Serre’s theorem and the N il_{l} filtration of Lionel Schwartz

Jesper Grodal

Abstract. We give three different cohomological characterizations of classifying spaces of p-compact toral groups amongst finite Postnikov systems satisfying mild conditions. This leads to a unifying generalization of previous versions of Serre’s theorem on the homotopy groups of a finite complex.

1. Introduction

In the heart of all generalizations of Serre’s theorem on the homotopy groups of a finite complex (cf. e.g. [17, 12, 10, 11, 4, 6]) there have implicitly or explicitly been statements about the ‘non-finiteness’ of the mod p cohomology of a finite Postnikov system. The negations of these statements say that a space with ‘finite’ mod p cohomology has infinitely many nontrivial mod p homotopy groups (that is, infinitely many homotopy groups are not uniquely p-divisible). Such statements can then in some cases (but not all cf. Remark 4.5) afterwards be improved to showing that the space has infinitely many homotopy groups containing p-torsion, by employing a method of McGibbon and Neisendorfer [12]. We put these generalizations into a common framework by offering the following cohomological characterizations of classifying spaces of p-compact toral groups amongst finite Postnikov systems.

Theorem 1.1. Let X be a connected finite Postnikov system with π_{1}X a finite pgroup and H^{∗}(X; F_{p}) of finite type. The following conditions are equivalent.

1. QH^{∗}(X; F_{p}) ∈ \overline{N il}_{2}.

2. H^{∗}(X; F_{p}) is of finite transcendence degree.

3. H^{∗}(X; F_{p}) is noetherian.

4. X is F_{p}-equivalent to the classifying space of a p-compact toral group.

Here Q denotes the indecomposables, and \overline{N il}_{2} refers to the N il filtration of the category U of unstable modules over the Steenrod algebra of Schwartz [15]. (Note that locally finite modules over the Steenrod algebra lie in \overline{N il}_{2}.) Recall that two spaces are said to be F_{p}-equivalent if they become homotopy equivalent after Bousfield H_{∗}(−; F_{p}) localization [1]. Furthermore, recall that a classifying space of a p-compact toral group is a space which fits as the total space in a fibration sequence over K(P, 1) with fiber K(Ẑ_{p}^{n}, 2) for some n < ∞, where Ẑ_{p} denotes the p-adic integers and P is a finite p-group (cf. [5]).

1991 Mathematics Subject Classification. Primary: 55Q52 ; Secondary: 55S45, 55S10.

Key words and phrases. Postnikov system, unstable module, N il filtration, Serre’s theorem, p-toral group, p-compact group.

The theorem extends results in [6], and also the generalization of Serre’s theorem of [4] can be easily recovered and generalized (Theorem 4.4).

To prove the results we study the N il_{l} filtration and its relation to the Eilenberg-Moore spectral sequence, using the methods of Schwartz [15], and use this to reduce the theorem to results of [6].

I am grateful to W. Dwyer and B. Shipley for helpful discussions and suggestions, and to B. Shipley for making an early thesis proposal [18] on a similar topic available to me.

2. Notation and preliminaries

By a space we mean, for simplicity, an object in the homotopy category of finite pointed CW-complexes. A connected finite Postnikov system is a connected space such that π_{i}X = 0 for i large. Throughout we abbreviate H^{∗}(X; F_{p}) to H^{∗}X where p is a fixed but arbitrary prime.

We now state some basic facts about the Eilenberg-Moore spectral sequence and the N il_{l} filtration.

2.1. The Eilenberg-Moore spectral sequence The cohomological Eilenberg-Moore spectral sequence of a fibration D → E → B over a connected space B is a second quadrant spectral sequence with E_{2}- term given by E_{2}^{s,∗} = Tor_{H^{∗}B}^{s}(F_{p}, H^{∗}E) as an unstable module over the Steenrod algebra A [14]. The columns E_{r}^{s,∗}, s ≤ 0, r ≥ 2 are likewise unstable A-modules, and the differentials d_{r} : E_{r}^{s,∗} → E_{r}^{s+r,∗} are A-linear of degree −(r − 1). The spectral sequence converges strongly to H^{∗}D when H^{∗}E and H^{∗}B are of finite type and π_{1}B acts nilpotently on H^{∗}D [3]. Specifically, there is a cocomplete filtration of A-modules of H^{∗}D:

H^{∗}D ⊃ ⋯ F_{s} ⊃ F_{s+1} ⊃ ⋯ ⊃ F_{0} ⊃ F_{1} = 0

such that Σ^{−s}F_{s}/F_{s+1} ≃ E_{∞}^{s,∗}.

2.2. The N il_{l} filtration An unstable module M is said to be l-nilpotent if it is the colimit of unstable modules each having a finite filtration whose filtration quotients are l-suspensions. (For some equivalent definitions of l-nilpotency see for example [16], which is also a general reference for other facts concerning unstable modules over the Steenrod algebra.) Let N il_{l} denote the full subcategory of U of l-nilpotent modules. We hence get a decreasing filtration

⋯ ⊂ N il_{2} ⊂ N il_{1} = N il ⊂ N il_{0} = U

Serre’s theorem and the N il_{l} filtration 3

of U with the property that N il_{1} equals the usual subcategory of nilpotent modules N il and ∩_{l}N il_{l} = 0. It is often more convenient to work with the slightly larger subcategory \overline{N il}_{l} which is the smallest Serre class in U that contains N il_{l} and B, the category of locally finite modules, and is closed under colimits. A more concrete description of \overline{N il}_{l} is given by observing that M ∈ \overline{N il}_{l} if and only if M^{≥l} ∈ N il_{l}.

3. Relations between QH^{∗}X and H^{∗}ΩX

In this section, we study the relation between the N il_{l} filtration and the Eilenberg-Moore spectral sequence using the methods of Schwartz [15]. Our Theorem 3.1 can fairly easily be derived directly from [15, 16], but since we need the results under slightly weaker assumptions than the ones used in [15, 16], and think that insight is gained from a direct proof, we give one.

Define the nilpotency degree of an unstable module M as the largest l, such that M ∈ \overline{N il}_{l}. If M is locally finite we say that the nilpotency degree of M is infinite. We start by giving a proof of the key result about the nilpotency degree.

Theorem 3.1. Let X be a connected space with π_{1}X a finite p-group and H^{∗}X of finite type. For any l ≥ 0, QH^{∗}X ∈ \overline{N il}_{l+1} if and only if H^{∗}ΩX ∈ \overline{N il}_{l}.

Proof. Since we assume that π_{1}X is a finite p-group, it acts nilpotently on the F_{p}-vector space H^{∗}ΩX, since the group-ring F_{p}π_{1}X has nilpotent augmentation ideal. Hence the Eilenberg-Moore spectral sequence for the path-loop fibration over X converges strongly to H^{∗}ΩX.

For an unstable algebra K we have by [16, Lemma 8.7.6] that

QK ∈ \overline{N il}_{n} implies that Tor_{K}^{s}(F_{p}, F_{p}) ∈ \overline{N il}_{n−s−1}.  (3.1)

Assume that QH^{∗}X ∈ \overline{N il}_{l+1}. This implies that Tor_{H^{∗}X}^{s}(F_{p}, F_{p}) ∈ \overline{N il}_{l−s}, so Σ^{−s}F_{s}/F_{s+1} ∈ \overline{N il}_{l−s}. Therefore F_{s}/F_{s+1} ∈ \overline{N il}_{l} for all s which gives H^{∗}ΩX ∈ \overline{N il}_{l}.

To prove the other direction assume that H^{∗}ΩX ∈ \overline{N il}_{l}, so ΣH̄^{∗}ΩX ∈ \overline{N il}_{l+1}. If QH^{∗}X ∈ B we are done; else let n be the nilpotency degree of QH^{∗}X. Consider the canonical map

QH^{∗}X = E_{2}^{−1,∗} → E_{∞}^{−1,∗} ↪ ΣH̄^{∗}ΩX

We claim that this map has kernel in \overline{N il}_{n+1}. To see this, first note that since QH^{∗}X ∈ \overline{N il}_{n} we have that E_{r}^{−r−1} ∈ \overline{N il}_{n+r} by (3.1). Therefore the image of d_{r} : E_{r}^{−r−1,∗} → E_{r}^{−1,∗} is in \overline{N il}_{n+1} by the fact that d_{r} is A-linear of degree −(r − 1), and this establishes the claim. Since \overline{N il}_{l} is a Serre class we get that n ≥ min{n + 1, l + 1}, so n ≥ l + 1 as wanted.

Corollary 3.2. Let X be a connected space with π_{1}X a finite p-group and H^{∗}X of finite type. We have that QH^{∗}X is locally finite if and only if H^{∗}ΩX is locally finite. □

Remark 3.3. Note that one of the main results in [5] can be formulated as saying that if H^{∗}ΩX is finite over F_{p} then QH^{∗}X is finite over F_{p}—here the converse is however far from being true.

Theorem 3.1 makes it desirable to understand the relationship between the nilpotency degree of K and QK for an unstable algebra K. In general the nilpotency degree can differ radically if K is not in \overline{N il}_{1}, for example for any finite group G whose order is divisible by p, QH^{∗}BG ∈ B but H^{∗}BG ∉ \overline{N il}_{1}. However the next proposition shows that this is the only thing that can go wrong.

Proposition 3.4. Let K be a connected unstable algebra and assume that K ∈ \overline{N il}_{1}. Then K ∈ \overline{N il}_{l} if and only if QK ∈ \overline{N il}_{l}.

Proof. It is clear that QK ∈ \overline{N il}_{l} if K ∈ \overline{N il}_{l}. To see the converse suppose that QK ∈ \overline{N il}_{l}. The statement is trivially true for l = 0, 1, so suppose that l ≥ 2 and assume by induction that the statement is true for l − 1.

It is straight forward to see that K̄^{0} = 0 and K̄ ∈ N il_{l−1} ensures that K̄ ⊗ K̄ ∈ \overline{N il}_{l} (cf. [15, Cor. 1.9]). (Here K̄ denotes the augmentation ideal of K.) Hence both QK and K̄ ⊗ K̄ lie in \overline{N il}_{l}, so K̄ ∈ \overline{N il}_{l} as well, by the defining sequence of QK and the fact that \overline{N il}_{l} is a Serre class. This shows that K ∈ \overline{N il}_{l} as wanted.

4. Generalizations of Serre’s theorem

In this section we use the results of the preceding section to prove the promised generalizations of Serre’s theorem on the homotopy groups of a finite complex.

Before doing this, however, we prove a general result which says that the transcendence degree of the cohomology ring decreases when passing to covers. To get the result in its best form we use the Sullivan p-adic completion (see [13]), which coincides with the Bousfield-Kan F_{p}-completion on spaces with mod p cohomology of finite type [13, 3.4]. (We will actually only need the result for spaces with mod p cohomology of finite type, where references to [13] can be replaced by references to [9].)

Theorem 4.1. Let X be a Sullivan p-adically complete space. Then the transcendence degree of H^{∗}X⟨1⟩ is less than or equal to the transcendence degree of H^{∗}X.

Proof. First note that we may assume that X is connected. Recall that, by the work of Morel [13], for any Sullivan p-adically complete space X and any elementary abelian p-group V [BV, X] = Hom_{K}(H^{∗}X, H^{∗}BV), where [−, −] denotes free homotopy classes of maps and Hom_{K} denotes Hom in the category of unstable algebras over the Steenrod algebra. By [8] the transcendence degree of H^{∗}X is equal to the transcendence degree of the functor [B−, X] = Hom_{K}(H^{∗}X, H^{∗}B−) from elementary abelian p-groups to profinite sets. One definition of the transcendence degree of the functor [B−, X] is the rank of the largest elementary abelian p-group V such that there exists s ∈ [BV, X] which cannot be written as s = s′ ∘ Bφ, where φ ∈ End V is singular (see [8],[6, Prop. 5.8]). Since [BV, X] is obtained from [BV, X]_{pt} by taking the quotient under the action of π_{1}X we especially see that [B−, X] and [B−, X]_{pt} have the same transcendence degree.

The principal fibration π_{1}X → X⟨1⟩ → X shows that the map [BV, X⟨1⟩]_{pt} → [BV, X]_{pt} is injective, so the transcendence degree of [B−, X⟨1⟩] is less than or equal to that of [B−, X]. Since X⟨1⟩ is again Sullivan p-adically complete (see [13, 1.3,1.4]) we conclude that the transcendence degree of H^{∗}X⟨1⟩ is less than or equal to that of H^{∗}X.

Theorem 4.2. Assume that X is a connected finite Postnikov system, with π_{1}X a finite p-group and H^{∗}X of finite type. Then QH^{∗}X ∈ \overline{N il}_{2} if and only if X is F_{p}-equivalent to the classifying space of a p-compact toral group.

Proof. If X is F_{p}-equivalent to a the classifying space of a p-compact toral group, then ΩX is F_{p}-equivalent to a disjoint union of circles, so especially H^{∗}ΩX ∈ \overline{N il}_{1}, so QH^{∗}X ∈ \overline{N il}_{2} by Theorem 3.1.

To see the converse, assume that QH^{∗}X ∈ \overline{N il}_{2}. Since π_{1}X is a finite pgroup the Bousfield-Kan F_{p}-completion of X is F_{p}-complete and is again a finite Postnikov system by [2, II 5.1], so we may assume that X is F_{p}-complete. Hence ΩX is also F_{p}-complete [2] [5, 11.9]. By for example the Eilenberg-Moore spectral sequence, H^{∗}ΩX is also of finite type so both X and ΩX are Sullivan p-adically complete. By Theorem 3.1 H^{∗}(ΩX) ∈ \overline{N il}_{1}, so Theorem 4.1 implies that also H^{∗}((ΩX)⟨1⟩) ∈ \overline{N il}_{1}. Likewise, as above, H^{∗}((ΩX)⟨1⟩) is of finite type.

We therefore have that (ΩX)⟨1⟩ is a one-connected finite Postnikov system with cohomology in \overline{N il}_{1} and of finite type, which by [6, Thm. 1.1] implies that H̄^{∗}((ΩX)⟨1⟩) = 0. So X is homotopy equivalent to its second Postnikov stage P_{2}X, since both spaces are F_{p}-complete. Since H^{∗}ΩP_{2}X ∈ \overline{N il}_{1}, the abelian group π_{2}X cannot have p-torsion, as this would imply the existence of an element of infinite height in the mod p reduced cohomology ring of Ω_{0}P_{2}X = K(π_{2}X, 1) by standard group cohomology (or [9]). (Ω_{0} denotes the zero component of the loop space.) Hence X is an F_{p}-complete space with homotopy only in dimensions 1 and 2, and π_{2}X p-torsion free. But this means that π_{2}X is a torsion free Ext-p complete abelian group and hence isomorphic to Ẑ_{p}^{n} for some n (cf. [2, p. 181],[7]). Since H^{∗}X is of finite type we have n < ∞. This completes the proof.

Combining Theorem 4.2 with the results in [6] now enables us to give a proof of the main Theorem 1.1.

Proof of Theorem 1.1. By [5, Prop. 6.9, 12.1] 4. implies 3. Condition 3. obviously implies 1. and 2., and the implication 1. implies 4. follows from Theorem 4.2. The remaining 2. implies 4. follows easily from [6]. Namely, assume that the transcendence degree of H^{∗}X is finite. We can assume that X is F_{p}-complete. By Theorem 4.1, H^{∗}X⟨1⟩ likewise has finite transcendence degree. Now, X⟨1⟩ is an F_{p}-complete one connected finite Postnikov system with H^{∗}X⟨1⟩ of finite type, which by [6, Thm. 1.2] means that X⟨1⟩ is homotopy equivalent to K(Ẑ_{p}^{n}, 2) for some n < ∞. Hence X is F_{p}-equivalent to the classifying space of a p-compact toral group.

Remark 4.3. Note how 1. and 2. are complementary, in the sense that 2. is a statement about the ring structure of H^{∗}X, whereas 1. is a statement about the Steenrod algebra action on what is left when you kill all products.

Using the ‘McGibbon-Neisendorfer trick’, which basically says that an arbitrary space cannot have locally finite cohomology and have infinitely many nontrivial homotopy groups which are all p-torsion free, we can now easily strengthen the result of Dwyer and Wilkerson [4]. We show that the 2-connected assumption on their result was only necessary to exclude classifying spaces of p-compact toral groups.

Theorem 4.4. Assume that X is a connected space, with π_{1}X a finite p-group and H^{∗}X of finite type. If QH^{∗}X is locally finite as a module over the Steenrod algebra then X is either F_{p}-equivalent to the classifying space of a p-compact toral group, or it contains p-torsion in infinitely many of its homotopy groups.

Proof. Assume that X is not F_{p}-equivalent to the classifying space of a p-compact toral group. By Theorem 4.2, X cannot be F_{p}-equivalent to a finite Postnikov system, in other words π_{i}(X; Z/p) ≠ 0 for infinitely many i. But H^{∗}ΩX is also locally finite by Corollary 3.2 which by [12, p. 255] implies that π_{i}X actually has p-torsion for infinitely many i.

Remark 4.5. The above theorem is not a true generalization of Theorem 4.2 for a good reason. The assumptions in Theorem 4.4 cannot be weakened to QH^{∗}X ∈ \overline{N il}_{2} as is demonstrated by setting X = BSU, the classifying space of the infinite special unitary group. Indeed, H^{∗}SU ∈ \overline{N il}_{1}, so QH^{∗}BSU ∈ \overline{N il}_{2} by Theorem 3.1. But BSU is obviously neither the classifying space of a p-compact toral group nor does it contain any p-torsion in its homotopy groups by Bott periodicity. (See also [11].)

References

[1] A. K. Bousfield. The localization of spaces with respect to homology. Topology, 14:133–150, 1975.

[2] A. K. Bousfield and D. M. Kan. Homotopy Limits, Completions and Localization. Springer, 1972.

[3] W. G. Dwyer. Strong convergence of the Eilenberg-Moore spectral sequence. Topology, 13:255–265, 1974.

[4] W. G. Dwyer and C. W. Wilkerson. Spaces of null homotopic maps. Astérisque, 191:97–108, 1990. International Conference on Homotopy Theory (Marseille-Luminy, 1988).

[5] W. G. Dwyer and C. W. Wilkerson. Homotopy fixed point methods for Lie groups and finite loop spaces. Ann. of Math., 139:395–442, 1994.

[6] J. Grodal. The transcendence degree of the mod p cohomology of finite Postnikov systems. In Stable and unstable homotopy (Toronto, ON, 1996), pages 111–130. Amer. Math. Soc., Providence, RI, 1998.

[7] D. K. Harrison. Infinite abelian groups and homological methods. Ann. of Math., 69:366–391, 1959.

[8] H.-W. Henn, J. Lannes, and L. Schwartz. The categories of unstable modules and unstable algebras over the Steenrod algebra modulo nilpotent objects. Amer. J. Math., 115:1053–1106, 1993.

[9] J. Lannes. Sur les espaces fonctionnels dont la source est le classifiant d’un p-groupe abélien élémentaire. Inst. Hautes Études Sci. Publ. Math., 75:135–244, 1992.

[10] J. Lannes and L. Schwartz. A propos de conjectures de Serre et Sullivan. Invent. Math., 83:593–603, 1986.

[11] J. Lannes and L. Schwartz. Sur les groupes d’homotopie des espaces dont la cohomologie modulo 2 est nilpotente. Israel J. Math., 66:260–273, 1989.

[12] C. A. McGibbon and J. Neisendorfer. On the homotopy groups of a finite dimensional space. Comment. Math. Helv., 59:253–257, 1984.

[13] F. Morel. Quelques remarques sur la cohomologie modulo p continue des pro-p-espaces et les résultats de J. Lannes concernant les espaces fonctionnels hom(BV, X). Ann. Sci. École Norm. Sup. (4), 26(3):309–360, 1993.

[14] D. Rector. Steenrod operations in the Eilenberg-Moore spectral sequence. Comment. Math. Helv., 45:540–552, 1970.

[15] L. Schwartz. La filtration nilpotente de la categorie U et la cohomologie des espaces de lacets. In Algebraic topology—rational homotopy (Louvain-la-Neuve, 1986), volume 1318 of Lecture Notes in Math., pages 208–218. Springer, 1988.

[16] L. Schwartz. Unstable Modules over the Steenrod Algebra and Sullivan’s fixed Point Set Conjecture. The University of Chicago Press, 1994.

[17] J.-P. Serre. Cohomologie modulo 2 des complexes d’Eilenberg-Mac Lane. Comment. Math. Helv., 27:198–232, 1953.

[18] B. Shipley. On Serre’s conjecture. 1992 MIT Thesis proposal.

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

E-mail address: jg@math.mit.edu