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. References [Bou70] A. K. Bousfield. Nice homology coalgebras. Trans. Amer. Math. Soc., 148* *:473-489, 1970. [Bou94] _____. Localization and periodicity in unstable homotopy theory. J. Ame* *r. Math. 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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