NOETHERIAN LOOP SPACES

NATÀLIA CASTELLANA, JUAN A. CRESPO, AND JÉRÔME SCHERER^{∗}

Abstract. The class of loop spaces whose mod p cohomology is Noetherian is much larger than the class of p-compact groups (for which the mod p cohomology is required to be finite). It contains Eilenberg-Mac Lane spaces such as ℂP^{∞} and 3-connected covers of compact Lie groups. We study the cohomology of the classifying space BX of such an object and prove it is as small as expected, that is, comparable to that of BℂP^{∞}. We also show that BX differs basically from the classifying space of a p-compact group in a single homotopy group. This applies in particular to 4-connected covers of classifying spaces of compact Lie groups and sheds new light on how the cohomology of such an object looks like.

Introduction

From the point of view of homotopy theory compact Lie groups are finite loop spaces, i.e. triples (X, BX, e) where X is a finite complex, and e : X → ΩBX is a homotopy equivalence. Most of their geometric features are captured p-locally in homotopy theory, where p is any prime, as shown by Dwyer and Wilkerson in [17]. They introduced the notion of p-compact group, replacing the finiteness condition by a cohomological one, namely that H^{∗}(X; 𝔽_{p}) must be finite, and requiring the additional property that BX be local with respect to mod p homology, or equivalently pcomplete, [5]. It is the “classifying space” BX which carries all of the information about the loop space. Amazingly enough, apart from compact Lie groups, there are only a few families of exotic p-compact groups and they have been recently completely classified by Andersen, Grodal, Møller, and Viruel, see [3] for the odd prime case and [2], [28], [29] for the prime 2 (the only exotic 2-compact group is basically the space DI(4) constructed by Dwyer and Wilkerson, [16]).

If one aims at an understanding not only of finite loop spaces, but larger ones, the next natural step to take is to relax the cohomological finiteness condition. We define thus a p-Noetherian group to be a loop space (X, BX, e) where BX is p-complete and H^{∗}(X; 𝔽_{p}) is a finitely generated (Noetherian) 𝔽_{p}-algebra. Relying on Bousfield localization techniques, [4], and Miller’s solution to the Sullivan conjecture, [27], more precisely on Lannes’ T -functor technology, [25], we describe the structure of p-Noetherian groups and their relation to p-compact groups. We compute qualitatively

2000 Mathematics Subject Classification. Primary 55P45; Secondary 13D03, 55S10, 55T20.

*: corresponding author. The first and third authors are partially supported by FEDER/MEC grant MTM2007-61545. The second author is partially supported by FEDER/MEC grant SEJ2007-67135. The third author has benefitted from the LMS scheme 2 grant number 2618 and the hospitality from the MPI, Bonn.

2 NATÀLIA CASTELLANA, JUAN A. CRESPO, AND JÉRÔME SCHERER^{∗}

the cohomology of the classifying space BX and obtain new general results when X is the 3connected cover of a compact Lie group, see Corollary 4.9.

Let us be more precise. In the case of p-compact groups, i.e. when the mod p cohomology of the loop space H^{∗}(X; 𝔽_{p}) is finite, Dwyer and Wilkerson’s main theorem in [17] shows that there are severe restrictions on the cohomology of the classifying space: H^{∗}(BX; 𝔽_{p}) is always a finitely generated (Noetherian) 𝔽_{p}-algebra. Likewise the cohomology of the classifying space of a p-Noetherian group cannot be arbitrarily large.

Theorem 4.1 Let (X, BX, e) be a p-Noetherian group. Then H^{∗}(BX; 𝔽_{p}) is finitely generated as an algebra over the Steenrod algebra.

This information is not optimal as it does not permit to decide for example when an Eilenberg-Mac Lane space of type K(ℤ/p, m) is a p-Noetherian group. The mod p cohomology of any of them is finitely generated as an algebra over A_{p}, but the only classifying spaces of a p-Noetherian group are K(ℤ/p, 1) and K(ℤ/p, 2). Schwartz’s Krull filtration of the category U of unstable modules is an established and convenient tool to measure how large an unstable algebra is, [32]. The Krull filtration U_{0} ⊂ U_{1} ⊂ … is defined inductively, starting with the full subcategory U_{0} of U of locally finite unstable modules (the span of every element under the action of the Steenrod algebra is finite). In fact the cohomology H^{∗}(X; 𝔽_{p}) is locally finite if and only if the space X is Bℤ/p-local, i.e. the evaluation map(Bℤ/p, X) → X is a weak equivalence ([26, Théorème 0.14]).

There are many Bℤ/p-local spaces, but there are none for which the cohomology lies in higher stages of the Krull filtration. This is the statement of Kuhn’s non-realizability conjecture [24], which has been settled by Schwartz in [33] and [34], and proved in its full generality by Dehon and Gaudens in [13]. Thus the cohomology of a space lies in U_{0} or it does not lie in any U_{n}. The cohomology H^{∗}(K(ℤ/p, m); 𝔽_{p}) for example does not lie in any U_{n}.

The situation changes drastically if one looks at the quotient module of indecomposable elements instead. For example QH^{∗}(K(ℤ/p, m); 𝔽_{p}) lies in U_{m−1} for any m ≥ 1 ([9, Example 2.2]). Moreover we observed in [9, Lemma 7.1] that if H^{∗}(BX; 𝔽_{p}) is finitely generated as an algebra over A_{p}, then QH^{∗}(BX; 𝔽_{p}) must be finitely generated as a module over A_{p}, and hence lies in U_{k} for some k. We remark that the condition that H^{∗}(X; 𝔽_{p}) be a Noetherian 𝔽_{p}-algebra is equivalent to saying that H^{∗}(X; 𝔽_{p}) is finitely generated as an algebra over A_{p} and that the unstable module QH^{∗}(X; 𝔽_{p}) of indecomposable elements lies in U_{0}. Our second result shows that the cohomology of the classifying space of a p-Noetherian group is as small as expected in terms of the Krull filtration.

Theorem 2.6 Let (X, BX, e) be a p-Noetherian group. Then QH^{∗}(BX; 𝔽_{p}) belongs to U_{1}.

The arguments to prove these results are the following. We start in Section 1 with the study of the structure of p-Noetherian groups. The most basic examples of p-Noetherian groups X are p-compact groups and Eilenberg-Mac Lane spaces K(ℤ/p^{r}, 1) and, K(ℤ_{p}^{∧}, 2). In the spirit of our deconstruction results for H-spaces, [9], we show that these are the basic building blocks for all p-Noetherian groups.

Theorem 1.9 Let (X, BX, e) be any p-Noetherian group. There exists then a fibration

K(P, 2)_{p}^{∧} ⟶ BX ⟶ BY ,

where P is a p-group which is a finite direct sum of copies of cyclic groups and Prüfer groups and Y is a p-compact group.

We stress that this fibration is functorial in BX as the base space is obtained by Bousfield localization. The understanding of the cohomology of BX goes through the analysis of the Serre spectral sequence of this fibration. Note that, by Dwyer and Wilkerson’s main theorem in [17], H^{∗}(BY; 𝔽_{p}) is finitely generated as an algebra. Also, the mod p cohomology of K(P, 2)_{p}^{∧} is finitely generated as an algebra over the Steenrod algebra by [35] and [7]. The spectral sequence is not nearly as nice as what happens for H-spaces though, compare with [10], and we must first tackle Theorem 2.6.

This we do in Sections ?? – 2 by giving first a geometric interpretation to T̄QH^{∗}(BX; 𝔽_{p}), where T̄ is Lannes’ reduced T -functor. Recall that for “nice” spaces (such as BX) the unreduced T functor T H^{∗}(BX; 𝔽_{p}) computes the cohomology of the mapping space map(Bℤ/p, BX) and we rely on Schwartz’s characterization [32, Theorem 6.2.4] of the Krull filtration in terms of T̄. In order to perform our reduced T functor calculation, we prove that the component map(Bℤ/p, BX)_{c} of the constant map splits as a product BX × map_{∗}(Bℤ/p, BX)_{c}. Using the properties of the T functor, this splitting yields a geometric interpretation of the reduced T functor in terms of the pointed mapping space, more precisely T̄QH^{∗}(X; 𝔽_{p}) ≅ H^{∗}(map_{∗}(Bℤ/p, BX)_{c}; 𝔽_{p}).

We finally come back to Theorem 4.1, which we prove in two steps. First we investigate in Section 3 the Serre spectral sequence for fibrations over spaces with finite cohomology and fibre a finite product of Eilenberg Mac–Lane spaces. We show that in this situation the cohomology of the total space is finitely generated as an algebra over A_{p}. Secondly, we reduce the situation in which the base space of the fibration is a p-compact toral group.

Let us conclude the introduction with a remark. The results in this article show that for a compact simply-connected Lie group G, the module QH^{∗}((BG)⟨4⟩; 𝔽_{p}) is finitely generated over A_{p} and belongs to U_{1}. This puts into context the calculations made by Harada and Kono, [21], and sheds new light on how the cohomology will look like even in the cases where an explicit description has not been obtained, compare with Example 4.10.

Acknowledgments. This project originated at the Mittag-Leffler Institute during the emphasis semester on homotopy theory in 2006. We would like to thank Kasper Andersen, Jesper Grodal,

4 NATÀLIA CASTELLANA, JUAN A. CRESPO, AND JÉRÔME SCHERER^{∗}

Frank Neumann, and Alain Jeanneret for their interest and Akira Kono for pointing out a problem in an earlier version of this article.

1. The structure of p-Noetherian groups

This first section is devoted to the description of the classifying space of p-Noetherian groups and the relation to p-compact groups. Let us start with the definition and the basic examples. The statements about the action of the Steenrod algebra and the Krull filtration will be explained and developed in Section 2.

Definition 1.1. A p-Noetherian group is a triple (X, BX, e) where H^{∗}(X; 𝔽_{p}) is a Noetherian algebra (it is finitely generated as an algebra), BX is a p-complete space, and e : X → ΩBX is a weak equivalence.

We will often use the word p-Noetherian group for the loop space X and refer to BX as the classifying space. Hence, we will say that a p-Noetherian group is n-connected if so is the loop space X, or equivalently if the classifying space is (n + 1)-connected. Note that if the integral cohomology of a space is Noetherian, as an algebra, then so is the mod p cohomology (just like finite loop spaces have finite mod p cohomology), [11, Lemma 1.5].

Remark 1.2. Since π_{1}(BX) ≅ π_{0}(X) and H^{∗}(X; 𝔽_{p}) is of finite type, it follows that π_{1}(BX) is finite and, therefore BX is a p-good space by [5, VII, Proposition 5.1]. In fact, π_{1}(BX) is a p-group by [5, VII, Proposition 4.3].

Example 1.3. In [17], Dwyer and Wilkerson introduced the notion of a p-compact group. A p-compact group is a loop space (X, BX, e) such that BX is p-complete and H^{∗}(X; 𝔽_{p}) is a finite 𝔽_{p}-vector space. It is clear from the definition that p-compact groups are p-Noetherian groups.

The most basic example of p-compact group is given by the p-completed circle and its classifying space K(ℤ_{p}^{∧}, 2). Our definition of p-Noetherian group allows to include not only all p-compact groups but also the following Eilenberg-Mac Lane spaces.

Example 1.4. Let X = K(ℤ_{p}^{∧}, 2), BX = K(ℤ_{p}^{∧}, 3), and e the obvious homotopy equivalence between ΩK(ℤ_{p}^{∧}, 3) and K(ℤ_{p}^{∧}, 2). This is a p-Noetherian group since H^{∗}(K(ℤ_{p}^{∧}, 2); 𝔽_{p}) ≅ 𝔽_{p}[u] is finitely generated as an algebra. Let us point out here that H^{∗}(K(ℤ_{p}^{∧}, 3); 𝔽_{p}) is finitely generated as an algebra over A_{p} and that the module of indecomposable elements QH^{∗}(K(ℤ_{p}^{∧}, 3); 𝔽_{p}) lives in U_{1}. For example QH^{∗}(K(ℤ_{2}^{∧}, 3); 𝔽_{2}) ≅ ΣF (1), where F (1) is the free unstable module on one generator in degree 1.

In fact, this is basically the only 1-connected p-Noetherian group X such that ΩX is 𝔽_{p}-finite.

Proposition 1.5. Let (X, BX, e) be a p-Noetherian group such that H^{∗}(ΩX; 𝔽_{p}) is finite. Then BX is 2-connected if and only if it is a product of a finite number of copies of K(ℤ_{p}^{∧}, 3).

NOETHERIAN LOOP SPACES 5

Proof. The loop space ΩX ≃ Ω^{2}BX is a connected homotopy commutative mod p finite H-space. Thus, by the mod p version of Hubbuck’s Torus Theorem, [22] and [1], we see that ΩX is equivalent to a p-completed torus.

Example 1.6. Let us consider the compact Lie group S^{3} and its 3-connected cover S^{3}⟨3⟩. Identifying B(S^{3}⟨3⟩) with (BS^{3})⟨4⟩, we have a fibration K(ℤ, 3) → (BS^{3})⟨4⟩ → BS^{3}. The triple (S^{3}⟨3⟩_{p}^{∧}, B(S^{3})⟨4⟩_{p}^{∧}, e) is then a p-Noetherian group since H^{∗}(S^{3}⟨3⟩; 𝔽_{p}) ≅ 𝔽_{p}[x] ⊗ E(y) where x has degree 2p, y has degree 2p + 1, and a Bockstein connects x and y, β(x) = y. This p-Noetherian group is an extension of the p-compact group (S^{3})_{p}^{∧} and the Eilenberg-Mac Lane space from Example 1.4.

It is not difficult to compute the mod p cohomology of (BS^{3})⟨4⟩ using the Serre spectral sequence. For example, H^{∗}(B(S^{3}⟨3⟩); 𝔽_{2}) ≅ 𝔽_{2}[z, Sq^{1}z, Sq^{4}z, Sq^{8,4}z, …], where z has degree 5. This is a subalgebra of the mod 2 cohomology of K(ℤ, 3) and z corresponds to Sq^{2}ι_{3}, where ι_{3} is the fundamental class. Again, we see that the module of indecomposable elements belongs to U_{1}, as it differs from ΣF (1) by only a few classes in low degrees.

This last example fits into a more general picture. One can consider 4-connected covers of classifying spaces of compact Lie groups.

Example 1.7. Let G be a simply connected compact Lie group and consider the 4-connected cover of its classifying space, (BG)⟨4⟩. Since the mod p cohomology G⟨3⟩ is Noetherian, this provides an infinite number of examples of p-Noetherian groups.

Harada and Kono studied in [21] and [20] the fibration K(ℤ, 3) → (BG)⟨4⟩ → BG. They were able to compute explicitly, as an algebra, the cohomology of the total space at odd primes and a few cases at the prime 2. In all of these computations the result is the tensor product of a quotient of H^{∗}(BG; 𝔽_{p}) with a certain subalgebra of H^{∗}(K(ℤ, 3); 𝔽_{p}), which turns out to be always finitely generated as an algebra over the Steenrod algebra.

In fact, p-Noetherian groups are closed under fibrations in the following sense (and this explains why G⟨3⟩ defines a p-Noetherian group).

Proposition 1.8. Let BX → E → BZ be a fibration of connected spaces where BX and BZ are classifying spaces of p-Noetherian groups. Then E is also the classifying space of a p-Noetherian group.

Proof. Since π_{1}(BZ) is a finite p-group and both BX and BZ are p-complete and p-good (see Remark 1.2), the fibre lemma [5, II.5.1] shows that E is also p-complete. It remains to show that the mod p cohomology of ΩE is a finitely generated algebra. Since π_{1}E is a finite (p-) group it is good enough to prove it for the connected component Ω_{0}E of the base point (the constant loop).

Looping the fibration, we obtain an H-fibration X → ΩE → Z where both X and Z have finitely generated mod p cohomology. By [10, Theorem 4.1], the cohomology of ΩE is finitely generated

6 NATÀLIA CASTELLANA, JUAN A. CRESPO, AND JÉRÔME SCHERER^{∗}

as an algebra over the Steenrod algebra, in other words QH^{∗}(ΩE; 𝔽_{p}) is finitely generated as an A_{p}-module. This unstable module, which is isomorphic to the module of indecomposable elements of Ω_{0}E, is thus finite if and only if it is locally finite and this last condition is equivalent to the pointed mapping space map_{∗}(Bℤ/p, ΩΩ_{0}E) being contractible. For simply connected spaces this was shown in [15, Theorem 3.2] and it is easy to remove the connectivity assumption; for H-spaces this was done in [9, Proposition 1.2] and for arbitrary spaces in [8, Lemma 1.1]. These results tell us that both map_{∗}(Bℤ/p, ΩX) and map_{∗}(Bℤ/p, ΩZ) are contractible and the result follows.

Let us analyze the structure of an arbitrary connected p-Noetherian group. The following theorem tells us that it always differs from a p-compact group in a single p-completed Eilenberg-MacLane space.

Theorem 1.9. Let (X, BX, e) be any p-Noetherian group. There exists then a fibration

K(P, 2)_{p}^{∧} → BX → BY ,

where P is a finite direct sum of copies of cyclic groups and Prüfer groups and Y is a p-compact group.

Proof. By assumption the mod p cohomology of X is finitely generated as an algebra. In other words, the module of indecomposable elements QH^{∗}(X; 𝔽_{p}) is finite. Therefore, by [9, Proposition 1.2], the loop space ΩX is Bℤ/p-local, or equivalently the classifying space BX is Σ^{2}Bℤ/plocal ([19, Theorem 3.A.1]). The analysis in [4] by Bousfield of the Postnikov like nullification tower shows then that the homotopy fiber of the nullification map BX → P_{ΣBℤ/p}BX is a single Eilenberg-Mac Lane space K(P, 2), where P is an abelian p-torsion group. Moreover, he also shows that the corresponding fibration is principal. In particular, it implies that P_{ΣBℤ/p}BX is a p-good space, and (P_{ΣBℤ/p}BX)_{p}^{∧} is p-complete by the fibre lemma [5, II.5.1].

From the equivalence P_{Bℤ/p}X ≃ ΩP_{ΣBℤ/p}BX, [19, Theorem 3.A.1], we obtain a loop fibration K(P, 1) → X → P_{Bℤ/p}X. Since X is a loop space with finitely generated mod p cohomology, we know from [9, Theorem 7.3], or directly from [12], that P is a finite direct sum of copies of cyclic groups and Prüfer groups and that H^{∗}(P_{Bℤ/p}X; 𝔽_{p}) is finite.

Let us consider the loop space P_{Bℤ/p}X. Notice that π_{1}P_{ΣBℤ/p}BX ≅ π_{1}BX, which must be a finite p-group by Remark 1.2. By p-completing we obtain hence the classifying space of a p-compact group BY = (P_{ΣBℤ/p}BX)_{p}^{∧}.

The fibration we have obtained allows us to give a precise description of the component of the constant in the pointed mapping space map_{∗}(Bℤ/p, BX).

Corollary 1.10. Let (X, BX, e) be a p-Noetherian group. Then map_{∗}(Bℤ/p, BX)_{c} is the classifying space of a finite elementary abelian p-group.

NOETHERIAN LOOP SPACES 7

Proof. Consider the fibration K(P, 2)_{p}^{∧} → BX → BY from Theorem 1.9. Since Y is a p-compact group, H^{∗}(ΩBY; 𝔽_{p}) is finite and map_{∗}(Bℤ/p, ΩBY) ≃ ∗ by [27]. Therefore the component map_{∗}(Bℤ/p, BY)_{c} is contractible and map_{∗}(Bℤ/p, BX)_{c} ≃ map_{∗}(Bℤ/p, K(P, 2)_{p}^{∧})_{c}. By [27, Theorem 1.5], map_{∗}(Bℤ/p, K(P, 2)_{p}^{∧}) ≃ map_{∗}(Bℤ/p, K(P, 2)), which has trivial homotopy groups in degrees ≥ 2. The component of the constant map is thus the classifying space of a finite elementary abelian p-group V = Hom(ℤ/p, P).

From Theorem 1.9 we deduce that many p-Noetherian groups are 3-connected covers of pcompact groups.

Corollary 1.11. Let (X, BX, e) be a p-Noetherian group. Then X is 3-connected if and only if BX is the 4-connected cover of the classifying space of a p-compact group.

Proof. One implication is obvious. Let us hence assume that X is 3-connected and consider the fibration K(P, 2)_{p}^{∧} → BX → BY from Theorem 1.9. We see that BY is 2-connected, hence 3connected [6, Theorem 6.10]. This shows that P must be a divisible abelian p-group, or equivalently that K(P, 2)_{p}^{∧} ≃ K(⊕ℤ_{p}^{∧}, 3).

To the Lie group Example 1.7 we can add now new examples of p-Noetherian groups, namely those given by 3-connected covers of exotic p-compact groups.

Example 1.12. Let X be a p-compact group such that BX is 3-connected and π_{4}(BX) ≅ ℤ_{p}^{∧}. By looking at the classification of p-compact groups, we observe that there are only two sporadic examples, namely numbers 23 and 30 in the Shephard-Todd list [36], and one infinite family, number 2b, corresponding to the dihedral groups D_{2m}. The triple (X⟨3⟩, (BX)⟨4⟩, e) is a p-Noetherian group by Corollary 1.11.

The two sporadic examples are defined at primes p ≡ 1, 4 mod 5, and they are non-modular since the only primes which divide the order of their Weyl group are 2, 3 and 5. The family of p-compact groups corresponding to the dihedral groups D_{2m} is defined for primes p ≡ ±1 mod m. Note that p = 2 occurs when m = 3 and corresponds to the exceptional Lie group G_{2}.

Remark 1.13. From Corollary 1.11 we obtain a classification of 3-connected p-Noetherian groups. They are given by the 3-connected covers of simply connected p-compact groups, which are known from the recent classification results, [28], [29], [2], and [3]. A general classification will be more difficult to obtain, even in the 2-connected case, as there are p-Noetherian groups fibering over a product of p-compact groups which do not split themselves as a product. Consider indeed the homotopy fiber of the composite map

f : BS^{3} × BS^{3} → K(ℤ, 4) × K(ℤ, 4) → K(ℤ, 4),

8 NATÀLIA CASTELLANA, JUAN A. CRESPO, AND JÉRÔME SCHERER^{∗}

where the first map is the fourth Postnikov section and the second is given by the sum. Let us complete this fiber at the prime 7 for example and call it BX. Even though X splits as a product (S^{3})_{7}^{∧} × (S^{3})_{7}^{∧}⟨3⟩, the classifying space BX does not split.

Assume that BX splits as a product (BS^{3})_{7}^{∧} × (BS^{3})_{7}^{∧}⟨4⟩. There exists then an essential map g : (BS^{3})_{7}^{∧} → (BS^{3})_{7}^{∧} × (BS^{3})_{7}^{∧} such that f ∘ g ≃ ∗ and p_{1} ∘ g is an equivalence, where p_{1} denotes the projection on the first factor. But g induces on the fourth homology group a map of degree n ≠ 0 on the first copy of (BS^{3})_{7}^{∧} and of degree m on the second. The composite f ∘ g will thus have degree n + m on H_{4}. We claim that this cannot be zero. Both m and n must be squares in ℤ_{7}^{∧} as a self-map of (BS^{3})_{7}^{∧} is induced by a self-map on the maximal torus (BS^{1})_{7}^{∧}. But the sum of two 7-adic squares is nul if and only both are so. Therefore BX cannot split as a product. We refer the reader to Dwyer and Mislin’s article [14] for a complete study of self-maps of BS^{3}.

2. Indecomposable elements and the Krull filtration

As mentioned in the introduction, a good way to understand the cohomology of a space as an algebra over the Steenrod algebra is to look at the module of indecomposable elements QH^{∗}(X; 𝔽_{p}) = H̃^{∗}(X; 𝔽_{p})/H̃^{∗}(X; 𝔽_{p})·H̃^{∗}(X; 𝔽_{p}). An important observation here is that this definition depends on the choice of a base point, or more exactly on the choice of a component X_{0} if X is not connected. Since H^{0}(X; 𝔽_{p}) is a p-Boolean algebra, it follows that QH^{∗}(X; 𝔽_{p}) is isomorphic to QH^{∗}(X_{0}; 𝔽_{p}).

There is a (Krull) filtration of the category U of unstable modules, U_{0} ⊂ U_{1} ⊂ … such that U_{0} consists in the locally finite unstable module. Schwartz established in [32, Theorem 6.2.4] a criterion to check whether (and where) an unstable module lives in the Krull filtration, namely M ∈ U_{n} if and only if T̄^{n+1}M = 0, where T̄ is Lannes’ reduced T -functor.

Therefore our objective in this section is to prove that T̄^{2}QH^{∗}(BX; 𝔽_{p}) = 0 for a p-Noetherian group. To do so, we need first to find a geometrical interpretation of the reduced T -functor.

Recall that “under some mild assumptions”, T H^{∗}(Z; 𝔽_{p}) ≅ H^{∗}(map(Bℤ/p, Z); 𝔽_{p}). Lannes’ standard mild assumptions on Z are that T H^{∗}(Z; 𝔽_{p}) is of finite type (or H^{∗}(map(Bℤ/p, Z); 𝔽_{p}) is of finite type), and that map(Bℤ/p, Z) is p-good, [25, Proposition 3.4.4]. We will not need to understand globally the mapping space, but restrict our attention to the component map(Bℤ/p, Z)_{c} of the constant map, the natural choice of base point in the full mapping space. We thus only consider the component T_{c}(H^{∗}(Z; 𝔽_{p})) of Lannes’ T -functor. Our first goal is to prove that the finite type conditions are satisfied in our situation.

Lemma 2.1. Let K(G, 1) → E → B be a fibration of connected spaces with a section where G is a discrete group. Then E ≃ K(G, 1) × B if and only if the induced action π_{1}(B) → Aut(G) is trivial.

Proof. Since aut_{∗}(K(G, 1)) ≃ Aut(G) the map classifying this split fibration is of the form B → B Aut G and factors through Bπ_{1}(B) → B Aut(G).

Let us have a closer look at the mapping space map(Bℤ/p, BX)_{c} for a p-Noetherian group X.

NOETHERIAN LOOP SPACES 9

Proposition 2.2. Let Z be a connected space such that Ω^{2}Z is Bℤ/p-local. Then the component of the mapping space map(Bℤ/p, Z)_{c} splits as a product Z ×map_{∗}(Bℤ/p, Z)_{c} and map_{∗}(Bℤ/p, Z)_{c} is the classifying space of an elementary abelian p-group (not necessarily finite).

Proof. By the work of Bousfield, [4, Theorem 7.2], the homotopy fiber of the nullification map Z → P_{ΣBℤ/p}Z is a single Eilenberg-Mac Lane space K(P, 2), where P is an abelian p-torsion group. He also shows that the fibration K(P, 2) → Z → P_{ΣBℤ/p}Z is principal. By adjunction, the component map_{∗}(Bℤ/p, P_{ΣBℤ/p}Z)_{c} is contractible. Therefore map_{∗}(Bℤ/p, Z)_{c} ≃ map_{∗}(Bℤ/p, K(P, 2))_{c}, which is the classifying space of the elementary abelian p-group W = Hom(ℤ/p, P).

Now, by Lemma 2.1, we only need to check that the action of π = π_{1}(Z) on W is trivial. By taking mapping spaces at the component of the constant map and evaluation, we obtain the following diagram of fibrations (since map_{∗}(Bℤ/p, P_{ΣBℤ/p}Z)_{c} is contractible):

[figure: BW; BW; map(Bℤ/p, K(P, 2))_{c}; map(Bℤ/p, Z)_{c}; map(Bℤ/p, P_{ΣBℤ/p}Z)_{c}; ≃; K(P, 2); Z; P_{ΣBℤ/p}Z]

The bottom and middle horizontal fibrations are principal, therefore the action of the fundamental group of the base space, π_{1}P_{ΣBℤ/p}Z ≅ π, is trivial on all homotopy groups of the fiber, in particular on the fundamental group of the fiber. This action can be seen as conjugation in the fundamental group of the total space map(Bℤ/p, Z)_{c}, but now it does not matter whether we look at the vertical fibration or the horizontal one (in both cases the induced morphism is surjective on the fundamental group).

Corollary 2.3. Let X be a p-Noetherian group. Then the mapping space map(Bℤ/p, BX)_{c} splits as a product BX × map_{∗}(Bℤ/p, BX)_{c} where map_{∗}(Bℤ/p, BX)_{c} is the classifying space of a finite elementary abelian p-group. In particular, the mapping space map(Bℤ/p, BX)_{c} is p-good, p-complete and H^{∗}(map(Bℤ/p, BX)_{c}; 𝔽_{p}) is of finite type.

Proof. The finiteness of the elementary abelian p-group follows from Corollary 1.10.

In particular, we see that map(Bℤ/p, BX)_{c} is again the classifying space of a p-Noetherian group.

Theorem 2.4. Let Z be a connected p-complete space such that H^{∗}(Z; 𝔽_{p}) and H^{∗}(map(Bℤ/p, Z)_{c}) are of finite type. If Ω^{2}Z is Bℤ/p-local, then

T̄QH^{∗}(Z; 𝔽_{p}) ≅ QH^{∗}(map_{∗}(Bℤ/p, Z)_{c}; 𝔽_{p}).

In particular the unstable module QH^{∗}(Z; 𝔽_{p}) lies in U_{1}.

10 NATÀLIA CASTELLANA, JUAN A. CRESPO, AND JÉRÔME SCHERER^{∗}

Proof. In Proposition 2.2 we obtained a splitting map(Bℤ/p, Z)_{c} ≃ map_{∗}(Bℤ/p, Z)_{c} × Z and an equivalence map_{∗}(Bℤ/p, Z)_{c} ≃ BW where W is an elementary abelian p-group. With the hypothesis of the theorem this splitting shows that H^{∗}(BW; 𝔽_{p}) is of finite type and therefore W is finite. Therefore map(Bℤ/p, Z)_{c} is p-good. Since moreover H^{∗}(map(Bℤ/p, Z)_{c}; 𝔽_{p}) is of finite type by assumption, we can apply Lannes’ result [25, Proposition 3.4.4] and deduce that the T -functor computes what it should: T_{c}H^{∗}(Z; 𝔽_{p}) ≅ H^{∗}(map(Bℤ/p, Z)_{c}; 𝔽_{p}).

Notice also that QH^{∗}(map(Bℤ/p, Z); 𝔽_{p}) ≅ QH^{∗}(map(Bℤ/p, Z)_{c}; 𝔽_{p}). Since Lannes’ T -functor commutes with taking the module of indecomposable elements, T QH^{∗}(Z; 𝔽_{p}) ≅ QT H^{∗}(Z; 𝔽_{p}). But in degree zero T H^{∗}(Z; 𝔽_{p}) is a Boolean algebra, [32, Section 3.8], so that QT H^{∗}(Z; 𝔽_{p}) ≅ Q(T_{c}H^{∗}(Z; 𝔽_{p})), which is isomorphic to QH^{∗}(map(Bℤ/p, Z)_{c}; 𝔽_{p}). The splitting yields next an isomorphism

T QH^{∗}(Z; 𝔽_{p}) ≅ QH^{∗}(map_{∗}(Bℤ/p, Z)_{c}; 𝔽_{p}) ⊕ QH^{∗}(Z; 𝔽_{p})

so that we have finally identified T̄QH^{∗}(Z; 𝔽_{p}) ≅ QH^{∗}(map_{∗}(Bℤ/p, Z)_{c}; 𝔽_{p}). This proves the first part of the theorem. For the second claim, use the fact that map_{∗}(Bℤ/p, Z)_{c} ≃ BW, the classifying space of a finite elementary abelian group. The cohomology of W is finitely generated as an algebra, so QH^{∗}(BW; 𝔽_{p}) is finite and lies in U_{0}. Therefore T̄QH^{∗}(BW; 𝔽_{p}) = 0, or equivalently T̄^{2}QH^{∗}(Z; 𝔽_{p}) = 0, and so QH^{∗}(Z; 𝔽_{p}) lies in U_{1}.

Let us now turn to an even finer analysis of the module of indecomposable elements. Let us denote by Q_{1} the unstable module QH^{∗}(Bℤ/p; 𝔽_{p}) of the cohomology of a cyclic group of order p. At the prime p = 2, the unstable module Q_{1} is isomorphic to Σ𝔽_{2} = ΣF (0). At an odd prime Q_{1} is an unstable module with one generator t in degree 1 and its Bockstein βt in degree 2.

Proposition 2.5. Let Z be a p-complete space such that H^{∗}(Z; 𝔽_{p}) and H^{∗}(map(Bℤ/p, Z)_{c}) are of finite type. Assume that Ω^{2}Z is Bℤ/p-local. Define Q_{1} = QH^{∗}(Bℤ/p; 𝔽_{p}). Then there exists a morphism QH^{∗}(Z; 𝔽_{p}) → F (1) ⊗ (Q_{1}^{⊕k}) with finite cokernel and locally finite kernel.

Proof. Schwartz characterizes in [34, Proposition 2.3] the unstable modules M in U_{1} as those sitting in an exact sequence 0 → K → M → F (1) ⊗ L → N → 0, where K, L, and N are locally finite (i.e. in U_{0}). In particular T̄M ≅ L since T̄F (1) = F (0) and T commutes with tensor products, [32, Theorem 3.5.1]. In our case we know from the previous theorem that T̄QH^{∗}(Z; 𝔽_{p}) ≅ QH^{∗}(BW; 𝔽_{p}) where W is an abelian elementary group, say of rank k. Thus L = Q_{1}^{⊕k}. The quotient N of F (1) ⊗ (Q_{1}^{⊕k}) will be finitely generated. As it is locally finite it must be finite.

We finally come back to p-Noetherian groups and prove that the module of indecomposable elements QH^{∗}(BX; 𝔽_{p}) is as small as expected.

Theorem 2.6. Let X be a p-Noetherian group. Then

T̄QH^{∗}(BX; 𝔽_{p}) ≅ QH^{∗}(map_{∗}(Bℤ/p, BX)_{c}; 𝔽_{p}).

In particular the unstable module QH^{∗}(BX; 𝔽_{p}) lies in U_{1}.

Proof. The assumptions in Theorem 2.4 are satisfied by Corollary 2.3.

3. Fibrations over spaces with finite cohomology

In our study of H^{∗}(BX; 𝔽_{p}), we have already managed to prove that QH^{∗}(BX; 𝔽_{p}) lives in U_{1}, that is only one stage higher than where QH^{∗}(X; 𝔽_{p}) lives. What is left to prove is that H^{∗}(BX; 𝔽_{p}) is finitely generated as an algebra over the Steenrod algebra. Therefore we analyze the fibration K(P, 2)_{p}^{∧} → BX → BY of Theorem 1.9.

Let F → E → B be a fibration where both H^{∗}(B; 𝔽_{p}) and H^{∗}(F; 𝔽_{p}) are finitely generated A_{p}-algebras. In this situation, we ask whether the same finiteness condition holds for H^{∗}(E; 𝔽_{p}). When the fibration is one of H-spaces and H-maps we proved in [10] that this is true. But in general some restrictions have to be imposed, even when the fiber is a single Eilenberg-Mac Lane space as shown by the following example.

Example 3.1. Consider the folding map K(ℤ, 3) ∨ K(ℤ, 3) → K(ℤ, 3). An easy application of Puppe’s theorem [31], shows that the homotopy fiber is ΣΩK(ℤ, 3) ≃ ΣK(ℤ, 2). Therefore there exists a fibration

K(ℤ, 2) → ΣK(ℤ, 2) → K(ℤ, 3) ∨ K(ℤ, 3).

The mod 2 cohomology of the fiber is finitely generated as an algebra, the cohomology of the base space is generated over A_{2} by the two fundamental classes in degree 3. However the cohomology of ΣK(ℤ, 2) is not finitely generated over A_{2} (as it is a suspension it would be finitely generated as an unstable module, and therefore would belong to some stage of the Krull filtration; by Schwartz’s solution [33] to Kuhn’s non-realizability conjecture this would imply that the cohomology were locally finite, which it is not).

This example indicates that we must impose stronger conditions on the base space of the fibration to make sure that the cohomology of the total space is finitely generated as an algebra over A_{p}. In this section we study fibrations F → E → B where π_{1}B acts trivially on the cohomology of the fiber. We will assume that H^{∗}(B; 𝔽_{p}) is finite and the fiber F is a finite product of Eilenberg Mac-Lane spaces ∏_{i=1}^{q} K(A_{i}, n_{i}) where A_{i} is a finitely generated abelian group for all i. Both assumptions will play an essential role in the analysis of the cohomology of the total space. The finiteness of the base forces the Serre spectral sequence to collapse at some finite stage and the hypothesis on A_{i} implies that the cohomology of K(A_{i}, n_{i}) is generated, as an algebra over the Steenrod algebra A_{p}, by a finite number of fundamental classes ι_{1}, …, ι_{m} of degree n, and possibly certain higher Bockstein on these classes. It is a free algebra by work of Serre at the prime 2, [35], and Cartan at odd primes, [7].

12 NATÀLIA CASTELLANA, JUAN A. CRESPO, AND JÉRÔME SCHERER^{∗}

Lemma 3.2. There exists a splitting H^{∗}(∏ K(A_{i}, n_{i}); 𝔽_{p}) ≅ F^{∗} ⊗ G^{∗} of algebras where F^{∗} is finitely generated as an algebra, and G^{∗} consists of permanent cycles in the Serre spectral sequence. Moreover G^{∗} is finitely generated as an algebra over A_{p}.

Proof. By Kudo’s transgression theorem, all classes obtained by applying Steenrod operations to transgressive operations are transgressive. Let us choose therefore an integer r larger than the dimension of the cohomology of the base. If {x_{1}, …, x_{k}} is a set of generators of H^{∗}(F; 𝔽_{p}) as an A_{p}-algebra, the elements P^{I}x_{k} generate freely H^{∗}(F; 𝔽_{p}) as an algebra, where I are certain admissible sequences, [32, p.184].

The elements 1 ⊗ P^{I}x_{k} are permanent cycles for any sequence I of degree larger than r − n − 1 and any k. We will say that such generators P^{I}x_{k} have large degree and the others, of which there is only a finite number, have small degree. We define now F^{∗} to be the subalgebra generated by the generators of small degree and G^{∗} by all other large degree generators. Then H^{∗}(F; 𝔽_{p}) ≅ F^{∗} ⊗G^{∗}.

The only claim left to prove is the finite generation of G^{∗}. Let us look at the inclusion of algebras G^{∗} ⊂ H^{∗}(F; 𝔽_{p}). At the level of modules of indecomposable elements it induces an inclusion QG^{∗} ⊂ QH^{∗}(F; 𝔽_{p}), because of the freeness of H^{∗}(F; 𝔽_{p}) and our choices of generators. Since the category U of unstable modules is locally Noetherian, [32, Theorem 1.8.1], the unstable module QG^{∗} is finitely generated. Therefore, G^{∗} is a finitely generated A_{p}-algebra.

The proof of the next proposition follows the lines of the Dwyer-Wilkerson result [17, Proposition 12.4], see also Evens, [18].

Proposition 3.3. The cohomology of the total space H^{∗}(E; 𝔽_{p}) is finitely generated as a module over H^{∗}(B; 𝔽_{p})[z_{1}, …, z_{k}] ⊗ G^{∗}.

Proof. The free algebra F^{∗} is finitely generated and we consider first all polynomial generators a_{1}, …, a_{k}. We define z_{i} = (a_{i})^{p^{n_{i}}} where n_{i} is the smallest integer such that this power of a_{i} is a permanent cycle (it exists since these powers are transgressive, compare with the proof of Lemma 3.2). Then F^{∗} is a finitely generated module over 𝔽_{p}[z_{1}, …, z_{k}]. Choose now a finite set of generators g_{1}, …, g_{r} of G^{∗} as an algebra over the Steenrod algebra.

The elements z_{i} and the elements g_{1}, …, g_{r} are permanent cycles in the vertical axis of the Serre spectral sequence, one can thus choose elements z′_{i} and g′_{j} in H^{∗}(E; 𝔽_{p}) whose images in H^{∗}( ∏ K(A_{i}, n_{i}); 𝔽_{p}) are the z_{i}’s and the g_{j}’s. Better said, since both 𝔽_{p}[z_{i}] and G^{∗} are free algebras, we choose a monomorphic algebra map s: 𝔽_{p}[z_{i}] ⊗ G^{∗} ↪ H^{∗}(E; 𝔽_{p}). The elements in H^{∗}(B; 𝔽_{p}) act on H^{∗}(E; 𝔽_{p}) via p^{∗}: H^{∗}(X; 𝔽_{p}) → H^{∗}(E; 𝔽_{p}). This explains the module structure.

We see that E_{∞} = E_{r} is finitely generated as a module over H^{∗}(B; 𝔽_{p})[z_{1}, …, z_{k}]⊗G^{∗}. Therefore so is H^{∗}(E; 𝔽_{p}) by [37, Corollary VII.3.3].

NOETHERIAN LOOP SPACES 13

The difficulty to infer information about the A_{p}-algebra structure from the module structure come from the fact that the algebra map s is not a map of A_{p}-algebras. To circumvent this problem we will appeal to the algebraic result proved in the Appendix A.

Theorem 3.4. Consider a fibration ∏_{i=1}^{q} K(A_{i}, n_{i}) → E → B where H^{∗}(B; 𝔽_{p}) is finite and A_{i} is a finitely generated abelian group for all i. The cohomology H^{∗}(E; 𝔽_{p}) is then finitely generated as an algebra over A_{p}.

Proof. In the notation of the appendix, H^{∗}(B; 𝔽_{p})[z_{1}, …, z_{k}] is the connected and commutative finitely generated algebra C^{∗}, and B^{∗} = H^{∗}(E; 𝔽_{p}). By Proposition 3.3, H^{∗}(E; 𝔽_{p}) is a finitely generated C^{∗} ⊗ G^{∗}-module. The action of C^{∗} ⊗ G^{∗} on B^{∗} has been defined in the previous proof via an algebra map (constructed from a section s : G^{∗} → B^{∗}), thus B^{∗} is a C^{∗} ⊗ G^{∗}-algebra. Define now π : H^{∗}(∏ K(A_{i}, n_{i}); 𝔽_{p}) ≅ F^{∗} ⊗ G^{∗} → G^{∗} to be the projection and p : B^{∗} → G^{∗} to be the composite π ∘ i^{∗}. This is a morphism of G^{∗}-modules so that Proposition A.2 applies. Hence H^{∗}(E; 𝔽_{p}) is finitely generated as an algebra over A_{p}.

Remark 3.5. The nature of Theorem 3.4 is purely cohomological. Therefore the same statement remains true if we relax the assumption on the fiber in the following way: The fiber F should be homotopic, up to p-completion, to ∏_{i=1}^{q} K(A_{i}, n_{i}) where each A_{i} is a finitely generated abelian group. This will allow us to include summands of the form ℤ_{p^{∞}} or ℤ_{p}^{∧}. In fact the same proof goes through with the assumption that H^{∗}(F; 𝔽_{p}) is a free algebra, finitely generated as an algebra over A_{p}.

4. The cohomology of p-Noetherian groups

We are about to conclude our study of the cohomology of classifying spaces of p-Noetherian groups. We have seen in Theorem 1.9 that any p-Noetherian group is the total space of a fibration over a p-compact group with fiber an Eilenberg-Mac Lane space. Recall from [17]’s main theorem that the mod p-cohomology of the classifying space of a p-compact group is finitely generated as an algebra. Our objective is to prove the following theorem, which together with Theorem 2.6 gives a very accurate description for the cohomology of classifying spaces of p-Noetherian groups.

Theorem 4.1. Let (X, BX, e) be a p-Noetherian group. Then H^{∗}(BX; 𝔽_{p}) is finitely generated as an algebra over the Steenrod algebra.

The strategy is to use the fibration K(P, 2)_{p}^{∧} → BX → BY of Theorem 1.9. As we do not know whether the Serre spectral sequence collapses at some finite stage, we reduce the problem in several steps to the study of a spectral sequence over a finite base (in order to apply our results from the previous section).

A p-compact toral group P is a p-compact group which is an extension of a p-compact torus ((S^{1})^{n})_{p}^{∧} by a finite p-group. Dwyer and Wilkerson in [17] show that any p-compact group Y

14 NATÀLIA CASTELLANA, JUAN A. CRESPO, AND JÉRÔME SCHERER^{∗}

admits a maximal p-compact toral subgroup N ≤ Y such that the homotopy fiber Y/N of the map Bi: BN → BY has finite mod p cohomology and Euler characteristic prime to p (see [17, Proof of Theorem 2.3]). An important property of Y/N for us is that it is connected. This is obvious when Y is connected and follows in general from the analysis of the relationship between the Weyl groups of Y and a connected component undertaken by Møller and Notbohm in [30, Theorem 3.4]. A transfer argument (see [17, Theorem 9.13]) then shows that Bi induces a monomorphism in mod p cohomology. Consider now the pullback diagram

\xymatrix{B̃N \ar[d] \ar[r]^{Bĩ} & BX \ar[d] \\ BN \ar[r]^{Bi} & BY.}

First note that, by Proposition 1.8, B̃N is the classifying space of a p-Noetherian group because we have a fibration K(P, 2)_{p}^{∧} → B̃N → BN. We will first show that H^{∗}(B̃N; 𝔽_{p}) is a finitely generated A_{p}-algebra by using this fibration, and then we will show that the cohomology H^{∗}(BX; 𝔽_{p}) is a finitely generated A_{p}-algebra by using the fibration Y/N → B̃N → BX. We start with a technical result which will be used in both steps of the proof.

Proposition 4.2. Let F → E →^{q} B be a fibration such that π_{1}(B) acts nilpotently on H_{∗}(F; 𝔽_{p}). Assume that q induces an isomorphism T̄QH^{∗}B ≅ T̄QH^{∗}E, H^{∗}(F; 𝔽_{p}) is locally finite, and H^{∗}(E; 𝔽_{p}) is a finitely generated A_{p}-algebra. Then, if Ker q^{∗} is a finitely generated ideal then H^{∗}(B; 𝔽_{p}) is a finitely generated A_{p}-algebra.

Proof. Let A be the algebra which is the quotient of H^{∗}(E; 𝔽_{p}) by the ideal generated by the image of q^{∗}, that is, 𝔽_{p} ⊗_{H^{∗}(B;𝔽_{p})} H^{∗}(E; 𝔽_{p}). There is a coexact sequence

H^{∗}(B; 𝔽_{p})// Ker q^{∗} →^{q^{∗}} H^{∗}(E; 𝔽_{p}) → A.

Since H^{∗}(E; 𝔽_{p}) is a finitely generated A_{p}-algebra, the same is true for A. Therefore QA is a finitely generated A_{p}-module, which is the the cokernel of Qq^{∗} by right-exactness of the functor Q. Moreover, since by assumption T̄QH^{∗}B ≅ T̄QH^{∗}E, it follows from exactness of the reduced T functor that T̄QA = 0. This means that QA belongs to U_{0}, i.e. it is locally finite, hence finite. Equivalently A is a finitely generated algebra.

We want to show that A is in fact a finite algebra. The fiber inclusion F → E of the fibration q induces a morphism ι^{∗}: A → H^{∗}(F; 𝔽_{p}). A careful study of the Eilenberg-Moore spectral sequence, [26, Theorem 0.5] (for the prime 2) or [32, Theorem 8.7.8], shows that this morphism is an F - monomorphism. Take now any element a ∈ A. Because H^{∗}(F; 𝔽_{p}) is locally finite, there exists M > 0 such that a^{p^{M}} ∈ Ker ι^{∗}, which is nilpotent. There exists thus N > 0 such that a^{p^{N+M}} = 0. Since all elements of A are nilpotent and it is a finitely generated algebra, A must be finite.

NOETHERIAN LOOP SPACES 15

Note next that, as an H^{∗}(B; 𝔽_{p})-module, the cohomology H^{∗}(E; 𝔽_{p}) is isomorphic to (H^{∗}(B; 𝔽_{p})// Ker q^{∗}){b_{1}, …, b_{k}}, where the b_{i}’s are the generators of A as a (graded) 𝔽_{p}-vector space. This description shows that the morphism Q(H^{∗}(B; 𝔽_{p})// Ker q^{∗}) → QH^{∗}(E; 𝔽_{p}) is an isomorphism in high degrees. That is, there exists K > 0 such that

(Q(H^{∗}(B; 𝔽_{p})// Ker q^{∗}))^{>K} ≅ (QH^{∗}(E; 𝔽_{p}))^{>K} ,

which is an unstable submodule of QH^{∗}(E; 𝔽_{p}). Since H^{∗}(E; 𝔽_{p}) is a finitely generated A_{p}- algebra, QH^{∗}(E; 𝔽_{p}) is a finitely generated A_{p}-module. But then, as the category of unstable modules over the Steenrod algebra is locally Noetherian, [32, Theorem 1.8.1], the same holds for (Q(H^{∗}(B; 𝔽_{p})// Ker q^{∗}))^{>K} . In particular, Q(H^{∗}(B; 𝔽_{p})// Ker q^{∗}) is a finitely generated A_{p}- module since it is of finite type, that is H^{∗}(B; 𝔽_{p})// Ker q^{∗} is a finitely generated A_{p}-algebra.

Finally, H^{∗}(B; 𝔽_{p}) is generated by H^{∗}(B; 𝔽_{p})// Ker q^{∗} and Ker q^{∗}, it is therefore also finitely generated as an algebra over the Steenrod algebra since Ker q^{∗} is a finitely generated ideal .

We apply next this result to the fibration Y/N → B̃N → BX.

Corollary 4.3. H^{∗}(BX; 𝔽_{p}) is a finitely generated algebra if H^{∗}(B̃N; 𝔽_{p}) is so.

Proof. Consider the fibration Y/N → B̃N →^{Bĩ} BX and the diagram of horizontal fibrations

\xymatrix{map(Bℤ/p, Y/N)_{c} \ar[r] \ar[d]^{ev}_{≃} & map(Bℤ/p, B̃N)_{c} \ar[d]^{ev_{r}} \ar[r] & map(Bℤ/p, BX)_{c} \ar[d]^{ev_{n}} \\ Y/N \ar[r] & B̃N \ar[r] & BX}

where the left vertical arrow is an equivalence since the fiber Y/N is connected and mod p finite. The connectedness of Y/N also implies that the right hand side square is a homotopy pull-back square and thus Bĩ induces an equivalence map_{∗}(Bℤ/p, B̃N)_{c} → map_{∗}(Bℤ/p, BX)_{c}. Both BX and B̃N are p-Noetherian groups and Theorem 2.6 implies then that T̄QH^{∗}BX → T̄QH^{∗}B̃N is an isomorphism.

We conclude now by Proposition 4.2, because π_{1}(BX) is a finite p-group and q^{∗} is injective by a transfer argument [17, Theorem 9.13] (the Euler characteristic χ(Y/N) is prime to p).

The key fact in the next argument is that any p-compact toral group satisfies the Peter-Weyl theorem. That is, it admits a homotopy monomorphism into U(n)_{p}^{∧} for some n. This is shown for example in [23, Proposition 2.2]. Let us choose then such a map ρ : BN → BU(n)_{p}^{∧} for the maximal p-compact toral group BN. The mod p cohomology of the fibre F is hence 𝔽_{p}-finite.

Proposition 4.4. Let E_{0} be the homotopy pull-back of the diagram F → BN ← B̃N. Then H^{∗}(E_{0}; 𝔽_{p}) is finitely generated as an algebra over the Steenrod algebra.

16 NATÀLIA CASTELLANA, JUAN A. CRESPO, AND JÉRÔME SCHERER^{∗}

Proof. The space E_{0} fits by construction into a fibration K(P, 2)_{p}^{∧} → E_{0} → F. Since the mod p cohomology of F is finite, Theorem 3.4 shows that H^{∗}(E_{0}; 𝔽_{p}) is a finitely generated A_{p}-algebra.

Let us now concentrate on the map E_{0} → B̃N. We will filter it by using the top left block diagonal inclusions U(r − 1) ⊂ U(r) for 1 ≤ r ≤ n. At the level of classifying spaces these inclusions induce oriented spherical fibrations

S^{2r−1} → BU(r − 1) → BU(r).

Let BU(r) → BU(n) be the appropriate composite and define then E_{r} to be the homotopy pullback of B̃N ⟶^{ρ} BU(n)_{p}^{∧} ← BU(r)_{p}^{∧}. Thus E_{n} = B̃N and, for 1 ≤ r ≤ n, we have spherical fibrations (S^{2r−1})_{p}^{∧} → E_{r−1} → E_{r} . We summarize next some important properties of these spaces.

Proposition 4.5. The component map(Bℤ/p, E_{r})_{c} splits as a product E_{r} × map_{∗}(Bℤ/p, E_{r})_{c} for any 0 ≤ r ≤ n.

Proof. Let us denote by ev_{r}: map(Bℤ/p, E_{r})_{c} → E_{r} the evaluation at the component of the constant map and consider the following commutative diagram of horizontal fibrations of connected spaces

\xymatrix{map(Bℤ/p, H)_{c} \ar[r] \ar[d]^{ev}_{≃} & map(Bℤ/p, E_{r})_{c} \ar[d]^{ev_{r}} \ar[r] & map(Bℤ/p, B̃N)_{c} \ar[d]^{ev_{n}} \\ H \ar[r] & E_{r} \ar[r] & B̃N}

The homotopy fiber H has finite cohomology, and is thus equivalent via the evaluation map to the homotopy fiber map(Bℤ/p, H)_{c} of the top fibration. Since H is connected (because U(n) is so) this shows that the right hand square is a pull-back square. But ev_{n} is a trivial fibration by Proposition 2.3. Hence so is ev_{r} and map(Bℤ/p, E_{r})_{c} must split as a product E_{r}×map_{∗}(Bℤ/p, E_{r})_{c}.

Proposition 4.6. The morphism of A_{p}-modules T̄QH^{∗}(B̃N; 𝔽_{p}) → T̄QH^{∗}(E_{r}; 𝔽_{p}) induced by E_{r} → B̃N is an isomorphism.

Proof. We have seen in the previous proposition that the map E_{r} → BX induces a weak equivalence on the connected component of the constant map in the pointed mapping space map_{∗}(Bℤ/p, −)_{c}. From this point on, the same argument as in the proof of Theorem 2.6 goes through.

Proof of Theorem 4.1. We know that H^{∗}(E_{0}; 𝔽_{p}) is finitely generated as an algebra over the Steenrod algebra. It is thus sufficient to prove that H^{∗}(E_{r}; 𝔽_{p}) is a finitely generated A_{p}-algebra if so is H^{∗}(E_{r−1}; 𝔽_{p}). Denote by q_{r} the map E_{r−1} → E_{r} turned into a fibration. Since Ker q_{r}^{∗} is generated by a single element, namely the Euler class, Proposition 4.2 applies.

NOETHERIAN LOOP SPACES 17

Remark 4.7. In general, an unstable algebra which is finitely generated as an algebra over A_{p} may contain unstable subalgebras which are not finitely generated over A_{p}. Such an example appears in [10, Remark 2.2] as an unstable subalgebra B of H^{∗}(BS^{1} × S^{2}; 𝔽_{p}) ≅ 𝔽_{p}[x] ⊗ E(y), which is not a finitely generated B-module. This B is the ideal generated by y turned into an unstable algebra by adding 1. The quotient is a polynomial algebra, in particular it is not finite.

Remark 4.8. To prove Theorem 4.1 one could also use the fact that p-compact groups satisfy the Peter-Weyl theorem (see [3, Theorem 1.6] and [2, Remark 7.3]). This would slightly shorten the proof and avoid the use of the maximal p-compact toral subgroup. But the Peter-Weyl theorem for p-compact groups is proved using the classification of p-compact groups and we want to emphasize that Theorem 4.1 does not depend on the classification.

Corollary 4.9. Let G be a simply connected, simple, compact Lie group. Then H^{∗}(BG⟨4⟩; 𝔽_{p}) is finitely generated as an algebra over the Steenrod algebra and QH^{∗}(BG⟨4⟩; 𝔽_{p}) belongs to U_{1}. In fact there exists a morphism QH^{∗}(BG⟨4⟩; 𝔽_{p}) → F (1) ⊗ Q_{1} with finite kernel and cokernel.

Proof. Recall that Q_{1} = QH^{∗}(Bℤ/p; 𝔽_{p}). Since BG is 3-connected and π_{4}BG ≅ ℤ, Proposition 2.5 yields a a morphism QH^{∗}(BG⟨4⟩; 𝔽_{p}) → F (1) ⊗ Q_{1} (the elementary abelian group W appearing there has rank one). The cokernel is finite and the kernel locally finite. But since QH^{∗}(BG⟨4⟩; 𝔽_{p}) is a finitely generated module over A_{p}, the kernel must be finite.

Example 4.10. Let X be a either any simply connected compact Lie group (such as Spin(10), which is one of the smallest examples Harada and Kono could not handle at the prime 2) or one of the p-compact groups number 2b, 23 or 30 in the Shephard-Todd list. For odd primes, the exotic p-compact groups arising from this construction are again non-modular. Hence, in all the cases, H^{∗}(X; ℤ_{p}^{∧}) is torsion free. The mod p cohomology is given by H^{∗}(BX_{23}; 𝔽_{p}) = 𝔽_{p}[x_{4}, x_{12}, x_{20}], H^{∗}(BX_{30}; 𝔽_{p}) = 𝔽_{p}[x_{4}, x_{24}, x_{40}, x_{60}], and H^{∗}(BX_{2b,m}; 𝔽_{p}) = 𝔽_{p}[x_{4}, x_{2m}]. Since all examples are torsion free, the same techniques used by Harada and Kono in [21] and [20] show that H^{∗}(BX⟨4⟩; 𝔽_{p}) ≅ H^{∗}(BX; 𝔽_{p})/J ⊗ R_{h} where J is the ideal generated by x_{4}, P^{1}x_{4}, …, P^{h}x_{4} for a certain h, and R_{h} is an unstable subalgebra of H^{∗}(K(ℤ_{p}^{∧}, 3); 𝔽_{p}) which is finitely generated over A_{p}. In fact Theorems 4.1 and 2.6 show directly that QH^{∗}(BX⟨4⟩; 𝔽_{p}) is finitely generated as a module over A_{p} and belongs to U_{1}. From the last corollary we see for example that QH^{∗}(BSpin(10)⟨4⟩; 𝔽_{2}) differs from ΣF (1) in only a finite number of “low dimensional” classes.

Appendix A. Modules and algebras

Let us consider an unstable algebra B^{∗}. Assume there is another unstable algebra G^{∗} which is finitely generated as an algebra over A_{p} and which acts on B^{∗}. Assume also that B^{∗} is finitely generated as a module over G^{∗}. When can we conclude that B^{∗} is finitely generated as an algebra over A_{p}? If the action of G^{∗} on B^{∗} is compatible with the action of the Steenrod algebra, it is obvious, but it is not true in general as illustrated by the following example. Set G^{∗} = H^{∗}(K(ℤ/2, 2); 𝔽_{2})

18 NATÀLIA CASTELLANA, JUAN A. CRESPO, AND JÉRÔME SCHERER^{∗}

and let B^{∗} be isomorphic to G^{∗} as an algebra, but define the action of A_{2} to be trivial. Then B^{∗} is finitely generated as a G^{∗}-module, but not as an A_{2}-algebra. We propose a weak notion of compatibility between the G^{∗}-module and A_{p}-algebra structures.

Proposition A.1. Let G^{∗} be a connected and commutative finitely generated A_{p}-algebra, and let B^{∗} be a connected commutative G^{∗}-algebra which is finitely generated as a G^{∗}-module. Assume there exists a morphism p: B^{∗} → G^{∗} of G^{∗}-modules such that p(θ(g · 1)) = θg for all θ ∈ A_{p} and g ∈ G^{∗}. Then B^{∗} is also a finitely generated A_{p}-algebra.

Proof. Since B^{∗} is finitely generated as a G^{∗}-module, let us choose a finite set of generators b_{1} = 1, …, b_{n}, where the degree of b_{i} is positive for i ≥ 2 by the connectivity assumption. In fact we will assume that b_{i} belongs to the kernel of p for i ≥ 2: replace b_{i} by b_{i} − p(b_{i}) · 1 if necessary.

Let x be an element in B^{∗}, so x can be written as a G^{∗}-linear combination x = ∑_{i=1}^{n} λ_{i} · b_{i}. If {z_{1}, …, z_{m}} denote the A_{p}-algebra generators of G^{∗}, then each λ_{i} can be expressed as a polynomial on Steenrod operations θ_{i} applied to the generators, θ_{i}z_{i} (use the Cartan formula). Our claim is that b_{1}, …, b_{n} and z_{1} · 1, …, z_{m} · 1 generate B^{∗} as an algebra over A_{p}. Since B^{∗} is a unital G^{∗}- algebra, we need to prove that an element of the form θz · 1, for θ ∈ A_{p} and z ∈ {z_{1}, …, z_{m}}, can be written as a polynomial in the θ(z_{i} · 1)’s and the b_{k}’s.

If the action of the Steenrod algebra were compatible with the module action, one would have that θz·1 = θ(z·1). This is not the case, but it is sufficient to deal with the element ξ = θz·1−θ(z·1). Note that ξ ∈ Ker p since p(θz·1) = θz. We will proceed by induction on the degree. If ξ is in degree zero, then the statement is clear since the algebras are connected. Assume that the statement is true for degrees < |ξ| and write ξ = λ_{1} · 1 + λ_{2} · b_{2} + … λ_{n} · b_{n}. By the induction hypothesis we know that the elements λ_{i} · 1 can be expressed as polynomials in the θ(z_{i} · 1)’s and the b_{k}’s. So we only need to deal with λ_{1} · 1. But λ_{1} = p(λ_{1} · 1) = p(ξ − λ_{2} · b_{2} − ⋯ − λ_{n} · b_{n}) which is zero because p is a morphism of G^{∗}-modules and b_{i} ∈ Ker p for i ≥ 2. This concludes the proof.

In Section 3 we need a slight generalization of this proposition.

Proposition A.2. Let G^{∗} be a connected and commutative finitely generated A_{p}-algebra, C^{∗} be a connected and finitely generated algebra, and let B^{∗} be a connected commutative G^{∗} ⊗ C^{∗}-algebra which is a finitely generated as a G^{∗} ⊗ C^{∗}-module. Assume there exists a morphism p: B^{∗} → G^{∗} of G^{∗}-modules such that p(θ(g · 1)) = θg for all θ ∈ A_{p} and g ∈ G^{∗}. Then B^{∗} is also a finitely generated A_{p}-algebra.

Proof. Just as in the previous proof, the only problem is to write an element of the form (θz) · 1, with θ ∈ A_{p} and z ∈ G^{∗}, in terms of the generators z_{i}, b_{k}, and generators c_{m} of the algebra C^{∗}. The proof is then basically the same.

References

1. J. Aguadé and L. Smith, On the mod p torus theorem of John Hubbuck, Math. Z. 191 (1986), no. 2, 325–326.

2. K. Andersen and J. Grodal, The classification of 2-compact groups, Preprint.

3. K. Andersen, J. Grodal, J.M. Møller, and A. Viruel, The classification of p-compact groups for p odd, Ann. of Math. (2) (2008), no. 167, 95–210.

4. A. K. Bousfield, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994), no. 4, 831–873.

5. A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Springer-Verlag, Berlin, 1972, Lecture Notes in Mathematics, Vol. 304.

6. W. Browder, Torsion in H-spaces, Ann. of Math. (2) 74 (1961), 24–51.

7. H. Cartan, Séminaire Henri Cartan de l’Ecole Normale Supérieure, 1954/1955. Algèbres d’Eilenberg-MacLane et homotopie, exposés 2 –16, Secrétariat mathématique, 11 rue Pierre Curie, Paris, 1955.

8. N. Castellana, J. A. Crespo, and J. Scherer, On the homotopy groups of p-completed classifying spaces, Manuscripta Math. 118 (2005), no. 3, 399–409.

9. , Deconstructing Hopf spaces, Invent. Math. 167 (2007), no. 1, 1–18.

10. , On the cohomology of highly connected covers of Hopf spaces, Adv. Math. 215 (2007), 250–262.

11. W. Chachólski, W. Pitsch, J. Scherer, and D. Stanley, Homotopy exponents for large H-spaces, Int. Math. Res. Not. IMRN (2008), no. 16, Art. ID rnn061, 5.

12. J.A. Crespo, Structure of mod p H-spaces with finiteness conditions, Cohomological methods in homotopy theory (Bellaterra, 1998), Progr. Math., vol. 196, Birkhäuser, Basel, 2001, pp. 103–130.

13. F.-X. Dehon and G. Gaudens, Espaces profinis et problèmes de réalisabilité, Algebr. Geom. Topol. 3 (2003), 399–433 (electronic).

14. W. G. Dwyer and G. Mislin, On the homotopy type of the components of map_{∗}(BS^{3}, BS^{3}), Algebraic topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, Springer, Berlin, 1987, pp. 82–89.

15. W. G. Dwyer and C. W. Wilkerson, Spaces of null homotopic maps, Astérisque (1990), no. 191, 6, 97–108, International Conference on Homotopy Theory (Marseille-Luminy, 1988).

16. , A new finite loop space at the prime two, J. Amer. Math. Soc. 6 (1993), no. 1, 37–64.

17. , Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. (2) 139 (1994), no. 2, 395–442.

18. L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224–239.

19. E. Dror Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics, vol. 1622, Springer-Verlag, Berlin, 1996.

20. M. Harada and A. Kono, Cohomology mod p of the 4-connective fibre space of the classifying space of classical Lie groups, Proc. Japan Acad. Ser. A Math. Sci. 60 (1984), no. 2, 63–65.

21. , Cohomology mod p of the 4-connected cover of the classifying space of simple Lie groups, Homotopy theory and related topics (Kyoto, 1984), Adv. Stud. Pure Math., vol. 9, North-Holland, Amsterdam, 1987, pp. 109–122.

22. J. R. Hubbuck, On homotopy commutative H-spaces, Topology 8 (1969), 119–126.

23. A. Jeanneret and A. Osse, The K-theory of p-compact groups, Comment. Math. Helv. 72 (1997), no. 4, 556–581.

24. N. J. Kuhn, On topologically realizing modules over the Steenrod algebra, Ann. of Math. (2) 141 (1995), no. 2, 321–347.

25. J. Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un p-groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. (1992), no. 75, 135–244, With an appendix by Michel Zisman.

20 NATÀLIA CASTELLANA, JUAN A. CRESPO, AND JÉRÔME SCHERER^{∗}

26. J. Lannes and L. Schwartz, À propos de conjectures de Serre et Sullivan, Invent. Math. 83 (1986), no. 3, 593–603.

27. H. Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. (2) 120 (1984), no. 1, 39–87.

28. J. M. Møller, N-determined 2-compact groups. I, Fund. Math. 195 (2007), no. 1, 11–84.

29. , N-determined 2-compact groups. II, Fund. Math. 196 (2007), no. 1, 1–90.

30. J. M. Møller and D. Notbohm, Centers and finite coverings of finite loop spaces, J. Reine Angew. Math. 456 (1994), 99–133.

31. V. Puppe, A remark on “homotopy fibrations”, Manuscripta Math. 12 (1974), 113–120.

32. L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1994.

33. , À propos de la conjecture de non-réalisation due à N. Kuhn, Invent. Math. 134 (1998), no. 1, 211–227.

34. , La filtration de Krull de la catégorie U et la cohomologie des espaces, Algebr. Geom. Topol. 1 (2001), 519–548 (electronic).

35. J.-P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg-MacLane, Comment. Math. Helv. 27 (1953), 198–232.

36. G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274–304.

37. O. Zariski and P. Samuel, Commutative algebra. Vol. II, Springer-Verlag, New York, 1975, Reprint of the 1960 edition, Graduate Texts in Mathematics, Vol. 29.

Natàlia Castellana and Jerôme Scherer Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain

   E-mail: Natalia@mat.uab.es,
          jscherer@mat.uab.es

Juan A. Crespo

Departamento de Economía Cuantitativa, Universidad Autónoma de Madrid, E-28049, Cantoblanco Madrid, Spain

                                                    E-mail: juan.crespo@uam.es,