TORSION IN THE COHOMOLOGY OF FINITE LOOP SPACES AND THE EILENBERG-MOORE SPECTRAL SEQUENCE

FRANK NEUMANN

Abstract. We use the Eilenberg-Moore spectral sequence to study torsion phenomena in the integral cohomology of finite loop spaces with maximal torus generalizing some results for compact Lie groups due to Kac.

1. Introduction

This paper is a sequel to our paper [19] on the cohomology of homogeneous spaces of finite loop spaces where we extend classical theorems of Borel and Bott for compact Lie groups to the category of finite loop spaces. This category was proposed by Rector [22] as an adequate setting for the study of compact Lie groups from a purely homotopy point of view. In modern terms using the language of Dwyer-Wilkerson [5] a loop space is a tripel X = (X, BX, e) consisting of a topological space X, a pointed topological space BX called the classifying space and a homotopy equivalence e : ΩBX ⟶^{≃} X. A morphism is just a pointed map Bf : BX → BY. And the homotopy fiber of Bf is called the homogeneous space X/Y of Bf . A topological space is finite if H^{∗}(X, ℤ) is a finitely generated graded abelian group. A subgroup Y of a connected finite loop space is just a morphism f : X → Y such that X/Y is finite. It has maximal rank if Y and X have the same number of exterior generators in their rational cohomology algebras. Very important is the case of a maximal torus. Rector showed in contrast with classical Lie group theory that not every finite loop space has a maximal torus and that this is a special rigid property. And it is actually a conjecture of Wilkerson [26] that any finite loop space with maximal torus is isomorphic to a compact Lie group, in the sense that their classifying spaces are homotopy equivalent. The category of finite loop spaces was intensively studied also by Møller and Notbohm in [17] and [18] where the analogy with the category of compact Lie groups was developed much further.

The purpose of this paper is to study the torsion in the integral cohomology of finite loop spaces with maximal torus along the lines of the fundamental paper of Kac [8] in which the same question was examined for compact Lie groups. In that paper Kac proposed a unified approach to the classical cohomological calculations of Cartan, Leray, Borel and others. Our main tool will be again the Eilenberg-Moore spectral sequence.

1991 Mathematics Subject Classification. 55 P 35, 55 T 10, 55 T 20, 57 T 25, 57 T 35 Key words and phrases. Eilenberg-Moore spectral sequence, finite loop spaces, classifying spaces, torsion

In [19] we studied the Eilenberg-Moore spectral sequence of the fibration X/T_{X} ⟶ BT_{X} ⟶ BX assuming that X is 1-connected by comparing it with a fibration X/T_{X} ⟶^{l} BT_{X} × L ⟶ BX̃ given as pullback of the original one. The space L is constructed in the same way as by Kane and Notbohm in [10] using properties of the Ω-spectrum associated with the first Johnson-Wilson Spektrum BP ⟨1⟩. The space BX̃ is the homotopy fiber of a certain map BX ⟶ BL. The cohomology algebras of BX̃ and L are polynomial. In order to construct this pullback fibration with the desired properties we needed some conditions on the module of indecomposable elements of the mod p cohomology algebra of X. Namely for a prime p we say that a connected, 1-connected finite loop space satisfies the condition Q_{p} if Q^{2n}H^{∗}(X, 𝔽_{p}) = 0 except the cases n = p + 1, p^{2} + 1 if p is odd or Q^{2n}ξH^{∗}(X, 𝔽_{2}) = 0 except the cases n = 2^{j} + 2 with j ≥ 2 if p = 2, where ξH^{∗}(X, 𝔽_{2}) is the image of the Frobenius morphism.

It is a conjecture of Lin that for odd primes p every connected, 1-connected 𝔽_{p}- finite H-space has this simple structure of indecomposable elements [15]. In the case of compact Lie groups this conjecture is true if p is odd and for p = 2 the only exceptions are the compact Lie groups Spin(n) for n ≥ 15 and E_{8} [10].

In [19] we proved for example that under these conditions on the module of indecomposable elements of H^{∗}(X, 𝔽_{p}) for a prime p the cohomology of the homgenous space X/T_{X} is concentrated in even degrees. This we will use now to study the torsion in the integral cohomology of H^{∗}(X, ℤ). We will first analyze the pullback diagram D and proof two theorems of Kac [8] for the fibration X ⟶^{π} X/T_{X} ⟶^{k} BT_{X}. Namely that ker k^{∗} is a Borel ideal generated by a regular sequence and that H^{∗}(X/T_{X}, 𝔽) is a free R^{∗}-module where R^{∗} is the algebra generated by im k^{∗}. This allows us to proof that the Eilenberg-Moore spectral sequence of the above fibration collapses in the E_{2}-term. As a corollary we get that the Leray-Serre spectral sequence of the fibration T_{X} ⟶^{i} X ⟶^{π} X/T_{X} collapses in the E_{3}-term as was stated for compact Lie groups by Kac for any field coefficients generalizing an old theorem of Leray [12] which establish this for rational coefficients. From this we are able in the same way as Kac to deduce several consequence for the existence of p-torsion in the integral cohomology H^{∗}(X, ℤ). The proofs given here are purely homotopic and do not rely on special cohomological calculations. Only the structure theory of Hopf algebras [16] and the results of Lin on the structure of indecomposable elements in the mod p cohomology are used here. At the end we examine the torsionindex of Demazure [4] and give some other equivalent conditions for the existence of p-torsion.

2. The Eilenberg-Moore spectral sequence and the theorems of Kac

We will work in the category of finite loop spaces with maximal torus which we introduce now using the terminology of Dwyer and Wilkerson

Definition 2.1. A loop space X = (X, BX, e) is a tripel consisting of a topological space X, a pointed topological space BX and a homotopy equivalence e : ΩBX ⟶^{≃} X. The space BX is called the classifying space of X. A loop space X is n-connected if X is n-connected. A morphism f : X → Y of loop spaces is a pointed continous map Bf : BX → BY. The homotopy fiber Y/X = hofib(Bf) of Bf over the basepoint BY is called the homogeneous space of Bf.

In general we will call a topological space X finite or ℤ-finite if the singular cohomology H^{∗}(X, ℤ) is a finitely generated ℤ-module. More generally a space X is called 𝔽_{p}-finite if H^{∗}(X, 𝔽_{p}) is a finite dimensional graded 𝔽_{p}-vector space. Any space X which is ℤ-finite is also 𝔽_{p}-finite for all primes p.

Definition 2.2. A morphism f : X → Y of loop spaces is an isomorphism if Bf is a homotopy equivalence. A morphism f : X → Y of loop spaces is a monomorphism if Y/X is ℤ-finite. In this case X is called a subgroup of Y. A morphism f : X → Y of loop spaces is an epimorphism if (Ω(Y/X), Y/X, id) is a loop space. A torus is a loop space T = (T, BT, e) with BT ≃ K(ℤ^{n}, 2). The rank rk(X) of a connected finite loop space X is the transcendence degree of the extension of the polynomial ring H^{∗}(BX, ℚ) over the field ℚ or in other words the number of generators in the polynomial algebra H^{∗}(BX, ℚ). A subgroup Y of a connected finite loop space X has maximal rank if rk(Y) = rk(X). If the subgroup Y is a torus, then Y is called a maximal torus of X.

Now let X be a connected, 1-connected finite loop space with maximal torus T_{X}. In [19] we studied the Eilenberg-Moore spectral sequence of the fibration X/T_{X} ⟶ BT_{X} ⟶ BX by comparing it with a fibration X/T_{X} ⟶^{l} BT_{X} × L ⟶ BX̃ given as a pullback of the original one. The space L is just given as the k-fold product of the space BP ⟨1⟩_{2p+2} of the Ω-spectrum associated with the first Johnson-Wilson Spektrum BP ⟨1⟩ where k is the rank of the free ℤ_{(p)}-module H^{∗}(X, ℤ_{(p)}). The space BX̃ is the homotopy fiber of a certain map BX ⟶ BL. In this pullbacked fibration we have the fundamental property that the cohomology algebras of BX̃ and L are polynomial, actually with infinitely many generators but in any fixed degree there is only a finite number of them. In order to construct the pullback fibration with the desired properties we needed some conditions on the module of indecomposable elements of the mod p cohomology algebra of X which we will recall now.

Notation. We will say that p is a prime if p ∈ ℙ ⋃{0} where ℙ is the set of primes and we let 𝔽 = 𝔽_{p} if p is a prime and 𝔽 = ℚ if p = 0.

We use this notation to cover also the case of rational coefficients ℚ. We also like to point out that what follows will work with any field coefficients, but it is enough to concentrate on the prime fields.

Definition 2.3. Let p be a prime and X be a connected, 1-connected 𝔽_{p}-finite H-space. We say that X satisfies the condition Q_{p} if Q^{2n}H^{∗}(X, 𝔽_{p}) = 0 except n = p + 1, p^{2} + 1 if p is odd or Q^{2n}ξH^{∗}(X, 𝔽_{2}) = 0 except n = 2^{k} + 2 with k ≥ 2 if p = 2. In the case p = 0 the condition Q_{p} is empty.

Any connected, 1-connected compact Lie group G satisfies the condition Q_{p} for any odd prime p. Actually this is also the case for any known connected, 1-connected 𝔽_{p}- finite H-space and it is a conjecture of Lin that any connected, 1-connected 𝔽_{p}-finite H-space satisfies the condition Q_{p} for odd primes p [15].

For p = 2 the condition is restrictive for compact Lie groups. But any connected, 1-connected compact Lie group G satisfies the condition Q_{2}, except in the cases G = Spin(n) for n ≥ 15 and G = E_{8} [10].

Let p be a prime and let us assume now that X is always a connected, 1-connected finite loop space with maximal torus T_{X} satisfying the condition Q_{p}. In [19] we constructed the following pullback diagram D

[figure: ∗; X/T_{X}; l; X/T_{X}; k; L; E ≃ BT_{X} × L; ϕ; μ; BT_{X}; Bi; L; BX̃; Bj; BX]

We then proved the following main theorem ([19], theorem 4.13) concerning the collapse of the Eilenberg-Moore spectral sequence

Theorem 2.4. Let p be a prime, X a connected, 1-connected finite loop space with maximal torus T_{X} satisfying the condition Q_{p}. Then the Eilenberg-Moore spectral sequence {E_{r}, d_{r}} of the fibration

X/T_{X} ⟶ E ⟶^{ϕ} BX̃

collapses in the E_{2}-term and there are isomorphisms of algebras (a) E_{2} ≅ E_{∞}

(b) E_{2} ≅ A/J where A = 𝔽[y_{1}, …, y_{k}] is a finitely generated subalgebra of H^{∗}(E, 𝔽) and the ideal J = (ϕ^{∗}(x_{1}), …, ϕ^{∗}(x_{k})) is a Borel ideal in A generated by the regular sequence ϕ^{∗}(x_{1}), …, ϕ^{∗}(x_{k}).

From this we derived also the following corollary ([19], corollary 4.14) about the Leray-Serre spectral sequence

Corollary 2.5. Let p be a prime, X a connected, 1-connected finite loop space with maximal torus T_{X} satisfying the condition Q_{p}. Then the Leray-Serre spectral sequence {E_{r}, d_{r}} of the fibration

X/T_{X} ⟶ E ⟶^{ϕ} BX̃

collapses in the E_{2}-term and there are isomorphisms of algebras (a) E_{2} ≅ E_{∞}

(b) E_{2} ≅ H^{∗}(BX̃, 𝔽) ⊗_{𝔽} H^{∗}(X/T_{X}, 𝔽) In addition, H^{∗}(E, 𝔽) is a free H^{∗}(BX̃, 𝔽)-module and as 𝔽-vector spaces

H^{∗}(E, 𝔽) ≅ H^{∗}(BX̃, 𝔽) ⊗_{𝔽} H^{∗}(X/T_{X}, 𝔽).

The fiber X/T_{X} is totally non-homologous to 0 with respect to the fibration ϕ.

Let p be a prime and let us assume now that X is always a connected, 1-connected finite loop space with maximal torus T_{X} satisfying the condition Q_{p}. We will now study the pullback diagram D and analyze the induced maps in 𝔽-cohomology. First we have the following important facts (see also [10]).

Proposition 2.6. The induced maps

ϕ^{∗} : H^{∗}(BX̃, 𝔽) ⟶ H^{∗}(E, 𝔽)
μ^{∗} : H^{∗}(BT_{X}, 𝔽) ⟶ H^{∗}(E, 𝔽)

are monomorphisms and ker(Bi)^{∗} = ker(Bj)^{∗}.

Proof. It is clear from the collapse of the Eilenberg-Moore spectral sequence of the fibration ϕ (see theorem 2.4) that the induced map ϕ^{∗} is a monomorphism. The result for the map μ^{∗} follows from the collapse of the Leray-Serre spectral sequence of the fibration

L ⟶ E ≃ BT_{X} × L ⟶ BT_{X}

in the E_{2}-term (see corollary 2.5). We have the following commutative diagram of fibrations

[figure: E; ϕ; μ; BT_{X}; Bi; BL; BX̃; Bj; BX; Bf; L]

and so

μ^{∗} ∘ (Bi)^{∗} = ϕ^{∗} ∘ (Bj)^{∗}

and the last assertion follows.

Proposition 2.7. The ideal ker l^{∗} is a Borel ideal.

Proof. From theorem 2.4 we know that the Eilenberg-Moore spectral sequence of the fibration

X/T_{X} ⟶^{l} E ⟶^{ϕ} BX̃

collapses in the E_{2}-term and from [1] corollary 5.2 it follows that the sequence

𝔽 ⟶ H^{∗}(BX̃, 𝔽) ⟶^{ϕ^{∗}} H^{∗}(E, 𝔽) ⟶^{l^{∗}} H^{∗}(X/T_{X}, 𝔽) ⟶ 𝔽

is coexact, i.e. ker l^{∗} = (\overline{im ϕ^{∗}}).

Again from theorem 2.4 we see that (\overline{im ϕ^{∗}}) is a Borel ideal generated by the regular sequence ϕ^{∗}(x_{1}), …, ϕ^{∗}(x_{k}), y_{k+1}, … where H^{∗}(E, 𝔽) ≅ 𝔽[y_{1}, …, y_{k}, y_{k+1}, …]. Therefore also ker l^{∗} is a Borel ideal.

Now we can derive two fundamental results concerning the fibration

X/T_{X} ⟶^{k} BT_{X} ⟶^{Bi} BX

which in the case of a compact Lie group are due to Kac [8]. We will give different and more homotopic proofs via comparision with the pullbacked fibration

X/T_{X} ⟶^{l} E ⟶^{ϕ} BX̃

and using propositions 2.6 and 2.7.

Theorem 2.8 (Kac). Let p be a prime and X a connected, 1-connected finite loop space with maximal torus T_{X} satisfying the condition Q_{p}. Then the Ideal

ker{k^{∗} : H^{∗}(BT_{X}, 𝔽) → H^{∗}(X/T_{X}, 𝔽)}

is a Borel ideal.

Proof. Following the process in [23] we choose a minimal ideal basis h_{1}, …, h_{r}, … of the ideal ker k^{∗}. After 2.6 the induced map μ^{∗} is a monomorphism and we can extend h_{1}, …, h_{r}, … to a minimal ideal basis of ker l^{∗}. The ideal ker l^{∗} is a Borel ideal by 2.7 and any minimal ideal basis therefore a regular sequence [23]. So the sequence h_{1}, …, h_{r}, … is a subsequence of a regular sequence and because H^{∗}(E, 𝔽) is a free H^{∗}(BT_{X}, 𝔽)-module we see that it is actually a regular sequence.

Kac [8] actually showed that in the case of compact Lie groups the Borel ideal ker k^{∗} is generated by the generalized invariants of the Weyl group. For a different proof of this fact and a careful investigation of rings of generalized invariants see also [20], [21]. It would be very interesting if this can also be generalized to finite loop spaces with maximal torus. Using explicit calculations in 𝔽-cohomology of compact Lie groups the fact that ker k^{∗} is a Borel ideal was proved earlier also by Toda [25].

Theorem 2.9 (Kac). Let p be a prime, X a connected, 1-connected finite loop space with maximal torus T_{X} satisfying the condition Q_{p} and R^{∗} be the subalgebra of H^{∗}(X/T_{X}, 𝔽) generated by

im{k^{∗} : H^{∗}(BT_{X}, 𝔽) → H^{∗}(X/T_{X}, 𝔽)}

Then H^{∗}(X/T_{X}, 𝔽) is a free R^{∗}-module.

Proof. In the pullback diagram D the induced maps μ^{∗} and ϕ^{∗} are monomorphisms after 2.6 and k^{∗} = l^{∗} ∘ μ^{∗}. From corollary 2.5 we know that H^{∗}(E, 𝔽) is a free H^{∗}(BX̃, 𝔽)-module and l^{∗} is an epimorphism. We have an isomorphism of graded 𝔽-vector spaces

H^{∗}(X/T_{X}, 𝔽) ≅ 𝔽 ⊗_{H^{∗}(BX̃,𝔽)} H^{∗}(E, 𝔽)

If we denote by P^{∗} the subalgebra of H^{∗}(X/T_{X}, 𝔽) generated by im l^{∗}, then we see that H^{∗}(X/T_{X}, 𝔽) is a free P^{∗}-module and again from proposition 2.6 we deduce that H^{∗}(X/T_{X}, 𝔽) is a free R^{∗}-module.

Kac proved this fact in [8], Appendix for compact Lie groups using ℤ-linear operators on the integral cohomology ring H^{∗}(G/T, ℤ) which where defined independly also by Demazure [4]. A somehow different construction of these operators in 𝔽cohomology can also be found in [21].

3. Chow Algebras and Torsion in Cohomology of Finite Loop Spaces

Let p be a prime and X a connected, 1-connected finite loop space with maximal torus T_{X} satisfying the condition Q_{p}. In this section we will study the integral cohomology H^{∗}(X, ℤ) and deduce criteria for the existence of p-torsion. Consider now the fibration

X ⟶^{π} X/T_{X} ⟶^{k} BT_{X}

and let {E_{r}, d_{r}} be the Eilenberg-Moore spectral sequence of this fibration over the field 𝔽

E_{2} ≅ Tor_{H^{∗}(BT_{X},𝔽)}(𝔽, H^{∗}(X/T_{X}, 𝔽))
E_{r} ⇒ H^{∗}(X, 𝔽)

We denote again by R^{∗} the subalgebra of H^{∗}(X/T_{X}, 𝔽) generated by

im{k^{∗} : H^{∗}(BT_{X}, 𝔽) → H^{∗}(X/T_{X}, 𝔽)}

and we let S^{∗} be the algebra defined as

S^{∗} = H^{∗}(X/T_{X}, 𝔽)//R^{∗}

Now we proof the main technical theorem of this section.

Theorem 3.1. Let p be a prime and X a connected, 1-connected finite loop space with maximal torus T_{X} satisfying the condition Q_{p}. Then the Eilenberg-Moore spectral sequence {E_{r}, d_{r}} of the fibration

X ⟶^{π} X/T_{X} ⟶^{k} BT_{X}

collapses in the E_{2}-term. Let f_{1}, …, f_{n} be a regular sequence generating the Borel ideal ker k^{∗}. Then as algebras

(a) E_{2} ≅ E_{∞}

(b) E_{2} ≅ S^{∗} ⊗_{𝔽} E(s^{−1}f_{1}, …, s^{−1}f_{n}) If char (𝔽) ≠ 2 there is an isomorphism of algebras

H^{∗}(X, 𝔽) ≅ S^{∗} ⊗_{𝔽} E(s^{−1}f_{1}, …, s^{−1}f_{n})

with deg(s^{−1}f_{i}) = deg(f_{i}) − 1 for all i = 1, 2, …, n.

Proof. On the one hand we know from theorem 2.9 that H^{∗}(X/T_{X}, 𝔽) is a free R^{∗}- module and therefore

H^{∗}(X/T_{X}, 𝔽) ≅ S^{∗} ⊗_{𝔽} R^{∗}

as modules over the algebra R^{∗}. For the E_{2}-term of the Eilenberg-Moore spectral sequence {E_{r}, d_{r}} we get then

E_{2} ≅ Tor_{H^{∗}(BT_{X},𝔽)}(𝔽, H^{∗}(X/T_{X}, 𝔽)) ≅ Tor_{H^{∗}(BT_{X},𝔽)}(𝔽, S^{∗} ⊗_{𝔽} R^{∗})
≅ Tor_{H^{∗}(BT_{X},𝔽)}(𝔽, R^{∗}) ⊗_{𝔽} S^{∗}

because Tor_{H^{∗}(BT_{X},𝔽)}(𝔽,?) is an additive functor and S^{∗} a trivial H^{∗}(BT_{X}, 𝔽)-module. On the other hand ker k^{∗} is a Borel ideal after theorem 2.8 so is generated by a regular sequence f_{1}, …, f_{n}. So the Koszul complex (K, ∂)

K = H^{∗}(BT_{X}, 𝔽) ⊗_{𝔽} E(s^{−1}f_{1}, …, s^{−1}f_{n})

with differential ∂ given by

∂(s^{−1}f_{i}) = f_{i}

and deg(s^{−1}f_{i}) = deg(f_{i}) − 1 for i = 1, 2, …, n is acyclic. It gives therefore a free resolution of R^{∗} ≅ H^{∗}(BT_{X}, 𝔽)/ ker k^{∗} as H^{∗}(BT_{X}, 𝔽)-module. So we get

Tor_{H^{∗}(BT_{X},𝔽)}(𝔽, R^{∗}) ≅ E(s^{−1}f_{1}, …, s^{−1}f_{n})

and for the E_{2}-term an isomorphism of algebras

E_{2} ≅ S^{∗} ⊗_{𝔽} E(s^{−1}f_{1}, …, s^{−1}f_{n})

Therefore E_{2}^{∗,∗} is generated as an algebra by the terms E_{2}^{0,∗} and E_{2}^{−1,∗}. The differentials are derivations of algebras and so the spectral sequence collapses in the E_{2}-term.

As a direct consequence we get a theorem of Kac [8] for compact Lie groups generalizing a theorem of Leray [12] for cohomology with rational coefficients (see also [21]). Actually Kac and Peterson [7] proved this fact also for the analogues fibration of flag varieties associated to Kac-Moody groups and used it to study the torsion phenomena for the integral cohomology of these groups. For rational coefficients this was also generalized by Kumar [11] for fibrations with a formal 1-connected base space and fiber a torus, containing the case of flag varieties of compact Lie groups and Kac-Moody groups as special cases.

Corollary 3.2 (Kac). Let p be a prime and X a connected, 1-connected finite loop space with maximal torus T_{X} satisfying the condition Q_{p}. Then the Leray-Serre spectral sequence {E_{r}, d_{r}} of the fibration

T_{X} ⟶^{i} X ⟶^{π} X/T_{X}

collapses in the E_{3}-term. Let f_{1}, …, f_{n} be a regular sequence generating the Borel ideal ker k^{∗}. Then as algebras

(a) E_{3} ≅ E_{∞}

(b) E_{3} ≅ S^{∗} ⊗_{𝔽} E(s^{−1}f_{1}, …, s^{−1}f_{n}) If char (𝔽) ≠ 2 there is an isomorphism of algebras

H^{∗}(X, 𝔽) ≅ S^{∗} ⊗_{𝔽} E(s^{−1}f_{1}, …, s^{−1}f_{n})

with deg(s^{−1}f_{i}) = deg(f_{i}) − 1 for all i = 1, 2, …, n.

Proof. The Leray-Serre spectral sequence {E_{r}, d_{r}} is given by

E_{2} ≅ H^{∗}(X/T_{X}, 𝔽) ⊗_{𝔽} H^{∗}(T_{X}, 𝔽)
E_{r} ⇒ H^{∗}(X, 𝔽).

We consider the following sequence of fibrations

T_{X} ⟶^{i} X ⟶^{π} X/T_{X} ⟶^{k} BT_{X}

As algebras we have isomorphisms

H^{∗}(T_{X}, 𝔽) ≅ E(u_{1}, …, u_{n})
H^{∗}(BT_{X}, 𝔽) ≅ 𝔽[t_{1}, …, t_{n}]

with n = rk(T_{X}) and deg(u_{i}) = 1. Let

σ^{∗} : H^{∗}(BT_{X}, 𝔽) ⟶ H^{∗}(T_{X}, 𝔽)

be the suspension homomorphism in cohomology. We have σ^{∗}(t_{i}) = u_{i} for all i = 1, 2, …, n. For the differential d_{2} we readily see that d_{2}(u_{i}) = k^{∗}(t_{i}) and we can identify the complex (E_{2}, d_{2}) with the Koszul complex (K, ∂) defined by

K = H^{∗}(X/T_{X}, 𝔽) ⊗_{𝔽} E(u_{1}, …, u_{n})

with the differential ∂ given as

∂(x) = 0
∂(u_{i}) = k^{∗}(t_{i}).

Because the cohomology of this Koszul complex is just given as

H(K, ∂) ≅ Tor_{H^{∗}(BT_{X},𝔽)}(𝔽, H^{∗}(X/T_{X}, 𝔽))

we see that the E_{3}-term of the Leray-Serre spectral sequence of the fibration π is up to regrading the E_{2}-term of the Eilenberg-Moore spectral sequence of the fibration k. So from theorem 3.1 we get therefore

dim_{𝔽}(Tot(E_{3})) = dim_{𝔽}(Tot(E^{0}H^{∗}(X, 𝔽))

which implies for the Poincaré series

P (E_{3}, t) = P (E^{0}H^{∗}(X, 𝔽), t) = P (H^{∗}(X, 𝔽), t)

and so the Leray-Serre spectral sequence collapses in the E_{3}-term.

Combining theorem 3.1 with [1] corollary 5.2 we get as another important corollary

Corollary 3.3. Let p be a prime and X a connected, 1-connected finite loop space with maximal torus T_{X} satisfying the condition Q_{p}. Then the sequence

𝔽 ⟶ H^{∗}(BT_{X}, 𝔽) ⟶^{k^{∗}} H^{∗}(X/T_{X}, 𝔽) ⟶^{π^{∗}} H^{∗}(X, 𝔽) ⟶ 𝔽

is coexact, i.e. ker π^{∗} = (\overline{im k^{∗}}).

Now we will use these results to find conditions for the existence of p-torsion in the cohomology of finite loop spaces. We will follow the lines of Kac in [8] where he studied the same question for compact Lie groups. First let us introduce the following charateristic algebra Definition 3.4. Let X be a finite loop space with maximal torus T_{X} and 𝔽 be a field. The algebra generated by

im{π^{∗} : H^{∗}(X/T_{X}, 𝔽) → H^{∗}(X, 𝔽)}.

is called the Chow algebra CH^{∗}(X, 𝔽) of X over 𝔽.

The Chow algebra CH^{∗}(X, 𝔽) is a sub Hopf algebra of the Hopf algebra H^{∗}(X, 𝔽). The terminology follows that of Kac [8]. For a connected compact Lie group according to Grothendieck [6] the ring π^{∗}(H^{∗}(G/T_{G}, 𝔽)) is just the Chow ring CH^{∗}(G_{ℂ}, 𝔽) of the complexification G_{ℂ} of G.

For the rest of this section let X be a connected, 1-connected finite loop space with maximal torus T_{X} satisfying the condition Q_{p} for any prime p. From corollary 3.3 we see imediately that

CH^{∗}(X, 𝔽) = π^{∗}(H^{∗}(X/T_{X}, 𝔽)) ≅ H^{∗}(X/T_{X}, 𝔽)/ ker π^{∗}
≅ H^{∗}(X/T_{X}, 𝔽)/(\overline{im k^{∗}}) ≅ H^{∗}(X/T_{X}, 𝔽)//R^{∗} = S^{∗}.

Because H^{∗}(X/T_{X}, 𝔽) ≅ R^{∗} ⊗_{𝔽} S^{∗} as R^{∗}-modules we get for the Poincaré series

P (H^{∗}(X/T_{X}, 𝔽), t) = P (R^{∗}, t) · P (S^{∗}, t).

First let 𝔽 = ℚ. Theorem 3.1 implies

H^{∗}(X, ℚ) ≅ CH^{∗}(X, ℚ) ⊗_{ℚ} E(s^{−1}f_{1}, …, s^{−1}f_{n})

with deg(s^{−1}f_{i}) = deg(f_{i}) − 1.

From the theorem of Hopf-Leray-Borel [16] we know that H^{∗}(X, ℚ) as a Hopf algebra over ℚ is isomorphic to an exterior algebra with n = rk(X) generators in odd degrees, therefore

H^{∗}(X, ℚ) ≅ E(s^{−1}f_{1}, …, s^{−1}f_{n}).

So the rational Chow algebra of X is trivial

CH^{∗}(X, ℚ) ≅ ℚ

and the homomorphism k^{∗} an epimorphism and therefore

H^{∗}(X/T_{X}, ℚ) ≅ R^{∗} ≅ H^{∗}(BT_{X}, ℚ)/ ker k^{∗}.

The Borel ideal ker k^{∗} is generated by a regular sequence f_{1}, …, f_{n}. We define the ℚ-degrees (d_{1}, …, d_{n}) as the integers

deg(f_{i}) = 2d_{i}

for all i = 1, 2, …, n. We have an isomorphism as algebras

H^{∗}(X/T_{X}, ℚ) ≅ \frac{ℚ[t_{1}, …, t_{n}]}{(f_{1}, …, f_{n})}

with deg(t_{i}) = 2 for all i = 1, 2, …, n. These results are actually true for any field 𝔽 with char (𝔽) = 0.

From [19], theorem 4.16 we know that H^{odd}(X/T_{X}, ℤ) = 0 and therefore the Poincaré series is independent of the field 𝔽 and so we get

P (H^{∗}(X/T_{X}, 𝔽), t) = P (H^{∗}(X/T_{X}, ℚ), t) = ∏_{i=1}^{n} \frac{1 − t^{2d_{i}}}{1 − t^{2}} .

Now let 𝔽 = 𝔽_{p}. The Hopf algebra CH^{∗}(X, 𝔽_{p}) is concentrated in even degrees as a quotient of H^{∗}(X/T_{X}, 𝔽_{p}) and by the theorem of Milnor-Moore [16]

CH^{∗}(X, 𝔽_{p}) ≅ \frac{𝔽_{p}[x_{1}, …, x_{m}]}{(x_{1}^{p^{s_{1}}}, …, x_{m}^{p^{sm}})}

where deg(x_{i}) = 2k_{i} for all i = 1, 2, …, m. So the Poincaré series has the form

P (CH^{∗}(X, 𝔽_{p}), t) = ∏_{i=1}^{m} \frac{1 − t^{2k_{i}p^{s_{i}}}}{1 − t^{2k_{i}}} (∗).

But also the following identity holds

P (CH^{∗}(X, 𝔽_{p}), t) = \frac{P (H^{∗}(X/T_{X}, 𝔽), t)}{P (R^{∗}, t)}

and for the algebra R^{∗} we have

R^{∗} ≅ \frac{H^{∗}(BT_{X}, 𝔽)}{ker k^{∗}} ≅ \frac{𝔽_{p}[t_{1}, …, t_{n}]}{(f_{1,p}, …, f_{n,p})}

where the Borel ideal ker k^{∗} is generated by the regular sequence f_{1,p}, …, f_{n,p}. We define the p-degrees (d_{1,p}, …, d_{n,p}) as the integers

deg(f_{i,p}) = 2d_{i,p}

for all i = 1, 2, …, n. Therefore it follows for the Poincaré series over 𝔽_{p}

P (R^{∗}, t) = ∏_{i=1}^{n} \frac{1 − t^{2d_{i,p}}}{1 − t^{2}} .

For the Poincaré series of the Chow algebra we get then finally the expression

P (CH^{∗}(X, 𝔽_{p}), t) = ∏_{i=1}^{n} \frac{1 − t^{2d_{i}}}{1 − t^{2d_{i,p}}} (∗∗).

From (∗) and (∗∗) it follows that m = n and so with the substitution z = t^{2} we get

∏_{i=1}^{n} \frac{1 − z^{k_{i}p^{s_{i}}}}{1 − z^{k_{i}}} = ∏_{i=1}^{n} \frac{1 − z^{d_{i}}}{1 − z^{d_{i,p}}} .

Therefore k_{i} = d_{i,p} for i = 1, 2, …, n and there exists an integer r ∈ ℕ_{0} and a permutation of the indices such that

d_{i} = p^{s_{i}}d_{i,p} with s_{i} > 0 for i = 1, 2, …, r.

d_{i} = d_{i,p} for i = r + 1, …, n.

With these notations the Poincaré series of the Chow algebra over 𝔽_{p} has the form

P (CH^{∗}(X, 𝔽_{p}), t) = ∏_{i=1}^{r} \frac{1 − t^{2p^{s_{i}}d_{i,p}}}{1 − t^{2d_{i,p}}} .

We now make the following definition according to Kac [8]

Definition 3.5. Let p be a fixed prime. The integers d_{1}, , …, d_{r} are called the p-exceptional degrees of X.

These numbers were intensively studied by Demazure [4] and Kac [8] in the case of compact Lie groups.

As a summary of the above considerations we get the following theorem generalizing a theorem of Kac ([8], theorem 3). By the universal coefficient theorem it is enough to work over the prime field 𝔽_{p}.

Theorem 3.6. Let X be a connected, 1-connected finite loop space with maximal torus T_{X} satisfying the condition Q_{p} for all primes p. Then we have the following:

(1) Let 𝔽 be a field with char 𝔽 = 0. Then the Eilenberg-Moore spectral sequence {E_{r}, d_{r}} of the fibration

X ⟶^{π} X/T_{X} ⟶^{k} BT_{X}

collapses in the E_{2}-term. If f_{2d_{1}}, …, f_{2d_{n}} is a regular sequence generating the Borel ideal ker k^{∗} then the Chow algebra over 𝔽 has the form

CH^{∗}(X, 𝔽) ≅ 𝔽

and there is an isomorphism as algebras

H^{∗}(X, 𝔽) ≅ E(s^{−1}f_{2d_{1}}, …, s^{−1}f_{2d_{n}}).

(2) Let 𝔽 be a field with char 𝔽 = p for a prime p. Then the Eilenberg-Moore spectral sequence {E_{r}, d_{r}} of the fibration

X ⟶^{π} X/T_{X} ⟶^{k} BT_{X}

collapses in the E_{2}-term. Let f_{2d_{1,p}}, …, f_{2d_{n,p}} be a regular sequence generating the Borel ideal ker k^{∗}. Then there is an integer r ∈ ℕ_{0} and a permutation of indices such that

d_{i} = p^{s_{i}}d_{i,p} with s_{i} > 0 for i = 1, 2, …, r.
d_{i} = d_{i,p} for i = r + 1, …, n.

and the Chow algebra over 𝔽 has the form

CH^{∗}(X, 𝔽) ≅ \frac{𝔽[x_{2d_{1,p}}, …, x_{2d_{r,p}}]}{(x_{2d_{1,p}}^{p^{s_{1}}}, …, x_{2d_{r,p}}^{p^{sr}})}.

There is an isomorphism of algebras

E_{2} ≅ CH^{∗}(X, 𝔽) ⊗_{𝔽} E(s^{−1}f_{2d_{1,p}}, …, s^{−1}f_{2d_{n,p}}).

If char 𝔽 ≠ 2 there is an isomorphism of algebras

H^{∗}(X, 𝔽) ≅ CH^{∗}(X, 𝔽) ⊗_{𝔽} E(s^{−1}f_{2d_{1,p}}, …, s^{−1}f_{2d_{n,p}}).

If char 𝔽 = 2 this is true additively as modules.

For the p-exceptional degrees we get also the following generalization of a theorem due to Kac ([8], theorem 4)

Theorem 3.7. Let p be a prime and X a connected, 1-connected finite loop space with maximal torus T_{X}. Then d_{i}/d_{i,p} = 1 or d_{i}/d_{i,p} = p^{ν_{p}(d_{i})} where ν_{p}(d_{i}) is the maximal power of p in d_{i}. In other words, if d_{i,p} is a proper divisor of d_{i}, then d_{i,p} is relatively prime to p.

Proof. Suppose that d_{i,p} ≡ 0 mod p for some i ∈ {1, 2, …, r}, where the integer r is choosen as in theorem 3.6. Then there exists a generator x_{i} of degree 2d_{i,p} in the Chow algebra CH^{∗}(X, 𝔽_{p}) and again by theorem 3.6

Q^{d_{i,p}}H^{∗}(X, 𝔽_{p}) ≠ 0

where Q^{∗}(?) is the functor of indecomposable elements. From the theorems of Lin [13], [14] (see also [9]) we know that for any connected finite H-space H^{odd}(ΩX, ℤ) = 0 and therefore by the universal coeficient theorem H^{odd}(ΩX, 𝔽_{p}) = 0 for any prime p.

We consider now the cohomology suspension in 𝔽_{p}-cohomology

σ^{∗} : H^{∗}(X, 𝔽_{p}) ⟶ H^{∗}(ΩX, 𝔽_{p})

From a theorem of Browder [3], theorem 5.14 it follows then that

σ^{∗} : Q^{l}H^{∗}(X, 𝔽_{p}) ⟶ P^{l−1}H^{∗}(ΩX, 𝔽_{p})

is a monomorphismus for l ≠ 2mp^{a} + 2. Here P^{∗}(?) is the functor of primitive elements.

Because d_{i,p} ≡ 0 mod p we get 2d_{i,p} ≠ 2mp^{a}+2 and therefore H^{2d_{i,p}−1}(ΩX, 𝔽_{p}) ≠ 0 contrary to Lin’s theorem. So d_{i,p} ≢ 0 mod p as claimed.

Now we can deduce equivalent conditions for the existence of p-torsion in the integral cohomology of finite loop spaces generalizing results of Borel [2] and Kac [8] for compact Lie groups.

Theorem 3.8. Let p be a prime and X a connected, 1-connected finite loop space with maximal torus T_{X} satisfying the condition Q_{p}. Let f_{2d_{1,p}}, …, f_{2d_{n,p}} be a regular sequence generating the Borel ideal ker k^{∗}. Then the following statements are equivalent:

(i) H^{∗}(X, ℤ) has no p-torsion. (ii) H^{∗}(X, 𝔽_{p}) ≅ E(s^{−1}f_{2d_{1,p}}, …, s^{−1}f_{2d_{n,p}}) (iii) H^{∗}(BX, 𝔽_{p}) ≅ 𝔽_{p}[f_{2d_{1,p}}, …, f_{2d_{n,p}}] (iv) H^{∗}(BX, ℤ) has no p-torsion.

(v) The Chow algebra is of the form CH^{∗}(X, 𝔽_{p}) ≅ 𝔽_{p}. (vi) k^{∗} : H^{∗}(BT_{X}, 𝔽_{p}) ⟶ H^{∗}(X/T_{X}, 𝔽_{p}) is an epimorphism. (vii) There are no p-exceptional degrees.

If p ∤ d_{i} for all i = 1, 2, …, n then the properties hold. With coefficients in ℚ the statements are also true.

Proof. The equivalence of (i) to (iv) is clear for any finite loop space [9] and the special form of the exterior and polynomial algebra follows from the above considerations.

If CH^{∗}(X, 𝔽_{p}) ≅ 𝔽_{p} then S^{∗} ≅ H^{∗}(X/T_{X}, 𝔽_{p})//R^{∗} ≅ 𝔽_{p} and therefore im k^{∗} ≅ H^{∗}(X/T_{X}, 𝔽_{p}). The other direction is analogue. So the equivalence of (v) and (vi) is clear.

If CH^{∗}(X, 𝔽_{p}) ≅ 𝔽_{p} then

\frac{𝔽_{p}[x_{2d_{1,p}}, …, x_{2d_{r,p}}]}{(x_{2d_{1,p}}^{p^{s_{1}}}, …, x_{2d_{r,p}}^{p^{sr}})} ≅ 𝔽_{p}

and therefore s_{1} = … = s_{r} = 0, so d_{i} = d_{i,p} for all i = 1, 2, …, n. This means that there are no p-exceptional degrees. The other direction is similiar and we established the equivalence of (v) and (vii).

If H^{∗}(X, ℤ) has no p-torsion, then

H^{∗}(X, 𝔽_{p}) ≅ E(s^{−1}f_{2d_{1,p}}, …, s^{−1}f_{2d_{n,p}}).

Because CH^{∗}(X, 𝔽_{p}) = π^{∗}(H^{∗}(X/T_{X}, 𝔽_{p})) is a sub Hopf algebra of H^{∗}(X, 𝔽_{p}) concentrated in even degrees it follows that CH^{∗}(X, 𝔽_{p}) ≅ 𝔽_{p}. If CH^{∗}(X, 𝔽_{p}) ≅ 𝔽_{p} then we know from theorem 3.6 that

H^{∗}(X, 𝔽_{p}) ≅ E(s^{−1}f_{2d_{1,p}}, …, s^{−1}f_{2d_{n,p}})

and therefore H^{∗}(X, ℤ) has no p-torsion. This finally gives the equivalence of (i) and (v). The rest is again clear from theorem 3.6.

We can now define like Kac in [8]:

Definition 3.9. Let p be a prime and X a connected finite loop space with maximal torus T_{X}. The prime p is a good prime if H^{∗}(X, ℤ) has no p-torsion. Otherwise it is called a bad prime.

In [8] Kac analized the torsion phenomena for simple compact Lie groups. From the classical calculations of their 𝔽_{p}-cohomology he detected the good and bad primes and the associated p-exceptional degrees.

Let p be a prime and X any connected, 1-connected finite loop space with maximal torus T_{X}. From [22] we know that H^{∗}(X/T_{X}, 𝔽_{p}) is always a Poincaré duality algebra, i.e. a graded complete intersection over 𝔽_{p} [24], theorem 6.5.1. This gives another characterization for the existence of p-torsion in the cohomology H^{∗}(X, ℤ).

Theorem 3.10. Let p be a prime, X a connected, 1-connected finite loop space with maximal torus T_{X} and [X/T_{X}]_{p} ∈ H^{2d}(X/T_{X}, 𝔽_{p}) the 𝔽_{p} fundamental class. Then

im{k^{∗} : H^{2d}(BT_{X}, 𝔽_{p}) ⟶ H^{2d}(X/T_{X}, 𝔽_{p})} = (D(𝔽_{p}) · [X/T_{X}]_{p})

for an integer D(𝔽_{p}) ∈ ℕ. H^{∗}(X, ℤ) has p-torsion if and only if p divides D(𝔽_{p}).

Proof. Consider the fibration

X/T_{X} ⟶^{k} BT_{X} ⟶^{Bi} BX.

Because H^{∗}(X/T_{X}, 𝔽_{p}) is a Poincaré duality algebra there exists an integer d with H^{k}(X/T_{X}, 𝔽_{p}) = 0 for all k > 2d and an integer D(𝔽_{p}) ∈ ℕ such that

im{k^{∗} : H^{2d}(BT_{X}, 𝔽_{p}) ⟶ H^{2d}(X/T_{X}, 𝔽_{p})} = (D(𝔽_{p}) · [X/T_{X}]_{p})

We have (D(𝔽_{p})·[X/T_{X}]_{p}) ≠ 0 if and only if p ∤ D(𝔽_{p}). From a theorem of Borel ([2], lemme 3.1) it follows that (Bi)^{∗} is a monomorphism if and only if p ∤ D(𝔽_{p}). But if (Bi)^{∗} is a monomorphism, then H^{∗}(BX, 𝔽_{p}) is a polynomial algebra as a sub algebra of the polynomial algebra H^{∗}(BT_{X}, 𝔽_{p}) and therefore H^{∗}(X, ℤ) has no p-torsion and vice versa.

We can make these kind of observations also integrally and get the so-called torsionindex of X defined by Demazure [4], which is given as the order of the cokernel of the homomorphism

k^{∗} : H^{2d}(BT_{X}, ℤ) ⟶ H^{2d}(X/T_{X}, ℤ).

We have the following theorem:

Theorem 3.11. Let X be a connected, 1-connected finite loop space with maximal torus T_{X} and [X/T_{X}] ∈ H^{2d}(X/T_{X}, ℤ) the fundamental class. Then

im{k^{∗} : H^{2d}(BT_{X}, ℤ) ⟶ H^{2d}(X/T_{X}, ℤ)} = (D · [X/T_{X}])

with an integer D ∈ ℕ. The integer D is the torsionindex of X and has the prime factor decomposition

D = p_{1}^{l_{1}} ⋯ p_{m}^{l_{m}}

where p_{1}, …, p_{m} are the bad primes of X. H^{∗}(X, ℤ) is torsionfree if and only if D = 1.

Proof. Because X/T_{X} satiesfies Poincaré duality it follows from the universal coefficient theorem that H^{k}(X/T_{X}, ℤ) = 0 for all k > 2d und

H^{2d}(X/T_{X}, ℤ) ≅ ℤ

generated by the fundamental class [X/T_{X}]. Furthermore the cup product pairing

∪ : H^{i}(X/T_{X}, ℤ) ⊗ H^{2d−i}(X/T_{X}, ℤ) ⟶ H^{2d}(X/T_{X}, ℤ)

is non-singular and therefore it follows from theorem 3.10 and the universal coefficient theorem that

im{k^{∗} : H^{2d}(BT_{X}, ℤ) ⟶ H^{2d}(X/T_{X}, ℤ)} = (D · [X/T_{X}])

for an integer D ∈ ℕ. Actually D(𝔽_{p}) is the mod p reduction of D and H^{∗}(X, ℤ) has p-torsion if and only if p ∤ D. Let p_{1}, …, p_{m} be the bad primes of X. Then D must have a prime factor decomposition of the form

D = p_{1}^{l_{1}} ⋯ p_{m}^{l_{m}}

and so H^{∗}(X, ℤ) is torsionfree if and only if D = 1.

Let X be a connected 1-connected finite loop space with maximal torus T_{X} satisfying the condition Q_{p}. We can also look at the following index associated to X. Let p_{1}, …, p_{m} be the bad primes of X. For any of these bad primes p_{j} there exist p_{j}-exceptional degrees d_{1}^{(j)}, …, d_{r_{j}}^{(j)}. Now we take the product

∏_{j=1}^{m} ∏_{i=1}^{r_{j}} d_{i}^{(j)}/d_{i,p_{j}}^{(j)} = ∏_{j=1}^{m} ∏_{i=1}^{r_{j}} p_{j}^{ν_{p_{j}} (d_{i}^{(j)})}.

This product has a decomposition in the same primes as the torsionindex D. The explicit calculations of Demazure [4] and Kac [8] for compact Lie groups lead us to the following conjecture

D = ∏_{j=1}^{m} ∏_{i=1}^{r_{j}} d_{i}^{(j)}/d_{i,p_{j}}^{(j)} = ∏_{j=1}^{m} ∏_{i=1}^{r_{j}} p_{j}^{ν_{p_{j}} (d_{i}^{(j)})}.

It is not clear if this in general is true or if there is a conceptional proof without using explicit cohomological calculations. This would share new light on the relation between the papers of Demazure [4] and Kac [8].

Acknowledgements. We are very grateful to D. Notbohm and L. Smith for many useful remarks and suggestions.

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Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3-5, D-37073 Göttingen, Fed. Rep. of Germany

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