Deconstructing Hopf spaces

⋆Natàlia Castellana^{1}, Juan A. Crespo^{2}, Jérôme Scherer^{1}

1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain

  e-mail: natalia@mat.uab.es; jscherer@mat.uab.es

2 Departamento de Economía, Universidad Carlos III de Madrid, E–28903 Getafe, Spain

  e-mail: jacrespo@eco.uc3m.es

Abstract. We characterize Hopf spaces with finitely generated cohomology as an algebra over the Steenrod algebra. We “deconstruct” the original space into an H-space Y with finite mod p cohomology and a finite number of p-torsion Eilenberg-Mac Lane spaces. We give a precise description of homotopy commutative H-spaces in this setting.

Introduction

Since their introduction in the 50’s by Serre, H-spaces have produced some of the most beautiful results in Algebraic Topology. Some examples are Adams’ solution of the Hopf invariant one conjecture [1], the criminal of Hilton-Roitberg [22], the construction of DI(4) by Dwyer and Wilkerson [18], the recent proof that a finite loop space is of the homotopy type of a manifold [4], and the new example of a finite loop space in [3].

The structure of finite H-spaces is rather well understood. In one of the most important articles on finite H-spaces, [23], Hubbuck shows that there are no other finite connected homotopy commutative H-spaces than products of circles, which was proved for compact Lie groups by James. It was not until the early 90’s that this result was extended by Slack to H-spaces with finitely generated mod 2 cohomology. With the aid of secondary operations, he shows in [37] that such homotopy commutative H-spaces are products of circles and other Eilenberg-Mac Lane spaces. In fact, using the modern techniques of Lannes’ T functor, Broto et al. obtain in [11] a structure theorem for all H-spaces with noetherian mod p cohomology. They “deconstruct” such an H-space into mod p finite ones and copies of K(ℤ/p^{r}, 1) and ℂP^{∞} in a functorial way. Recall that an H-space is said to be mod p finite if it is p-complete with finite mod p cohomology, which we denote simply by H^{∗}(−).

⋆ The authors are partially supported by MEC grant MTM2004-06686 and the third author by the program Ramón y Cajal, MEC, Spain.

Our goal is to extend such results to an even larger class of infinite dimensional spaces and understand which are the basic pieces permitting to reconstruct the original H-space.

Natural examples of H-spaces, arising in connection with those which are finite, are their Postnikov sections and connected covers. The mod p cohomology of the n-connected cover of a finite H-space is not finite in general, but is finitely generated as an algebra over the Steenrod algebra A_{p} (we refer to the article [14] for more details). Up to p-completion, the basic examples of H-spaces satisfying this cohomological finiteness condition are all finite H-spaces, and Eilenberg-Mac Lane spaces of type K(ℤ/p^{r}, n) and K(ℤ_{p^{∞}}, n).

We show that one can deconstruct any such H-space in terms of these basic H-spaces. We call an H-space which has only finitely many non-trivial homotopy groups an H-Postnikov piece.

Theorem 7.3. Let X be a connected H-space such that H^{∗}(X) is a finitely generated 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 ptorsion H-Postnikov piece whose homotopy groups are finite direct sums of copies of cyclic groups ℤ/p^{r} and Prüfer groups ℤ_{p^{∞}}.

The above fibration behaves well with respect to loop structures and a similar result holds for loop spaces. Our deconstruction theorem enables us to reduce questions on infinite dimensional H-spaces to finite ones. For instance, we use this technique to give a generalization of Hubbuck’s Torus Theorem.

Corollary 7.4. Let X be a connected homotopy commutative Hspace such that the mod 2 cohomology H^{∗}(X) is finitely generated as an algebra over the Steenrod algebra A_{2}. Then, up to 2-completion, X is homotopy equivalent to (S^{1})^{n}×F, where F is a connected 2-torsion H-Postnikov piece.

The above splitting is not a splitting of H-spaces in general, think of S^{1} × K(ℤ/2, 2), [40, Section 1.4]. When H^{∗}(X) is finitely generated as an algebra, we get back Slack’s result [37], as well as their generalization by Lin and Williams in [27].

The arguments to prove our main theorem are the following. When H^{∗}(X) is finitely generated over A_{p}, we show in Lemma 7.1 that the unstable module of indecomposable elements QH^{∗}(X) belongs to some stage U_{n} of the Krull filtration in the category of unstable modules. This filtration has been studied in [33] by Schwartz and then used in [35] in order to prove Kuhn’s non-realizability conjecture [24].

The stage U_{0} of the Krull filtration is particularly interesting since it consists exactly of all locally finite modules (direct limits of finite modules). In fact, the condition that QH^{∗}(X) is locally finite is equivalent to requiring that the loop space ΩX be Bℤ/p-local, i.e. the space of pointed maps map_{∗}(Bℤ/p, ΩX) is contractible, see [19, Prop 3.2] and [33, Proposition 6.4.5].

We extend this topological characterization to H-spaces X with QH^{∗}(X) ∈ U_{n}. We use the standard notation T_{V} for Lannes’ T functor and say that H^{∗}(X) is of finite type if H^{n}(X) is a finite 𝔽_{p}-vector space for any integer n ≥ 0.

Theorem 5.3. Let X be a connected H-space such that T_{V} H^{∗}(X) is of finite type for any elementary abelian p-group V. Then QH^{∗}(X) is in U_{n} if and only if Ω^{n+1}X is Bℤ/p-local.

We apply now Bousfield’s results on the Postnikov-like tower associated to the Bℤ/p-nullification functor P_{Bℤ/p} (relying on his “Key Lemma”, [7, Chapter 7]). They enable us to reconstruct those Hspaces such that Ω^{n+1}X is Bℤ/p-local from P_{Bℤ/p}X in a finite number of principal H-fibrations over p-torsion Eilenberg-Mac Lane spaces. When n = 0, we recover the results of Broto et al. from [10,15, 11].

Theorem 5.5. Let X be an H-space such that T_{V} H^{∗}(X) is of finite type for any elementary abelian p-group V. Then QH^{∗}(X) is in U_{n} if and only if X is the total space of an H-fibration

F  X  P_{Bℤ/p}X

where F is a p-torsion H-Postnikov piece whose homotopy groups are finite direct sums of copies of cyclic groups ℤ/p^{r} and Prüfer groups ℤ_{p^{∞}} concentrated in degrees from 1 to n + 1.

It is worthwhile to mention that working with H-spaces is crucial as illustrated by the space BS^{3}. Its loop space is Bℤ/p-local, but the fiber of the nullification map has infinitely many non-trivial homotopy groups (see Dwyer’s computations [17, Theorem 1.7, Lemma 6.2]).

Acknowledgements. 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 generosity. We warmly thank Carles Broto for his questions which regularly opened new perspectives and Jesper Grodal for many useful comments. Finally, we are indebted to the referee for suggesting to use André-Quillen cohomology to fix an earlier proof of Theorem 6.1.

1. Lannes’ T functor

Lannes’ T functor, [26], 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 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 the Steenrod algebra. The functor T_{V} is the left adjoint of − ⊗ H^{∗}(BV) in U and K, where V is an elementary abelian p-group. The left adjoint of − ⊗ H̃^{∗}(BV) is called the reduced T functor and denoted by \overline{T}_{V}. For each unstable module M ∈ U, we have a splitting of modules over the Steenrod algebra T_{V} M = M ⊕ \overline{T}_{V} M. We will use T to denote T_{ℤ/p} and \overline{T} to denote \overline{T}_{ℤ/p}.

If M = H^{∗}(X), the evaluation map BV × map(BV, X) → X induces by adjunction a map λ_{V} : T_{V} H^{∗}(X) H^{∗}(map(BV, X)) of unstable algebras over A_{p}, which is often an isomorphism. When working with H-spaces, it is often handy to deal with the pointed mapping space instead of the full mapping space.

Proposition 1.1. Let X be an H-space such that H^{∗}(X) is of finite type. Then, T_{V} H^{∗}(X) is of finite type if and only if H^{∗}(map_{∗}(BV, X)) is of finite type. In this case T_{V} H^{∗}(X) ≅ H^{∗}(map(BV, X)), as algebras over A_{p}.

Proof. If 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 [40]). Moreover, when X is connected, all connected components of the mapping space have the same homotopy type. Since the evaluation map is an H-map and has a section, there is a splitting

map(BV, X) ≃ X × map_{∗}(BV, X) .

which allows to work with the pointed mapping space. By [29, Theorem 1.5] there is a weak equivalence map_{∗}(BV, X) ≃ map_{∗}(BV, X̂_{p}) for any elementary abelian p-group V. Since X is p-good, we can work with X̂_{p}. Now T_{V} H^{∗}(X̂_{p}) ≅ H^{∗}(map(BV, X̂_{p})) by [25, Proposition 3.4.4]. ⊓ ⊔

When X is connected, the evaluation map(BV, X) → X is a homotopy equivalence if T_{V} H^{∗}(X) ≅ H^{∗}(X) (for finite spaces, this is the Sullivan conjecture proved by Miller [29, Theorems A,C]). Actually, spaces for which this happens have been cohomologically characterized by Lannes and Schwartz in [26]: their mod p cohomology is locally finite.

When one restricts the evaluation map to the connected component of the constant map, the module of indecomposable elements QH^{∗}(X) comes into play as observed by Dwyer and Wilkerson in [19, Proposition 3.2] (see also [33, 3.9.7 and 6.4.5]).

Proposition 1.2. Let X be a connected H-space of finite type. Then QH^{∗}(X) is a locally finite A_{p}-module if and only if map_{∗}(BV, ΩX) is contractible for some elementary abelian p-group V.

Proof. Since X̂_{p} is a connected p-complete H-space, QH^{∗}(X) is a locally finite A_{p}-module if and only if map_{∗}(BV, X̂_{p}) is homotopically discrete for any elementary abelian p-group V by [19, Proposition 3.2] and [33, Proposition 6.4.5].

The weak equivalence map_{∗}(BV, X) ≃ map_{∗}(BV, X̂_{p}) given by [29, Theorem 1.5] shows that this is equivalent to map_{∗}(BV, ΩX) being contractible, i.e. the loop space ΩX is BV -local. ⊓ ⊔

2. The Krull filtration of U

In [35], Schwartz proves the “strong realization conjecture” extending his previous results from [34]. This conjecture, given by Kuhn in [24], states that if the cohomology of a space lies in some stage of the Krull filtration of the category U of unstable modules, then it must be locally finite. The Krull filtration U_{0} ⊂ U_{1} ⊂ … is defined inductively, see [33, Section 6.2]. It starts with the full subcategory U_{0} of U of locally finite unstable modules and the modules in U_{n} can be characterized as follows by means of the functor \overline{T}:

Theorem 2.1. [33, Theorem 6.2.4] Let M be an unstable module. Then M ∈ U_{n} if and only if \overline{T}^{n+1}M = 0. ⊓ ⊔

The proof of Kuhn’s conjecture by Schwartz shows that under the usual finiteness conditions the cohomology of a space either lies in U_{0} or it is not in any U_{n}. Instead of looking at when the full cohomology of a space is in U_{n}, we will study the module of the indecomposable elements QH^{∗}(X). The Krull filtration induces a filtration of the category of H-spaces by looking at those H-spaces X for which QH^{∗}(X) ∈ U_{n}. There exist many spaces lying in each layer of this filtration, the most obvious ones being Eilenberg-Mac Lane spaces.

Example 2.2. The module QH^{∗}(K(ℤ/2, n + 1)) is isomorphic to the suspension of the free unstable module F (n) on one generator in degree n. In particular, the formula \overline{T} F (n) = ⊕_{0≤i≤n−1}F (i) (see [33, Lemma 3.3.1]) yields that QH^{∗}(K(ℤ/2, n + 1)) ∈ U_{n}. For an odd prime p, an easy induction argument based on the next lemma shows that QH^{∗}(K(ℤ/p, n + 1)) ∈ U_{n} as well.

More generally, let G be any abelian discrete group such that H^{∗}(K(G, n + 1)) is of finite type. Then QH^{∗}(K(G, n + 1)) ∈ U_{n} (see [33, Section 9.8] for the explicit computations of the T functor).

From the above example, we see that the filtration is not exhaustive, since the infinite product ∏_{n≥1} K(ℤ/p, n) does not belong to any stage of the filtration. As we explain in Example 5.7 nor does BU.

Next lemma shows, by means of the reduced T functor, how QH^{∗}(X) is related to QH^{∗}(map_{∗}(Bℤ/p, X)).

Lemma 2.3. Let X be an H-space such that T H^{∗}(X) is of finite type. Then \overline{T} QH^{∗}(X) ≅ QH^{∗}(map_{∗}(Bℤ/p, X)).

Proof. Under such assumptions, Proposition 1.1 applies and we know that the T functor computes the cohomology of the mapping space. Thus QT H^{∗}(X) is isomorphic to

QH^{∗}(map(Bℤ/p, X)) ≅ Q(H^{∗}(map_{∗}(Bℤ/p, X)) ⊗ H^{∗}(X))

Since T commutes with taking indecomposable elements [33, Lemma 6.4.2], it follows that T QH^{∗}(X) ≅ QH^{∗}(X)⊕QH^{∗}(map_{∗}(Bℤ/p, X)). This is equivalent to \overline{T} QH^{∗}(X) ≅ QH^{∗}(map_{∗}(Bℤ/p, X)). ⊓ ⊔

We end the section with a proposition which will allow us to perform an induction in the Krull filtration. Observe that Kuhn’s strategy to move in the Krull filtration is to consider the cofiber of the inclusion X → map(Bℤ/p, X) in the component of the constant map, see [24]. 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) ∈ U_{n} if and only if QH^{∗}(map_{∗}(Bℤ/p, X)) is in U_{n−1}.

Proof. By Theorem 2.1, the unstable module QH^{∗}(X) belongs to U_{n} if and only if \overline{T}^{n+1}QH^{∗}(X) = 0. By Lemma 2.3, \overline{T}^{n}\overline{T} QH^{∗}(X) ≅ \overline{T}^{n}(QH^{∗}(map_{∗}(Bℤ/p, X)) , and we obtain that QH^{∗}(X) ∈ U_{n} if and only if QH^{∗}(map_{∗}(Bℤ/p, X)) ∈ U_{n−1}. ⊓ ⊔

By repeatedly applying the previous proposition, one can give a more geometrical formulation to the condition QH^{∗}(X) ∈ U_{n}. This happens if and only if the pointed mapping space out of an (n + 1)- fold smash product map_{∗}(Bℤ/p ∧ ⋯ ∧ Bℤ/p, X) is homotopically discrete.

3. Bousfield’s Bℤ/p-nullification filtration.

The plan of this section follows the preceding one step-by-step, replacing the algebraic filtration defined with the module of indecomposables by a topological one. Recall (cf. [20]) that, given a pointed connected space A, a space X is A-local if the evaluation at the base point in A induces a weak equivalence of mapping spaces map(A, X) ≃ X. When X is an H-space, it is sufficient to require that the pointed mapping space map_{∗}(A, X) be contractible.

Dror-Farjoun and Bousfield have constructed a localization functor P_{A} from spaces to spaces together with a natural transformation l : X → P_{A}X which is an initial map among those having an A-local space as target (see [20] and [5]). This functor is known as the Anullification. It preserves H-space structures since it commutes with finite products. Moreover, when X is an H-space, the map l is an H-map and its fiber is an H-space.

Bousfield has determined the structure of the fiber of the nullification map l : X → P_{A}X under certain assumptions on A. We are interested in the situation in which A an iterated suspension of Bℤ/p.

Theorem 3.1. [6, Theorem 7.2] Let n ≥ 1 and X be a connected H-space such that Ω^{n}X is Bℤ/p-local. The homotopy fiber of the localization map X → P_{Σ^{n−1}Bℤ/p}X is then an Eilenberg-Mac Lane space K(P, n) where P is an abelian p-torsion group (possibly infinite). ⊓ ⊔

As mentioned by Bousfield in [6, p. 848], an inductive argument allows to obtain a precise description of the fiber of the Bℤ/p-nullifica-tion map for H-spaces for which some iterated loop space is local.

Theorem 3.2. Let n ≥ 0 and X be a connected H-space such that Ω^{n}X is Bℤ/p-local. Then there is an H-fibration

F  X  P_{Bℤ/p}X,

where F is a p-torsion H-Postnikov piece whose homotopy groups are concentrated in degrees from 1 to n. ⊓ ⊔

We introduce a “nullification filtration” by looking at those Hspaces X such that the iterated loop space Ω^{n}X is Bℤ/p-local. The example of the 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 3.3. Let G be an abelian discrete group with non-trivial mod p cohomology. Then, the Eilenberg-Mac Lane space K(G, n) enjoys the property that its n fold iterated loop space is Bℤ/p-local (it is even discrete). The infinite product ∏_{n≥1} K(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 connected covers of finite H-spaces.

Example 3.4. Let X be a finite connected H-space and consider its n-connected cover X⟨n⟩. Then Ω^{n−1}(X⟨n⟩) is Bℤ/p-local.

For a connected H-space X such that Ω^{n}X is Bℤ/p-local, the study of the homotopy type of map_{∗}(Bℤ/p, X) is drastically simplified by Theorem 3.2, since this space is equivalent to map_{∗}(Bℤ/p, F) where F is a Postnikov piece, as we explain in the proof below. A complete study of the Bℤ/p-homotopy theory of such H-spaces is undertaken in [13].

We prove now the topological analogue of Lemma 2.3.

Proposition 3.5. Let X be a connected H-space such that Ω^{n}X is Bℤ/p-local, then Ω^{n−1}map_{∗}(Bℤ/p, X) is Bℤ/p-local.

Proof. Under the hypothesis that Ω^{n}X is Bℤ/p-local, Theorem 3.2 tells us that we have a fibration

F  X  P_{Bℤ/p}X,

where F is a p-torsion Postnikov system with homotopy concentrated in degrees from 1 to n. Thus, map_{∗}(Bℤ/p, X) ≃ map_{∗}(Bℤ/p, F) because P_{Bℤ/p}X is a Bℤ/p-local space. Now, Ω^{n−1}map_{∗}(Bℤ/p, F) is Bℤ/p-local (in fact, it is a homotopically discrete space) and thus so is Ω^{n−1}map_{∗}(Bℤ/p, X). ⊓ ⊔

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 explain 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 Γ -spaces and follow his notation from [36], which is better adapted to our needs than that of Bousfield and Friedlander, see [8]. Recall that the category Γ is the category of finite sets, and a morphism θ : S → T between two finite sets is a partition of a subset of T into |S| disjoint subsets { θ(α) }_{α∈S}. A Γ -space is a contravariant functor from Γ to the category of spaces with some extra conditions. We first construct a covariant functor A_{•} : Γ → Spaces for any pointed space A by setting A_{n} = A^{n} (so in particular A_{0} = ∗) and a morphism θ : [n] → [m] induces the map θ_{∗} : A^{n} → A^{m} sending (a_{1}, …, a_{n}) to the element (b_{1}, …, b_{m}) with b_{j} = a_{i} if and only if j ∈ θ(i) and b_{j} = ∗ otherwise.

Hence, we get a contravariant functor for any pointed space X by taking the pointed mapping space map_{∗}(−, X). For map_{∗}(A_{•}, X) to be a Γ -space one needs to check it is special, i.e. the n inclusions i_{k} : [1] → [n] sending 1 to k must induce a weak equivalence map_{∗}(A^{n}, X) → map_{∗}(A, X)^{n}.

Lemma 4.1. Let A and X be pointed spaces and assume that the inclusion A^{n} ∨A ↪ A^{n} ×A induces for any n ≥ 1 a weak equivalence map_{∗}(A^{n} × A, X) → map_{∗}(A^{n} ∨ A, X). Then, map_{∗}(A_{•}, X) is a Γ - space. ⊓ ⊔

Proposition 4.2. Let A be a pointed connected space and X an Hspace. Assume that map_{∗}(A, X) is A-local. Then map_{∗}(A_{•}, X) is a Γ -space.

Proof. The cofiber sequence A^{n} ∨ A → A^{n} × A → A^{n} ∧ A yields a fibration of pointed mapping spaces

map_{∗}(A^{n} ∧ A, X)  map_{∗}(A^{n} × A, X)  map_{∗}(A^{n} ∨ A, X).

By adjunction, the fiber map_{∗}(A^{n} ∧ A, X) ≃ map_{∗}(A^{n}, map_{∗}(A, X)) is contractible since any A-local space is also A^{n}-local (A^{n} is Acellular or use Dwyer’s version of Zabrodsky’s Lemma in [17, Proposition 3.4]). Moreover, the inclusion A^{n} ∨ A → A^{n} × A induces a bijection on sets of homotopy classes [A^{n} × A, X] → [A^{n} ∨ A, X] by [40, Lemma 1.3.5]. Since all components of these pointed mapping spaces have the same homotopy type, we have a weak equivalence map_{∗}(A^{n} × A, X) ≃ map_{∗}(A^{n} ∨ A, X) and conclude by the preceding proposition. ⊓ ⊔

Theorem 4.3. Let A be a pointed connected space and let 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 constant map.

Proof. The Γ -space structure constructed above yields classifying spaces B^{n}map_{∗}(A, X) and weak equivalences ΩB^{n+1}map_{∗}(A, X) ≃ B^{n}map_{∗}(A, X) for any n ≥ 1. In our situation X is a loop space, and so is the mapping space map_{∗}(A, X). Therefore, Segal’s result [36, Proposition 1.4] applies and shows that map_{∗}(A, X) is equivalent to the loop space ΩBmap_{∗}(A, X). ⊓ ⊔

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}. In the context of Proposition 5.2 it will turn out to be contractible.

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-vector spaces.

Proof. Since π_{n}map_{∗}(Bℤ/p, X)_{c} ≅ [Bℤ/p, Ω^{n}X], consider a map Bℤ/p → Ω^{n}X. We claim that it is homotopic to an H-map. Indeed, by [40, Proposition 1.5.1], the obstruction lives in the set [Bℤ/p ∧ Bℤ/p, Ω^{n}X], which is trivial since map_{∗}(Bℤ/p, X) is Bℤ/plocal. 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-spaces whose module of indecomposable elements lives in some stage of the Krull filtration. This is achieved by comparing this algebraic filtration with the topological one and by making use of Bousfield’s result 3.2.

Proposition 5.1. Let X be an H-space such that T_{V} H^{∗}(X) is of finite type for any elementary abelian p-group V. Assume that Ω^{n}X is Bℤ/p-local. Then QH^{∗}(X) ∈ U_{n−1}.

Proof. We proceed by induction. For n = 1, assume that ΩX is Bℤ/p-local, that is, Ωmap_{∗}(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 homotopy type. Hence, QH^{∗}(map_{∗}(Bℤ/p, X)) = 0 and, by Lemma 2.3, QH^{∗}(X) ∈ U_{0}.

If n > 1, let X be an H-space such that Ω^{n}X is Bℤ/p-local. We see by Proposition 3.5 that Ω^{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 Ω^{n−1}map_{∗}(Bℤ/p, X)_{c} is Bℤ/p-local. Moreover, by Proposition 1.1, T_{V} 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}) ∈ U_{n−2}. Since all components have the same homotopy type, QH^{∗}(map_{∗}(Bℤ/p, X)) ∈ U_{n−2}, and we conclude that QH^{∗}(X) ∈ U_{n−1} by Proposition 2.4. ⊓ ⊔

Proposition 5.2. Let X be a connected H-space such that T_{V} H^{∗}(X) is of finite type for any elementary abelian p-group V. Suppose that QH^{∗}(X) ∈ U_{n}. Then Ω^{n+1}X is Bℤ/p-local.

Proof. As in the proof of Proposition 1.1 we can assume without loss of generality that X is p-complete.

Let us proceed by induction. The case n = 0 is given by Proposition 1.2. Now assume that the result is true for n − 1, and consider a space X such that QH^{∗}(X) ∈ U_{n}. Then, by Proposition 2.4, QH^{∗}(map_{∗}(Bℤ/p, X)) ∈ U_{n−1} and the induction hypothesis ensures that Ω^{n}map_{∗}(Bℤ/p, X)_{c} ≃ map_{∗}(Bℤ/p, Ω^{n}X) is Bℤ/p-local. Apply now Theorem 4.3 to deduce that the space map_{∗}(Bℤ/p, Ω^{n}X)_{c} is an infinite loop space, with a p-torsion fundamental group by Proposition 4.4.

These are precisely the conditions of McGibbon’s main theorem in [28]: the Bℤ/p-nullification of connected infinite loop spaces with p-torsion fundamental group is trivial, up to p-completion. Moreover, our infinite loop space is Bℤ/p-local, so

(map_{∗}(Bℤ/p, Ω^{n}X)_{c})_{p}^{∧} ≃ (P_{Bℤ/p}(map_{∗}(Bℤ/p, Ω^{n}X)_{c}))_{p}^{∧} ≃ ∗

As we assume that X is p-complete, so are the loop space Ω^{n}X and the pointed mapping space map_{∗}(Bℤ/p, Ω^{n}X)_{c}. Thus, we see that map_{∗}(Bℤ/p, Ω^{n}X)_{c} must be contractible. Since all components of the pointed mapping space have the same homotopy type as the component of the constant map, we infer that map_{∗}(Bℤ/p, Ω^{n}X) is homotopically discrete. Looping once again, one obtains the equivalence map_{∗}(Bℤ/p, Ω^{n+1}X) ≃ ∗, i.e. Ω^{n+1}X is Bℤ/p-local, as we wanted to prove. ⊓ ⊔

Let us sum up these two results in one single statement, which extends widely Dwyer and Wilkerson’s [19, Proposition 3.2] when X is assumed to be an H-space.

Theorem 5.3. Let X be a connected H-space such that T_{V} H^{∗}(X) is of finite type for any elementary abelian p-group V. Then, QH^{∗}(X) is in U_{n} if and only if Ω^{n+1}X is Bℤ/p-local. ⊓ ⊔

Combining these results with Theorem 3.2 (about the nullification functor P_{Bℤ/p}) enables us to give a topological description of the Hspaces X for which the indecomposables QH^{∗}(X) live in some stage of the Krull filtration. Recall that the Prüfer group ℤ_{p^{∞}} is defined as the union of all ℤ/p^{n}, n ≥ 1. It is a p-torsion divisible abelian group.

Theorem 5.4. Let X be a connected H-space such that T_{V} H^{∗}(X) is of finite type for any elementary abelian p-group V. Then QH^{∗}(X) is in U_{n} if and only if X fits into a principal H-fibration

K(P, n + 1)  X  Y,

where Y is a connected H-space such that QH^{∗}(Y) ∈ U_{n−1}, and P is a p-torsion abelian group which is a finite direct sum of copies of cyclic groups ℤ/p^{r} and Prüfer groups ℤ_{p^{∞}}.

Proof. Assume that QH^{∗}(X) is in U_{n}. Let F be the homotopy fiber of the nullification map X → P_{Σ^{n}Bℤ/p}(X). By Theorem 3.1, F ≃ K(P, n+1) where P is an abelian p-group. Moreover, the equivalence map_{∗}(Σ^{n}Bℤ/p, K(P, n+1)) ≃ map_{∗}(Σ^{n}Bℤ/p, X) shows that the set

π_{n}map_{∗}(Bℤ/p, X) ≅ π_{0}map_{∗}(Σ^{n}Bℤ/p, X) ≅ Hom(ℤ/p, P)

is finite since all homotopy groups of map_{∗}(Bℤ/p, X) are p-torsion and its cohomology is of finite type. Thus, P is isomorphic to a finite direct sum of copies of cyclic groups ℤ/p^{r} and Prüfer groups ℤ_{p^{∞}} by Lemma 5.8, which we prove at the end of the section.

We conclude by taking Y = P_{Σ^{n}Bℤ/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 Ω^{n}Y ≃ P_{Bℤ/p}(Ω^{n}X) is Bℤ/p-local, Theorem 5.3 implies that QH^{∗}(Y) ∈ U_{n−1}. ⊓ ⊔

Equivalently, one can reformulate this result by describing the fiber of the Bℤ/p-nullification map.

Theorem 5.5. Let X be an H-space such that T_{V} H^{∗}(X) is of finite type for any elementary abelian p-group V. Then, QH^{∗}(X) is in U_{n} if and only if X is the total space of an H-fibration

F  X  P_{Bℤ/p}X,

where F is a p-torsion H-Postnikov piece whose homotopy groups are finite direct sums of copies of cyclic groups ℤ/p^{r} and Prüfer groups ℤ_{p^{∞}} concentrated in degrees 1 to n + 1. ⊓ ⊔

In other words, the only H-spaces such that QH^{∗}(X) lies in U_{n} for some n are the Bℤ/p-local H-spaces, the p-torsion Eilenberg-MacLane spaces introduced in Example 2.2, and extensions of the previous type.

Recall that the Bℤ/p-nullification of a loop space is again a loop space. Moreover, by [20, Lemma 3.A.3], the nullification map is a loop map, and hence its homotopy fiber is also a loop space. Thus we obtain automatically the following result about loop spaces.

Corollary 5.6. Let X be a loop space such that T_{V} H^{∗}(X) is of finite type for any elementary abelian p-group V. Then QH^{∗}(X) is in U_{n} if and only if X is the total space of loop fibration

F  X  P_{Bℤ/p}X,

where the loop space F is a p-torsion Postnikov piece whose homotopy groups are finite direct sums of copies of cyclic groups ℤ/p^{r} and Prüfer groups ℤ_{p^{∞}} concentrated in degrees from 1 to n + 1. ⊓ ⊔

If we restrict our attention to the case n = 0 in Theorem 5.5, our result reproves in a more conceptual way the theorems about Hspaces with locally finite module of indecomposable elements given by Broto, Saumell and the second named author in [10,15,11].

What do we learn from our study about H-spaces which do not belong to any stage 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. Let us discuss the example of BU.

Example 5.7. The module QH^{∗}(BU) is isomorphic to Σ^{2}H^{∗}(BS^{1}). In particular, it is not a finitely generated A_{p}-module. A computation of the value of the T functor on this module can be done using [33, Section 9.8] and shows that QH^{∗}(BU) does not belong to any U_{n}.

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 into elementary pieces. In fact BU is K(ℤ/p, 2)-local by a result of Mislin (see [31, Theorem 2.2]).

Finally, we prove the lemma about abelian p-torsion groups which was used in the proof of Theorem 5.4.

Lemma 5.8. Let P be an abelian p-torsion group. If Hom(ℤ/p, P) is finite then P is a finite direct sum of copies of cyclic groups ℤ/p^{r} and Prüfer groups ℤ_{p^{∞}}.

Proof. By Kulikov’s theorem (see [32, 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 [32, Corollary 10.41]) shows now that this subgroup is a direct summand. Since the quotient is divisible and Hom(ℤ/p, P) is finite, P is a finite direct sum of copies of cyclic groups ℤ/p^{r} and Prüfer groups ℤ_{p^{∞}}. ⊓ ⊔

6. Fibrations over Eilenberg-MacLane spaces

In the next section we concentrate on H-spaces whose mod p cohomology is finitely generated as an algebra over the Steenrod algebra. Therefore we will need to establish a closure property under certain H-fibrations.

Theorem 6.1. Let A be a finite direct sum of copies of cyclic groups ℤ/p^{r} and Prüfer groups ℤ_{p^{∞}}, and n ≥ 2. Consider an H-fibration F ⟶^{i} E ⟶^{π} K(A, n). If H^{∗}(F) is a finitely generated A_{p}-algebra, then so is H^{∗}(E).

The proof relies mainly on the Eilenberg-Moore spectral sequence for an H-fibration over an Eilenberg-Mac Lane space, a situation studied by Smith in [39, Chapter II]. Following the terminology used in [39, Section 6], we say that a sequence of (Hopf) algebras

𝔽_{p}  A  B  C  𝔽_{p}

is coexact if the morphism A → B is a monomorphism and its cokernel B//A is isomorphic to C as a (Hopf) algebra.

Proposition 6.2. Let n ≥ 2 and consider a non-trivial H-fibration F ⟶^{i} E ⟶^{π} K(A, n) where A is either ℤ/p or a Prüfer group ℤ_{p^{∞}}. Then there is a coexact sequence of algebras

𝔽_{p}  H^{∗}(E)//π^{∗} ^{i^{∗}} H^{∗}(F)  Λ ⊗ S  𝔽_{p} ,

where Λ is an exterior algebra and S ⊆ H^{∗}(K(A, n − 1)) is a Hopf subalgebra.

Proof. The E_{2}-term of the Eilenberg-Moore spectral sequence is given by E_{2} = Tor_{H^{∗}(K(A,n))}(H^{∗}(E), 𝔽_{p}) and converges to H^{∗}(F). Since we deal with an H-fibration, an argument based on the change of rings spectral sequence, [12, XVI.6.1], allows Smith to identify E_{2} with H^{∗}(E)//π^{∗} ⊗ Tor_{H^{∗}(K(A,n))\\π^{∗}}(𝔽_{p}, 𝔽_{p}) as algebras, see [39, Theorem 2.4]. Here H^{∗}(K(A, n))\\π^{∗} is the (Hopf algebra) kernel of π^{∗}, an unstable Hopf subalgebra of the abelian Hopf algebra H^{∗}(K(A, n)), as explained in [39, Remark 3.2].

By [39, Proposition 7.3^{∗}], i^{∗} : H^{∗}(E)//π^{∗} ↪ H^{∗}(F) is a monomorphism and the cokernel R = H^{∗}(F)//i^{∗} is described by a coexact sequence of Hopf algebras

𝔽_{p}  Λ  R  S  𝔽_{p},

with Λ an exterior algebra and S ⊆ H^{∗}(K(A, n − 1)). Under our assumptions for A, H^{∗}(K(A, n − 1)) is a free commutative algebra. Therefore the previous coexact sequence splits, i.e. R is isomorphic to Λ ⊗ S as an algebra. ⊓ ⊔

To prove Theorem 6.1, we will show that the module of indecomposable elements QH^{∗}(E) is a finitely generated unstable module. When p is an odd prime, the coexact sequence in the proposition splits, which identifies Q(H^{∗}(E)//π^{∗}) as a submodule of QH^{∗}(F).

When p = 2, the coexact sequence does not split. The functor Q is not left exact and one is then naturally led to studying the left derived functors of Q, i.e. André-Quillen homology. Good references are [33, Chapter 7], [21], and [29]. We write H_{i}^{Q}(A) for the i-th derived functor. This is an unstable module when A is an unstable algebra.

Lemma 6.3. Let Λ be an exterior algebra over 𝔽_{2}, which is finitely generated as an algebra over the Steenrod algebra. Then H_{1}^{Q}(Λ) is a finitely generated unstable module.

Proof. In [21, Section 10], Goerss identifies the first André-Quillen homology group H_{1}^{Q}(Λ) with the indecomposable elements of degree 2 in Tor_{Λ}(𝔽_{2}, 𝔽_{2}). As an 𝔽_{2}-vector space it is generated by the elements [x|x] where x runs through all exterior generators of Λ. Since the Steenrod algebra acts via the Cartan formula, it follows that H_{1}^{Q}(Λ) is also a finitely generated A_{2}-module. Alternatively one could perform this computation using the simplicial resolution given by the symmetric algebra comonad described in [33]. ⊓ ⊔

Proof of Theorem 6.1. Given a group extension A′ → A → A, there is a pullback diagram of fibrations:

[figure: E′; E; K(A′′, n); K(A′, n); K(A, n); π; K(A′′, n); ‖]

If the statement is true for the first vertical fibration and the top horizontal fibration, then it follows for π. Therefore, we can assume that A = ℤ/p or ℤ_{p^{∞}}. If π is null-homotopic, the statement is obvious, and we can hence work under the hypothesis of Proposition 6.2.

Since H^{∗}(K(A, n)) is finitely generated as algebra over A_{p}, so is its image Im(π^{∗}) ⊆ H^{∗}(E). Hence, to prove the theorem, it is enough to show that H^{∗}(E)//π^{∗} is a finitely generated A_{p}-algebra, or equivalently that the module of indecomposable elements Q(H^{∗}(E)//π^{∗}) is a finitely generated A_{p}-module.

When p is odd, the coexact sequence in Proposition 6.2 splits (as algebras) because an exterior algebra is free commutative. Hence H^{∗}(F) ≅ H^{∗}(E)//π^{∗} ⊗ Λ ⊗ S (compare with [38, Theorem 5.7]). In particular, Q(H^{∗}(E)//π^{∗}) ⊆ QH^{∗}(F). Since U is a locally noetherian category, [33, Theorem 1.8.1], and QH^{∗}(F) is a finitely generated A_{p}- module, so is Q(H^{∗}(E)//π^{∗}).

The case p = 2 is less straightforward since an exterior algebra is not free commutative, so that the coexact sequence in Proposition 6.2 does not split in general. The inclusion A ⊂ B of a sub-Hopf algebra is not necessarily a cofibration (seen as a constant simplicial object). However, when B is of finite type, it is always a free A-module by the Milnor-Moore result [30, Theorem 4.4]). Therefore the argument in [21, Section 10] applies and the homotopy cofiber of the inclusion is weakly equivalent to B//A. Since cofibrations of simplicial algebras induce long exact sequences in André-Quillen cohomology, we have in our situation an exact sequence

H_{1}^{Q}(Λ ⊗ S) → Q(H^{∗}(E)//π^{∗}) → QH^{∗}(F).

As S is a free commutative algebra, H_{1}^{Q}(Λ ⊗ S) ≅ H_{1}^{Q}(Λ) by [21, Lemma 4.10]. The previous lemma tells us that this is a finitely generated unstable module. So is QH^{∗}(F), and we conclude since U is locally noetherian. ⊓ ⊔

Remark 6.4. Theorem 6.1 is actually true for fibrations over an Eilenberg-Mac Lane space K(A, 1) as well. The Eilenberg-Moore spectral sequence converges in this case by work of Dwyer, [16], and the above proof only needs minor modifications.

7. H-spaces with finitely generated cohomology over A_{p}

We will assume in this section that H^{∗}(X) is finitely generated as an algebra over 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 7.2.

We show first that, under this finiteness condition, the H-spaces considered in this section satisfy the hypothesis of Theorem 5.3 (they belong to some stage of the filtration we study in this paper).

Lemma 7.1. Let K be a finitely generated unstable A_{p}-algebra. Then there exists some integer n such that the module of indecomposables QK belongs to U_{n}. Moreover T_{V} K is a finitely generated unstable A_{p}-algebra for any elementary abelian group V.

Proof. First of all, QK is a finitely generated module over A_{p}, i.e. it is a quotient of a finite direct sum of free modules. Hence, there exists an epimorphism ⊕_{i=1}^{k}F (n_{i}) ↠ QK. Since \overline{T} is an exact functor, it follows that \overline{T}^{m}(QK) = 0, where m is the largest of the n_{i}’s, and so QK ∈ U_{m−1}.

Moreover, T_{V} commutes with taking indecomposables elements [35, Lemma 6.4.2]. Therefore, Q(T_{V} K) is a finitely generated unstable module. Then, the above discussion shows that T_{V} K is a finitely generated A_{p}-algebra. ⊓ ⊔

We can now state our main finiteness result. It enables us to understand better the Bℤ/p-nullification, which is the first building block in our deconstruction process (Theorem 5.5).

Theorem 7.2. Let X be a connected H-space such that H^{∗}(X) is finitely generated as algebra over the Steenrod algebra. Then, P_{Bℤ/p}X is an H-space with finite mod p cohomology.

Proof. By Lemma 7.1, there exists an integer n such that QH^{∗}(X) lies in U_{n−1}, so Theorem 5.3 applies and we know that Ω^{n}X is Bℤ/plocal.

We will show that if H^{∗}(X) is finitely generated as an algebra over A_{p} and Ω^{n}X is Bℤ/p-local, then H^{∗}(P_{Bℤ/p}X) is finitely generated as an algebra over A_{p}. We proceed by induction on n. When n = 0 the statement is clear. Assume the statement holds for n − 1.

We know from Theorem 5.4 that there is a principal H-fibration

K(P, n)  X  P_{Σ^{n−1}Bℤ/p}X,

where P is a p-torsion abelian group which is a finite direct sum of copies of cyclic groups ℤ/p^{r} and Prüfer groups ℤ_{p^{∞}}. It follows from Theorem 6.1 that H^{∗}(P_{Σ^{n−1}Bℤ/p}X) is finitely generated as an algebra over A_{p}. Moreover, Ω^{n−1}P_{Σ^{n−1}Bℤ/p}X is weakly equivalent to P_{Bℤ/p}Ω^{n−1}X, which is Bℤ/p-local, so the induction hypothesis applies. The cohomology of P_{Bℤ/p}X ≃ P_{Bℤ/p}P_{Σ^{n−1}Bℤ/p}X is finitely generated as an algebra over the Steenrod algebra.

Finally, since H^{∗}(P_{Bℤ/p}X) is locally finite, this implies that the space P_{Bℤ/p}X has finite mod p cohomology. ⊓ ⊔

Combining this last result with Theorem 5.5 we obtain:

Theorem 7.3. Let X be a connected H-space such that H^{∗}(X) is a finitely generated 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 ptorsion H-Postnikov piece whose homotopy groups are finite direct sums of copies of cyclic groups ℤ/p^{r} and Prüfer groups ℤ_{p^{∞}}. ⊓ ⊔

The analogous result for loop spaces follows from Corollary 5.6. We propose finally an extension of Hubbuck’s Torus Theorem on

homotopy commutative H-spaces. At the prime 2, we have:

Corollary 7.4. Let X be a connected homotopy commutative H-space such that the mod 2 cohomology H^{∗}(X) is finitely generated as algebra over the Steenrod algebra A_{2}. Then, up to 2-completion, X is homotopy equivalent to (S^{1})^{n} × F, where F is a connected 2-torsion H-Postnikov piece.

Proof. Consider the fibration F → X → P_{Bℤ/2}X. We know from the preceding theorem that the fiber is a 2-torsion H-Postnikov piece and the basis is an H-space with finite mod 2 cohomology. Both are homotopy commutative. In particular, the mod 2 Torus Theorem of Hubbuck, [23], implies that P_{Bℤ/2}X is, up to 2-completion, a finite product of circles (S^{1})^{n}. Since the fiber is 2-torsion, the above fibration splits (not necessarily by an H-map) and the result follows. ⊓ ⊔

When X is a mod 2 finite H-space, this corollary is the original Torus Theorem due to Hubbuck. When X is an H-space with noetherian cohomology, QH^{∗}(X) ∈ U_{0}, the Postnikov piece F is an Eilenberg-Mac Lane space K(P, 1) where P is a 2-torsion abelian group, and we get back Slack’s results [37], as well as their generalization by Lin and Williams in [27]: up to 2-completion, X is the product of a finite number of S^{1}’s, K(ℤ/2^{r}, 1)’s, and K(ℤ, 2)’s. Of course, in our setting it is no longer true that the fiber F in Theorem 7.3 is a product of Eilenberg-Mac Lane spaces.

At odd primes, there are many more finite H-spaces which are homotopy commutative (all odd dimensional spheres for example). However, Hubbuck’s result still holds for finite loop spaces of Hspaces, as was shown in [2] by Aguadé and Smith. Therefore, replacing the original Torus Theorem by the Aguadé-Smith version, the same conclusion as in Corollary 7.4 holds at odd primes for the loop space on an H-space.

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