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