*
*ve that both
QR and QS are trivial in degrees in An. First, since Q(-) is right exact, we h*
*ave a surjection
QH*(X[n-1]; F2) i QR. Now, QaH*(X[n-1]; F2) = 0 for any a 2 An-1 by induction h*
*ypothesis
and we conclude that QaR = 0 for any a 2 An since An An-1. Second, we deal w*
*ith QS.
Let us apply the preceding lemma to the inclusion S H*(X; F2). We see that QS*
* ! QH is a
monomorphism in odd degrees. Therefore QaS = 0 for all a 2 An by Lemma 3.2.
6 ALAIN CL'EMENT AND J'ER^OME SCHERER
4.Transverse elements in Eilenberg-Mac Lane spaces
Now begins the study of the 2-torsion in Postnikov pieces. In this section we*
* deal with the first
step of the induction, namely the analysis of the case of Eilenberg-Mac Lane sp*
*aces. Recall that
{B*r, dr} denote the mod-2 cohomology Bockstein spectral sequence of a space X.
Definition 4.1. Let n and r be two positive integers. An element x 2 Bnris said*
* to be `-transverse
if dr+lx2l 6= 0 2 B2lnr+lfor all 0 l `. An element x 2 Bnris said to be tr*
*ansverse if it is
`-transverse for all ` 0. We will also speak of transverse implications of an*
* element x 2 Bnr.
Every transverse element gives rise to 2-torsion of arbitrarily high order in*
* the integral coho-
mology of X. This definition, introduced in [Cl'e02], adapts Browder's "infinit*
*e implications" from
[Bro61] to our purpose. To us, the fact that the elements die in increasing pag*
*es of the Bockstein
spectral sequence is crucial, whereas Browder was merely interested to know tha*
*t the degrees of
the elements was increasing.
Our strategy for disproving the existence of a homology exponent for a space *
*will consist in
exhibiting a transverse element in its mod-2 cohomology Bockstein spectral sequ*
*ence. Note that
in principle the absence of transverse elements does not imply the existence of*
* an exponent for the
2-torsion part in H*(X; Z). An easy example if given by the infinite wedge _M(Z*
*=2n, n).
In the special case of Eilenberg-Mac Lane spaces, we have the following resul*
*t, taken from the
first author PhD thesis, [Cl'e02, Theorem 1.3.2].
Theorem 4.2. Let H be an abelian group of finite type and let n 2. Consider t*
*he Eilenberg-
Mac Lane space K(H, n) and its mod-2 cohomology Bockstein spectral sequence {B**
*r, dr}. Suppose
that one of the following assumptions holds:
o n is even and x 2 Bnsjis 0-transverse for some 1 j l,
o x 2 P evenB*1is 0-transverse (Sq1x 6= 0).
Then x is transverse.
Note that the abelian group H is isomorphic to Zs Z=2s1 . . .Z=2sl A, whe*
*re A is a
20-torsion group, which is therefore invisible to the mod 2 Bockstein spectral *
*sequence. Hence the
first type of 0-transverse elements correspond basically to the fundamental cla*
*sses un introduced in
Section 2, one for each copy of Z=2sj(the fundamental classes coming from the c*
*opies of Z survive
to Bn1).
Remark 4.3. In general a 0-transverse implication does not imply transverse imp*
*lications. More
precisely, the fact that x 2 P evenH*(X; F2) is such that Sq1x 6= 0 does not al*
*ways force x to be
transverse. A counter-example is given by X = BSO and x = w2, the second Stiefe*
*l-Withney class
in H2(BSO; F2).
From Theorem 4.2 it is not difficult to prove that most Eilenberg-Mac Lane sp*
*aces have no
homology exponent.
Proposition 4.4. Let H be a non-trivial 2-torsion abelian group and let n 2. *
*The Eilenberg-
Mac Lane space K(H, n) has no homology exponent.
Proof.Accordingly to the K"unneth formula, it is sufficient to establish the re*
*sult when H = Z=2sfor
some s 1. If n is even, consider the reduction of the fundamental class un 2 *
*Hn(K(Z=2s, n); F2).
This class survives to Bnsand is 0-transverse. Then un 2 Bnsis transverse. If n*
* is odd, consider the
admissible sequence (2, 1). Its excess is exactly 1 and therefore Sq2,1sun 2 P *
*evenH*(K(Z=2s, n); F2)
when n 3. Moreover we have Sq1Sq2,1sun = Sq3,1sun by Adem relations, which m*
*eans that
Sq2,1sun is 0-transverse. Hence Sq2,1sun 2 Bn+31is transverse.
Proposition 4.5. Let H be a finitely generated abelian group and n 4. The Eil*
*enberg-Mac Lane
space K(H, n) is then either mod 2 acyclic, or has no homology exponent.
HOMOLOGY EXPONENTS FOR H-SPACES *
* 7
Proof.By the K"unneth formula and Proposition 4.4, it is sufficient to analyze *
*the case H = Z.
Consider the reduction of the fundamental class un 2 Hn(K(Z, n); F2). If n is e*
*ven, then Sq2un is
transverse. If n is odd, then Sq6,3un is transverse.
5. Transverse elements in Postnikov pieces
We are now ready to prove our main result: Most Postnikov pieces do not have *
*a homology ex-
ponent. The strategy to prove this relies on the crucial observation that the t*
*ransverse implications
of certain element in the cohomology of the total space of a fibration can be r*
*ead in the cohomology
of the fibre.
Lemma 5.1. Let j : F ! X be a continuous map. If x 2 H*(X; F2) is such that j*(*
*x) 6= 0 2
H*(F ; F2) is transverse, then x itself is transverse.
Proof.It follows from the naturality of the Bockstein spectral sequence.
Theorem 5.2. Let X be an H-space of finite type which admits a homology exponen*
*t. Then either
X is, up to 2-completion, a product of spaces of the form BZ=2r, S1, CP 1 and K*
*(Z, 3), or X
admits infinitely many non-trivial k-invariants and homotopy groups.
Proof.Let us assume that X is a Postnikov piece. By Proposition 1.4, there is a*
*n H-fibration of
the form
Y ____//_X___//_Bss1X x K(Zr, 2),
where Y is a simply connected H-space of finite type such that ss2Y is a torsio*
*n abelian group.
Moreover, Y admits a homology exponent. It is also clearly a Postnikov piece. L*
*et us show that Y
is, up to 2-completion, a product of copies of K(Z, 3). By Proposition 1.2, thi*
*s will imply that X
itself splits as the announced product.
Assume that ssnY = H is the highest non-trivial homotopy group of Y , up to 2*
*-completion. If
n = 2, since ss2Y is a torsion abelian group, we deduce from Proposition 4.4 th*
*at H is 20-torsion.
In other words Y2^is contractible. We can therefore assume that n 3. The spac*
*e Y fits into the
fibration sequence
K(H, n)__j__//Y_i__//Y [n -_1]k//_K(H, n + 1),
where k denotes the last k-invariant. We analyze the situation in two steps, d*
*epending on the
presence of 2-torsion in H.
Let us first assume that H contains 2-torsion, let us say bZ=2tb. Choose an *
*index b and consider
the projection ss : H ! Z=2tbon the corresponding cyclic subgroup. Pick vn 2 Hn*
*(K(H, n); F2),
the image via ss* of the class un 2 Hn(K(Z=2tb, n); F2).
Set , = (2n-1 - 2, 2n-2 - 1, 2n-3 - 1, . .,.3, 1). The degree deg(Sq,tvn) = 2*
*n - 2 is even and
Sq1Sq,tvn 6= 0 since e(,) = n - 2. By Theorem 4.2, Sq,tvn is transverse. Since *
*Y is an H-space and
thenk-invariant is an H-map, the element dn+1vn isnprimitive, and so is d2n-1Sq*
*,tvn = Sq,tdn+1vn 2
P 2 -1H*(Y [n - 1]; F2). By Proposition 3.5, P 2 -1H*(Y [n - 1]; F2) = 0 since*
* 2n - 1 2 An-1.
Therefore, Sq,tvn survives in the Serre spectral sequence and by the previous l*
*emma, H*(Y ; F2)
contains a transverse element. In particular it has no homology exponent.
Hence, H must be 2-torsion free and is thus isomorphic to Zs A, where A is a*
* torsion group,
for some s 1. By Proposition 2.3, Y fits in the following H-fibration:
K(H, n) x K(ssn-1Y, n -_1)_//_Y____//Y [n - 2].
Choose now vn 2 Hn(K(H, n); F2) to be the image of the class un 2 Hn(K(Z, n); F*
*2) given by
projection on the first copy of Z in H.
If n 4, then set j = (2n-2 + 2n-3 - 2, 2n-3 + 2n-4 - 1, 2n-4 + 2n-5 - 1, . *
*.,.5, 2). The degree
deg(Sqjvn) = 2n-1+2n-2-2 is even and Sq1Sqjvn 6= 0 since e(j) = n-2. Thus Sqjvn*
* is transverse
8 ALAIN CL'EMENT AND J'ER^OME SCHERER
and survives in the Serre spectral sequence of the above fibration since 2n-1 +*
* 2n-2 - 1 2 An-2.
In this case, H*(Y ; F2) contains a transverse element and has no homology expo*
*nent.
Therefore, n = 3 and Y ' K(H, 3) x K(ss2Y, 2). Since Y admits a homology exp*
*onent, the
torsion group ss2Y is trivial and H is torsion free.
The proof of the theorem predicts explicit degrees in which to find higher an*
*d higher torsion in
the integral cohomology of the space.
Corollary 5.3. Let X be a simply connected H-Postnikov piece of finite type, sa*
*y X ' X[n].
Assume that ss2X is torsion and that X is not equivalent up to 2-completion to *
*a product of copies
of K(Z, 3). Then, for any integer k, there is a copy of Z=2k in H*(X; Z)
(1) in degree 2k(2n - 2) if ssnX contains 2-torsion,
(2) in degree 2k(2n-1 + 2n - 2- 2) if not.
Proof.Since X is different from K(Zm , 3), we know from Theorem 5.2 that X has *
*no exponent.
The higher and higher torsion is detected by the consecutive powers of the elem*
*ents Sq,vn and
Sqjvn constructed in the above proof.
Any finite H-space has obviously a homology exponent. Our second corollary a*
*pplies to its
Postnikov sections. As soon as it has at least two homotopy groups, it cannot h*
*ave a homology
exponent.
Corollary 5.4. Let X be a simply connected finite H-space and n 3. Then X[n] *
*has a homology
exponent if and only if X[n] ' X[3] ' K(Zr, 3) for some r 0.
Proof.The fact that the H-space X is finite and simply connected forces it to b*
*e 2-connected,
[Bro61, Theorem 6.10]. Moreover, ss3X ~=Zr for some integer r, by work of Hubbu*
*ck and Kane,
[HK75 ]. The result now follows directly from Theorem 5.2.
This corollary applies in particular to S3. The Postnikov section S3[3] ' K(Z*
*, 3) has a homology
exponent, but all higher Postnikov sections X[n], n 4, have none. The followi*
*ng proof of a result
obtained by Levi in [Lev95] is, to our knowledge, the first one not based on Mi*
*ller's solution of the
Sullivan's conjecture [Mil84]. Let us mention in this context the work of Klaus*
*, [Kla02], who proves
the statement about the k-invariants for BG^2, not for the loop space.
Corollary 5.5. Let G be a 2-perfect finite group. Then (BG^2) has infinitely m*
*any non-trivial
k-invariants and homotopy groups.
Proof.Suppose BG^2is a Postnikov piece. Following Levi [Lev95], there is a homo*
*logy exponent for
(BG^2) and therefore this space has to be a product of copies of BZ=2r, S1, CP*
* 1 or K(Z, 3). Since
(BG^2) has torsion homotopy groups, the only copies that can occur are of the *
*form BZ=2r. Thus
BG^2' K(A, 2), where A is a 2-torsion abelian group. By the Evens-Venkov theor*
*em, [Eve61],
H*(BG^2; F2) is Noetherian. Hence A is trivial, and so is BG^2.
6. Comparison with other forms of Serre's theorem
In this section we compare our theorem to the other results we mentioned in t*
*he introduction. We
show that the existence of a homology exponent is stronger than all previously *
*established criteria,
except possibly [FHLT89 ], which seems difficult to relate directly to cohomolo*
*gical statements.
Therefore, when X is an H-space, our result provides new proofs of those. They *
*are very different
in spirit, since they do not require the Sullivan conjecture. For simplicity we*
* deal here with simply
connected spaces.
Proposition 6.1. Let X be a simply connected H-Postnikov piece. Then
(1) (Serre [Ser53]) H*(X; F2) is not finite,
HOMOLOGY EXPONENTS FOR H-SPACES *
* 9
(2) (Lannes-Schwartz [LS86]) H*(X; F2) is not locally finite,
(3) there exists an element of infinite height in H*(X; F2),
(4) (Grodal [Gro98]) the transcendence degree of H*(X; F2) is infinite unles*
*s X is homotopy
equivalent, up to 2-completion, to K(Z, 2)s,
(5) (Dwyer-Wilkerson [DW90 ]) the unstable module QH*(X; F2) is not locally *
*finite unless X
is homotopy equivalent, up to 2-completion, to K(Z, 2)s.
Proof.Notice first that K(Z, 2) and K(Z, 3) satisfy (1) - (5). Assume now that *
*X is a Postnikov
piece, say X ' X[n]. In the proof of Theorem 5.2 we first considered the coveri*
*ng fibration (S1)r !
Y ! X. The map Y ! X induces isomorphisms in homology in high degrees. We can t*
*herefore
assume that ss2X is torsion. Our proof then provides a transverse element x 2 H*
**(X; F2) in even
degree whose image in H*(K(ssnX, n)) is a transverse element of the form SqItun*
* for some admissible
sequence (i1, . .,.im ). In particular all powers x2k are non-zero, which prove*
*s (1) - (3). Moreover
the elements x, Sq2i1x, Sq4i1,2i1x, . .a.re non-zero, indecomposable, and algeb*
*raically independent
because so are the corresponding images in H*(K(ssnX, n)). This proves (4) and *
*(5).
7. Cohomological finiteness conditions
The strategy we followed to analyze the integral homology of Postnikov pieces*
* can be applied in
a more general context. We work in this last section with simply connected H-sp*
*aces X such that
H*(X; F2) is finitely generated as an algebra over the Steenrod algebra. This s*
*ection relies on the
Sullivan conjecture. As it may be considered thus as less elementary than the p*
*art about Postnikov
pieces, we have decided to postpone it till the end of the article.
From the assumption on the mod 2 cohomology, we infer by [CCSa , Lemma 7.1] t*
*hat there exists
an integer n such that the module QH*(X; F2) of indecomposable elements lies in*
* the (n-1)-st stage
of the Krull filtration for unstable modules, [Sch94]. Therefore there exists b*
*y [CCSa , Theorem 7.3]
a simply connected H-space Y = PBZ=2X with finite mod 2 cohomology and a series*
* of principal
H-fibrations
X = Xn pn-!Xn-1 ! . .!.X1 p1-!X0 = Y
of simply connected spaces such that the homotopy fiber of pi is an Eilenberg-M*
*ac Lane space
K(Pi, i), where Pi splits as a product of a finite direct sum Pi0of cyclic grou*
*ps Z=2r and a finite
direct sum Pi00of Pr"ufer groups Z21. Let us recall here that Xk is obtained a*
*s the kBZ=p-
nullification of X (the above tower is Bousfield's nullification tower, [Bou94]*
*). Since K(Z21, i) and
K(Z, i + 1) are mod 2 equivalent, we alter slightly the way in which the Pi's a*
*re added to Y in
order to work in a more familiar setting. Then we can recover X from the tower
X = Yn qn-!Yn-1 ! . .!.Y1 q1-!Y0 = Y
of simply connected spaces and principal H-fibrations, where the homotopy fiber*
* of qiis the product
of Eilenberg-Mac Lane spaces K(Pi0+1, i + 1) x K(Pi00, i). Notice that Q1 = P10*
*is trivial because
we assume that X is simply connected (Y2^is therefore 2-connected, [Bro61]). We*
* have a splitting
result, just like in Proposition 2.3.
Lemma 7.1. Let X be a simply connected H-space such that H*(X; F2) is finitely *
*generated as an
algebra over the Steenrod algebra. Assume that ss2X^2is torsion. Then there is *
*an H-fibration
X _____//Yn-2___//_K(Pn00 Pn0, n + 1) x K(Pn00, n).
Proof.The proof is based on the Zabrodsky lemma, as in Proposition 2.3.
Our next result is the analog in the present setting of Proposition 3.5. Reca*
*ll from Section 3
that the set An consists in those integers for which the 2-adic length is stric*
*tly larger than n.
10 ALAIN CL'EMENT AND J'ER^OME SCHERER
Proposition 7.2. Let X be a simply connected H-space such that H*(X; F2) is fin*
*itely generated as
an algebra over the Steenrod algebra. There exists then integers m and N such t*
*hat QaH*(X; F2) =
0 = P aH*(X; F2) for all a 2 An with a N.
Proof.The integer m is determined by the stage of the Krull filtration in which*
* QH*(X; F2) lives,
i.e. by the degrees in which the homotopy groups of the homotopy fiber of X ! P*
*BZ=2X = Y are
non-trivial. With the above notation, m = n if Pn00is trivial, and m = n + 1 if*
* Pn00is not. The
proof goes then by induction on m. When m = 0, choose N to be larger than the c*
*ohomological
dimension of Y . The proof of Proposition 3.5 goes through.
Lemma 7.3. Let X be a simply connected H-space which fits, up to 2-completion, *
*in an H-fibration
of the form
K( tZ, 2)____//_X____//Y
where H*(Y ; F2) is finite. Then X has no homology exponent unless the fibrati*
*on splits up to
2-completion, i.e X '^2Y x K( tZ, 2).
Proof.Let us omit the 2-completions in the proof and write the details of the p*
*roof when t = 1. By
the result of Hubbuck and Kane, [HK75 ], ss3Y is isomorphic to a direct sum of *
*say s copies of Z.
The map classifying the fibration factors through Y [3] ' K( sZ, 3) ! K(Z, 3). *
*The E2-term of
the Serre spectral sequence has the form Z[u] H*(Y ; Z), where u has degree 2 a*
*nd the cohomology
of Y is of finite dimension N, and of exponent 2a for some integer a. The diffe*
*rential d3(u) = x for
some non-zero element x 2 H3(Y ; Z) ~= sZ. Therefore d3(un) = nx un-1. At worst*
* d3(x un-1)
is non-zero and then hits a torsion element, of order at most 2a. Hence, on the*
* third column of the
E4-term, we have a group covering Z=2n-a in vertical degree 2n. From the finite*
*ness of Y we see
that the spectral sequence collapses at EN-3 . An iteration of the above argume*
*nt shows therefore
that the third column of the E1 -term contains a group covering Z=2n-(N-5)ain v*
*ertical degree 2n,
for any n 1. In particular there is arbitrarily high torsion in H*(X; Z). The*
*refore, for X to have
an exponent, the fibration must split.
Remark 7.4. We point out that the preceding lemma provides simple examples of f*
*ibrations, such
as K(Z, 2) ! S3<3> ! S3, where both the fiber and the base have an exponent, bu*
*t the total space
has none.
Theorem 7.5. Let X be a simply connected H-space of finite type such that H*(X;*
* F2) is finitely
generated as an algebra over the Steenrod algebra. Assume that X admits a homol*
*ogy exponent.
Then X is, up to 2-completion, the product of a mod 2 finite H-space Y with cop*
*ies of K(Z, 2)
and K(Z, 3).
Proof.We follow the proof of Theorem 5.2. Let us thus assume that X admits a ho*
*mology exponent.
By killing the copies of Z in ss2X just like in Proposition 1.4, we can assume *
*that ss2X^2is torsion.
We also see by inspection of the tower that ss2(Yi)^2is torsion for any i 0. *
*Therefore the splitting
in Lemma 7.1 holds and we work with a fibration
K(Pn00 Pn0, n) x K(Pn00, n_-_1)//_X_//_Yn-2.
If Pn06= 0, it must contain a copy of Z=2r as direct summand. Choose a power of*
* the corresponding
element Sq,tvn, of degree larger than the integer N given in Proposition 7.2. F*
*rom the Serre spectral
sequence for the above fibration we see that this provides a transverse element*
* in H*(X; F2).
Therefore Pn0= 0 (and so Pn00is not trivial).
If n 3 we choose a copy of Z21 in Pn00and a suitable power of the correspon*
*ding element Sqjvn
to detect a transverse element in H*(X; F2). Since we assume that X has a homol*
*ogy exponent, we
see that n 2, i.e. X is the homotopy fiber of a map k : Y ! K(P100, 2) x K(P2*
*00, 3). To conclude
the proof we must show that this map is trivial.
HOMOLOGY EXPONENTS FOR H-SPACES *
*11
The mod 2 cohomology of the H-space Y is finite. Rationally it is thus a pr*
*oduct of odd
dimensional spheres and, in particular, ss4Y2^ is torsion. This implies that t*
*he projection of k
on the second factor Y ! K(P200, 3) is the trivial map, up to 2-completion. Hen*
*ce the copies of
K(Z^2, 3) split off X^2. We are left with the analysis of a fibration X -! Y -!*
* K(P100, 2). If the
map Y ! K(P100, 2) is not trivial, we conclude from Lemma 7.3 that X cannot hav*
*e a homology
exponent. Hence, the fibration must split and this concludes the proof.
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12 ALAIN CL'EMENT AND J'ER^OME SCHERER
Rue Louis-Meyer 9,
CH - 1800 Vevey,
Switzerland.
E-mail address: alain.clement@bluewin.ch
Departament de Matem`atiques,
Universitat Aut`onoma de Barcelona,
E-08193 Bellaterra,
Spain.
E-mail address: jscherer@mat.uab.es