ON THE COHOMOLOGY OF GENERALIZED HOMOGENEOUS SPACES

J.P. MAY AND F. NEUMANN

Abstract. We observe that work of Gugenheim and May on the cohomology of classical homogeneous spaces G/H of Lie groups applies verbatim to the calculation of the cohomology of generalized homogeneous spaces G/H, where G is a finite loop space or a p-compact group and H is a “subgroup” in the homotopical sense.

We are interested in the cohomology H^{∗}(G/H; R) of a generalized homogeneous space G/H with coefficients in a commutative Noetherian ring R. Here G is a “finite loop space” and H is a “subgroup”. More precisely, G and H are homotopy equivalent to ΩBG and ΩBH for path connected spaces BG and BH, and G/H is the homotopy fiber of a based map f : BH ⟶ BG. We always assume this much, and we add further hypotheses as needed. Such a framework of generalized homogeneous spaces was first introduced by Rector [10], and a more recent framework of p-compact groups has been introduced and studied extensively by Dwyer and Wilkerson [4] and others.

We ask the following question: How similar is the calculation of H^{∗}(G/H; R) to the calculation of the cohomology of classical homogeneous spaces of compact Lie groups? When R = 𝔽_{p} and H is of maximal rank in G, in the sense that H^{∗}(H; ℚ) and H^{∗}(G; ℚ) are exterior algebras on the same number of generators, the second author has studied the question in [8, 9]. There, the fact that H^{∗}(BG; R) need not be a polynomial algebra is confronted and results similar to the classical theorems of Borel and Bott [2, 3] are nevertheless proven. The purpose of this note is to begin to answer the general question without the maximal rank hypothesis, but under the hypothesis that H^{∗}(BG; R) and H^{∗}(BH; R) are polynomial algebras.

In fact, we shall not do any new mathematics. Rather, we shall merely point out that work of the first author [7] that was done before the general context was introduced goes far towards answering the question. Essentially the following theorem was announced in [7] and proven in [5]. We give a brief sketch of its proof and then return to a discussion of its applicability to the question on hand. Let BT^{n} be a classifying space of an n-torus T^{n}.

Theorem 1. Assume the following hypotheses.

(i) π_{1}(BG) acts trivially on H^{∗}(G/H; R). (ii) R is a PID and H_{∗}(BG; R) is of finite type over R. (iii) H^{∗}(BG; R) is a polynomial algebra. (iv) There is a map e : BT^{n} ⟶ BH such that H^{∗}(BT^{n}; R) is a free H^{∗}(BH; R)-

module via e^{∗}.

Date: April 20, 2000.

1991 Mathematics Subject Classification. Primary 55P35, 57T35; Secondary 57T15.

The first author was partially supported by the NSF.

Then H^{∗}(G/H; R) is isomorphic as an R-module to Tor_{H^{∗}(BG;R)}(R, H^{∗}(BH; R)), regraded by total degree. Moreover, there is a filtration on H^{∗}(G/H; R) such that its associated bigraded R-algebra is isomorphic to Tor_{H^{∗}(BG;R)}(R, H^{∗}(BH; R)).

Proof. The first two hypotheses ensure that H^{∗}(G/H; R) is isomorphic to the differential torsion product Tor_{C^{∗}(BG;R)}(R, C^{∗}(BH; R)). See, for example, [5, p. 21-25]. The second hypothesis allows Lemma 3.2 there to be applied with ℤ replaced by R, thus allowing the finite type over ℤ hypothesis assumed there to be replaced by the finite type over R hypothesis assumed here. Therefore there is an Eilenberg-Moore spectral sequence that converges from Tor_{H^{∗}(BG;R)}(R, H^{∗}(BH; R)) to H^{∗}(G/H; R). The conclusion of the theorem is that this spectral sequence collapses at E_{2} with trivial additive extensions, but not necessarily trivial multiplicative extensions. The last hypothesis and a comparison of spectral sequences argument essentially due to Baum [1] shows that the conclusion holds in general if it holds when BH = BT^{n}. See [5, p. 37-38]. Here the strange result [5, 4.1] gives that there is a morphism

g : C^{∗}(BT^{n}; R) ⟶ H^{∗}(BT^{n}; R)

of differential algebras such that g induces the identity map on cohomology and annihilates all ∪_{1}-products.

Now the general theory of differential torsion products of [5] kicks in. In modern language, implicit in the discussion of [6, p. 70], there is a model category structure on the category of A-modules for any DGA A over R such that every right A-module M admits a cofibrant approximation of a very precise sort. Namely, for any HA-free resolution X ⊗_{R} HA ⟶ HM of HM, there is a cofibrant approximation P = X ⊗_{R} A ⟶ M. Grading is made precise in the cited sources. The essential point is that P is not a bicomplex but rather has differential with many components. When HA is a polynomial algebra and M = R, we can take X to be an exterior algebra with one generator for each polynomial generator of HA. Here, asssuming that A has a ∪_{1}-product that satisfies the Hirsch formula (∪_{1} is a graded derivation), [5, 2.2] specifies the required differential explicitly in terms of ∪_{1}- products. Using g to replace C^{∗}(BT^{n}; R) by H^{∗}(BT^{n}; R) in our differential torsion product, we see that the differential torsion product Tor_{C^{∗}(BG;R)}(R, H^{∗}(BT^{n}; R)) is computed by exactly the same chain complex as the ordinary torsion product Tor_{H^{∗}(BG;R)}(R, H^{∗}(BT^{n}; R)). See [5, 2.3]. The conclusion follows.

Hypotheses (i) and (ii) in the theorem are reasonable and not very restrictive. Hypothesis (iii) is intrinsic to the method at hand. Note that H^{∗}(BG; R) can have infinitely many polynomial generators, so that G need not be finite. The key hypothesis is (iv). Here the following homotopical version of a theorem of Borel is relevant. It was first noticed by Rector [10, 2.2] that Baum’s proof [1] of Borel’s theorem is purely homotopical. A generalized variant of Baum’s proof is given in [5, p. 40-42]. That proof applies directly to give the following theorem. We state it for H and G as in the first paragraph. However, we are interested in its applicability to T^{n} and H in Theorem 1, and we restate it as a corollary in that special case.

Theorem 2. Let R be a field and assume the following hypotheses.

(i) π_{1}(BG) acts trivially on H^{∗}(G/H; R). (ii) H^{∗}(BH; R) and H^{∗}(BG; R) are polynomial algebras on the same finite num-

ber of generators. (iii) H^{∗}(G/H; R) is a finite dimensional R-module.

Then H^{∗}(G/H; R) ≅ R ⊗_{H^{∗}(BG;R)} H^{∗}(BH; R) as an algebra and

H^{∗}(BH; R) ≅ H^{∗}(BG; R) ⊗_{R} H^{∗}(G/H; R)

as a left H^{∗}(BG; R)-module. In particular, H^{∗}(BH; R) is H^{∗}(BG; R)-free.

Corollary 3. Let R be a field and assume given a map e : BT^{n} ⟶ BH that satisfies the following properties, where H/T^{n} is the fiber of e.

(i) π_{1}(BH) acts trivially on H^{∗}(H/T^{n}; R). (ii) H^{∗}(BH; R) is a polynomial algebra on n generators. (iii) H^{∗}(H/T^{n}; R) is a finite dimensional R-module. Then H^{∗}(H/T^{n}; R) ≅ R ⊗_{H^{∗}(BH;R)} H^{∗}(BT^{n}; R) as an algebra and

H^{∗}(BT^{n}; R) ≅ H^{∗}(BH; R) ⊗_{R} H^{∗}(H/T^{n}; R)

as a left H^{∗}(BH; R)-module. In particular, H^{∗}(BT^{n}; R) is H^{∗}(BH; R)-free.

When Corollary 3 applies, its conclusion gives hypothesis (iv) of Theorem 1. We comment briefly on applications to the integral and p-compact settings for the study of generalized homogeneous spaces.

Remark 4. A counterexample of Rector [10] shows that not all finite loop spaces H have (integral) maximal tori. When H does have a maximal torus, hypothesis (iii) of the Corollary holds by definition. Assuming that H is simply connected, [9, 3.11] describes for which primes p H^{∗}(BH; ℤ) is p-torsion free, so that H^{∗}(BH; 𝔽_{p}) is a polynomial algebra. If R is the localization of ℤ at the primes p for which H^{∗}(H; ℤ) is p-torsion free, then H^{∗}(BH; R) is also a polynomial algebra, and H^{∗}(BT; R) is a free H^{∗}(BH; R)-module. That is, hypothesis (iv) of Theorem 1 holds for the localization of ℤ away from the finitely many “bad primes” for which H^{∗}(BH; 𝔽_{p}) is not a polynomial algebra on n generators.

Remark 5. In the p-compact setting, taking R = 𝔽_{p}, Dwyer and Wilkerson [4, 8.13, 9.7] prove that if H is connected, BH is 𝔽_{p}-complete, H^{∗}(H; 𝔽_{p}) is finite dimensional, and H^{∗}(H; ℤ_{p}) ⊗_{ℤ_{p}} ℚ is an exterior algebra on n generators, then there is a map e : BT^{n} ⟶ BH such that H^{∗}(H/T^{n}; 𝔽_{p}) is finite dimensional. Here Corollary 3 applies whenever H^{∗}(BH; 𝔽_{p}) is a polynomial algebra on n generators.

References

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[8] F. Neumann. On the cohomology of homogeneous spaces of finite loop spaces and the Eilenberg-Moore spectral sequence. J. Pure and Applied Algebra 140(1999), 261-287.

[9] F. Neumann. Torsion in the cohomology of finite loop spaces and the Eilenberg-Moore spectral sequence. Topology and its Applications 100(2000), 133–150.

[10] D. Rector. Subgroups of finite dimensioanl topological groups. J. Pure Appl. Algebra 1(1971), 253–273.

Department of Mathematics, The University of Chicago, Chicago, IL 60637

E-mail address: may@math.uchicago.edu

Mathematisches Institut der Georg-August-Universität, Göttingen, Germany

E-mail address: neumann@uni-math.gwdg.de