HOMOTOPY EXPONENTS FOR LARGE H-SPACES

WOJCIECH CHACHÓLSKI, WOLFGANG PITSCH, JÉRÔME SCHERER, AND DON STANLEY

Abstract. We show that H-spaces with finitely generated cohomology, as an algebra or as an algebra over the Steenrod algebra, have homotopy exponents at all primes. This provides a positive answer to a question of Stanley.

Introduction

A simply connected space is elliptic if both its rational homotopy and rational homology are finite. Moore’s conjecture, see for example [9], predicts that elliptic complexes have an exponent at any prime p, meaning that there is a bound on the p-torsion in the graded group of all homotopy groups. Any finite H-space is known to be elliptic as it is rationally equivalent to a finite product of (odd dimensional) spheres. Relying on results by James [6] and Toda [11] about the homotopy groups of spheres, the fourth author (re)proved in [10] Long’s result that finite H-spaces have an exponent at any prime [7]. He proved in fact a stronger result which holds for example for Hspaces for which the mod p cohomology is finite. He also asked whether this would hold for finitely generated cohomology rings. The aim of this note is to give a positive answer to this question and provide a way larger class of H-spaces which have homotopy exponents.

Theorem 1.2 Let X be a connected and p-complete H-space such that H^{∗}(X; 𝔽_{p}) is finitely generated as an algebra over the Steenrod algebra. Then X has an exponent at p.

This class of H-spaces is optimal in the sense that H-spaces with a larger mod p cohomology, such as an infinite product of Eilenberg-Mac Lane spaces K(ℤ/p^{n}, n), will not have in general an exponent at p. As a corollary, we obtain the desired result. In fact we obtain the following global theorem.

Theorem 1.4 Let X be a connected H-space such that H^{∗}(X; ℤ) is finitely generated as an algebra. Then X has an exponent at each prime p.

The methods we use are based on the deconstruction techniques of the third author in his joint work with Castellana and Crespo, [3]. Our results on homotopy exponents should also be compared

2000 Mathematics Subject Classification. Primary 55P45; Secondary 55Q99, 55S10, 55U20.

The first author is supported in part by NSF grant DMS-0296117, Vetenskapsrådet grant 2001-4296, and Göran Gustafssons Stiftelse. The second and third authors are partially supported by FEDER/MEC grant MTM2007-61545. The second author is supported by the program Ramón y Cajal, MEC, Spain.

2 WOJCIECH CHACHÓLSKI, WOLFGANG PITSCH, JÉRÔME SCHERER, AND DON STANLEY

with the computations of homological exponents done with Clément, [4]. Whereas such H-spaces always have homotopy exponents, they almost never have homological exponents. The only simply connected H-spaces for which the 2-torsion in H_{∗}(X; ℤ) has a bound are products of mod 2 finite H-spaces with copies of the infinite complex projective space ℂP^{∞} and K(ℤ, 3).

Acknowledgments. This project originated during a workshop at the CRM, Barcelona, held in the emphasis year on algebraic topology (2007-08). We would like to thank the organizers for making it possible to meet in such a pleasant atmosphere.

1. Homotopy exponents

Our starting point is the fact that mod p finite H-spaces have always homotopy exponents. The following is a variant of Stanley’s [10, Corollary 2.9]. Whereas he focused on spaces localized at a prime, we will stick to p-completion in the sense of Bousfield and Kan, [2]. Since the p-localization map X → X_{(p)} is a mod p homology equivalence, his result implies the following.

Proposition 1.1 (Stanley). Let p be a prime and X be a p-complete and connected H-space such that H^{∗}(X; 𝔽_{p}) is finite. Then X has an exponent at p.

We will not repeat the proof, but let us sketch the main steps. Let us consider a decomposition of X by p-complete cells, i.e. X is obtained by attaching cones along maps from (S^{n})_{p}^{∧}. The natural map X → ΩΣX factors then through the loop spaces on a wedge W of a finite numbers of such p-completed spheres, up to multiplying by some integer N: the composite X → ΩΣX ⟶^{N} ΩΣX is homotopic to X → ΩW → ΩΣX. The proof goes by induction on the number of p-complete cells and the key ingredient here is Hilton’s description of the loop space on a wedge of spheres, [5]. Note that the suspension of a map between spheres is torsion except for the multiples of the identity. This idea to “split off” all the cells of X up to multiplication by some integer is dual to Arlettaz’ way to split off Eilenberg-Mac Lane spaces in H-spaces with finite order k-invariants, [1, Section 7]. The final step relies on the classical results by James, [6], and Toda, [11], that spheres do have homotopy exponents at all primes.

Theorem 1.2. Let X be a connected and p-complete H-space such that H^{∗}(X; 𝔽_{p}) is finitely generated as an algebra over the Steenrod algebra. Then X has an exponent at p.

Proof. A connected H-space such that H^{∗}(X; 𝔽_{p}) is finitely generated as an algebra over the Steenrod algebra can always be seen as the total space of an H-fibration F → X → Y where Y is an H-space with finite mod p cohomology and F is a p-torsion Postnikov piece whose homotopy groups are finite direct sums of copies of cyclic groups ℤ/p^{r} and Prüfer groups ℤ_{p^{∞}}, [3, Theorem 7.3]. This is a fibration of H-spaces and H-maps, so that we obtain another fibration F_{p}^{∧} → X → Y_{p}^{∧} by p-completing it. The base space Y_{p}^{∧} now satisfies the assumptions of Proposition 1.1. It has therefore an exponent at p. The homotopy groups of the fiber F_{p}^{∧} are finite direct sums of cyclic

HOMOTOPY EXPONENTS FOR LARGE H-SPACES 3

groups ℤ/p^{n} and copies of the p-adic integers ℤ_{p}^{∧}. Thus F_{p}^{∧} has an exponent at p as well. The homotopy long exact sequence of the fibration allows us to conclude.

We see here how the p-completeness assumption plays an important role. The space K(ℤ_{p^{∞}}, 1) for example has obviously no exponent at p, but its p-completion is K(ℤ_{p}^{∧}, 2) = (ℂP^{∞})_{p}^{∧}, which is a torsion free space. The mod p cohomology of K(ℤ_{p^{∞}}, 1) is a polynomial ring on one generator in degree 2, we must thus also work with p-complete spaces to give an answer to Stanley’s question [10, Question 2.10].

Corollary 1.3. Let X be a connected and p-complete H-space such that H^{∗}(X; 𝔽_{p}) is finitely generated as an algebra. Then X has an exponent at p.

In fact, when the mod p cohomology is finitely generated, the fiber F in the fibration described in the proof of Theorem 1.2 is a single Eilenberg-Mac Lane space K(P, 1). Thus the typical example of an H-space with finitely generated mod p cohomology is the 3-connected cover of a simply connected finite H-space (P is ℤ_{p^{∞}} in this case). Likewise, the typical example in Theorem 1.2 are highly connected covers of finite H-spaces. This explains why such spaces have homotopy exponents!

If one does not wish to work at one prime at a time and prefers to find a global condition which permits to conclude that a certain class of spaces have exponents at all primes, one must replace mod p cohomology by integral cohomology.

Theorem 1.4. Let X be a connected H-space such that H^{∗}(X; ℤ) is finitely generated as an algebra. Then X has an exponent at each prime p.

Proof. Since the integral cohomology groups are finitely generated it follows from the universal coefficient exact sequence (see [8]) that the integral homology groups are also finitely generated. Since X is an H-space we may use a standard Serre class argument to conclude that so are the homotopy groups. Therefore the p-completion map X → X_{p}^{∧} induces an isomorphim on the ptorsion at the level of homotopy groups. The theorem is now a direct consequence of the next lemma.

Lemma 1.5. Let X be a connected space. If H^{∗}(X; ℤ) is finitely generated as an algebra, then so is H^{∗}(X; 𝔽_{p}).

Proof. Let u_{1}, …, u_{r} generate H^{∗}(X; ℤ) as an algebra. Consider the universal coefficients short exact sequences

0 → H^{n}(X; ℤ) ⊗ ℤ/p ⟶ H^{n}(X; 𝔽_{p}) ⟶^{∂} Tor(H^{n+1}(X; ℤ); ℤ/p) → 0 .

Since H^{∗}(X; ℤ) is finitely generated as an algebra it is degree-wise finitely generated as a group and therefore Tor(H^{∗}(X; ℤ); ℤ/p) can be identified with the ideal of elements of order p in H^{∗}(X; ℤ).

This ideal must be finitely generated since H^{∗}(X; ℤ) is Noetherian. Choose generators a_{1}, …, a_{s}. Each a_{i} corresponds to a pair α_{i}, βα_{i} in H^{∗}(X; 𝔽_{p}), where β denotes the Bockstein.

We claim that the elements α_{1}, …, α_{s} together with the mod p reduction of the algebra generators, denoted by ū_{1}, …, ū_{r}, generate H^{∗}(X; 𝔽_{p}) as an algebra. Let x ∈ H^{∗}(X; 𝔽_{p}) and write its image ∂(x) = ∑ λ_{j}a_{j} with λ_{j} = λ_{j}(u) a polynomial in the u_{i}’s. Define now λ̄_{j} = λ_{j}(ū) ∈ H^{∗}(X; 𝔽_{p}) to be the corresponding polynomial in the ū_{i}’s. As the action of H^{∗}(X; ℤ) on the ideal Tor(H^{∗}(X; ℤ); ℤ/p) factors through the mod p reduction map H^{∗}(X; ℤ) → H^{∗}(X; 𝔽_{p}), the element x − ∑ λ̄_{j}α_{j} belongs to the kernel of ∂, i.e. it lives in the image of the mod p reduction. It can be written therefore as a polynomial μ̄ in the ū_{i}’s. Thus x = μ̄ + ∑ λ̄_{j}α_{j}.

References

1. D. Arlettaz, Algebraic K-theory of rings from a topological viewpoint, Publ. Mat. 44 (2000), no. 1, 3–84.

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

3. N. Castellana, J. A. Crespo, and J. Scherer, Deconstructing Hopf spaces, Invent. Math. 167 (2007), no. 1, 1–18.

4. A. Clément and J. Scherer, Homology exponents for h-spaces, Rev. Iberoam. Mat. (2009).

5. P. J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc. 30 (1955), 154–172.

6. I. M. James, The suspension triad of a sphere, Ann. of Math. (2) 63 (1956), 407–429.

7. J. Long, Thesis, Princeton university, Ph.D. thesis, 1978.

8. W. J. R. Mitchell, Dual modules, Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 1, 21–24.

9. P. Selick, Moore conjectures, Algebraic topology—rational homotopy (Louvain-la-Neuve, 1986), Lecture Notes in Math., vol. 1318, Springer, Berlin, 1988, pp. 219–227.

10. D. Stanley, Exponents and suspension, Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 1, 109–116.

11. H. Toda, On the double suspension E^{2}, J. Inst. Polytech. Osaka City Univ. Ser. A. 7 (1956), 103–145.

K.T.H., Matematik, S-10044 Stockholm, Sweden

  E-mail address: wojtek@math.kth.se

Universitat Autònoma de Barcelona, Departament de Matemàtiques, E-08193 Bellaterra, Spain

  E-mail address: pitsch@mat.uab.es, jscherer@mat.uab.es

Department of Mathematics and Statistics, University of Regina, College West 307.14, Regina, Saskatchewan, Canada S4S 0A2

  E-mail address: Donald.Stanley@uregina.ca