Homology, Homotopy and Applications, vol.4(1), 2002, pp.87–101

MODULE DERIVATIONS AND NON TRIVIALITY OF AN EVALUATION FIBRATION

KATSUHIKO KURIBAYASHI

(communicated by Charles A. Weibel)

Abstract We give a sufficient condition for the evaluation fibration, whose total space is the free iterated loop space, not to be totally non cohomologous to zero with respect to a given field.

1. Introduction

Let 𝔽_{p} the prime field with p elements if p ≠ 0 and let 𝔽_{0} denote the rational number field ℚ. We say that a fibration F ⟶^{i} E ⟶^{p} B is totally non cohomologous to zero (henceforth TNCZ) with respect to the field 𝔽_{p} if the induced map i^{∗} : H^{∗}(E; 𝔽_{p}) → H^{∗}(E; 𝔽_{p}) is surjective, equivalently, the Leray-Serre spectral sequence for the fibration collapses at the E_{2}-term.

Let X be an n-connected space of finite type with a base point and Λ^{m}X the m-fold free loop space map(S^{m}, X), namely, the space of all continuous maps from the m-dimensional sphere S^{m} to X. For 1 ⩽ m ⩽ n, let us consider the evaluation fibration

F_{m} : Ω^{m}X ⟶ Λ^{m}X ⟶^{ev} X,

where ev is the evaluation map defined by ev(γ) = γ(0) for γ ∈ Λ^{m}X. One may ask when the evaluation fibration F_{m} is TNCZ. The purpose of this paper is to give a sufficient condition for the evaluation fibration F not to be TNCZ with respect to 𝔽_{p}. To this end, we first take a note of some non-trivial relation between indecomposable elements in the cohomology algebra H^{∗}(X; 𝔽_{p}). Such a relation brings us a non-trivial relation in H^{∗}(Λ^{m}X; 𝔽_{p}) via the Eilenberg-Moore spectral

Acknowledgements: The author wishes to thank Sadok Kallel for his interest in this work and helpful comments on the proofs of Theorems 1.1 and 1.2. Received May 22, 2002, revised August 16, 2002; published on November 5, 2002. 2000 Mathematics Subject Classification: 55T20, 57T35, 55S05. Key words and phrases: Module derivation, evaluation fibration, the Eilenberg-Moore spectral sequence, Whitehead product. \begin{matrix}c \\ ◯ \end{matrix} 2002, Katsuhiko Kuribayashi. Permission to copy for private use granted.

sequences associated with fibre squares

Λ^{m}X →  X  ... ↓
↓_{ev_{m}}  ↓^{c_{m−1}}  → Λ^{2}X → X
X →^{c_{m−1}} Λ^{m−1}X →  X  ↓_{ev_{2}}  ↓^{c_{1}}
↓_{ev_{m−1}}  ↓  X →^{c_{1}} ΛX →  X
X  → Λ^{m−2}X →  ↓_{ev_{1}}  ↓^{Δ}
↓  ...  X →^{Δ} X × X,

in which Δ is the diagonal map and the map c_{i} is defined by carrying an element x of X to the constant loop at x. In consequence, we will see that there exists a non-trivial differential in the Leray-Serre spectral sequence. The module derivation, which has been introduced and studied in [10], plays an important role in the consideration. In fact our main theorem (Theorem 2.2) describes how to deduce the non-triviality of F_{m} from a non-trivial relation in H^{∗}(X; 𝔽_{p}) using the module derivation.

Let F ⟶^{i} E ⟶^{p} B be a fibration with a section. Recently, Kallel and Sjerve [8] have related the brace product of the fibration, which has been introduced by James [6], to some differential in the integral homology Leray-Serre spectral sequence for the fibration. Since the evaluation fibration F_{m} has a section c_{m} defined previously, we can detect a non-trivial differential in the spectral sequence for the fibration F_{m} if the brace product of the fibration is non-trivial. Observe that, in this case, the brace product is viewed as the Whitehead product up to the iterated adjoint isomorphism on the homotopy. As mentioned above, we obtain a way to deduce the non-triviality of F_{m} using the module derivation so that, conversely, such information on the non-triviality enables us to investigate the Whitehead products via the result due to Kallel and Sjerve.

Let [ , ] : π_{k}(X) ⊗ π_{l}(X) → π_{k+l−1}(X) be the Whitehead product, h : π_{s}(X) → H_{s}(X) the Hurewicz map and let ad^{m} : π_{n}(X) → π_{n−m}(Ω^{m}X) denote the iterated adjoint map.

Consider the composition map

ρ_{p} ∘ h ∘ ad^{m} ∘ [ , ] : π_{n}(X) ⊗ π_{n}(X) → H_{2n−m−1}(Ω^{m}X) ⟶^{ρ_{p}} H_{2n−m−1}(Ω^{m}X; 𝔽_{p}),

where ρ_{p} is the mod p reduction. Let ξ : H^{n}(X; 𝔽_{p}) ⊗ H^{n}(X; 𝔽_{p})/Im (1 − T) → H^{2n}(X; 𝔽_{p}) be the map induced by the cup product, in which T is the homomorphism on H^{n}(X; 𝔽_{p}) ⊗ H^{n}(X; 𝔽_{p}) defined by T (x ⊗ y) = (−1)^{n}y ⊗ x. We can then compare the dimension of the image by the composition map ρ_{p} ∘ h ∘ ad^{m} ∘ [ , ] with that of the kernel of the map ξ. More precisely, we will establish the following interesting inequality.

Theorem 1.1. Let p be an odd prime or zero and X an (n − 1)-connected space (n ⩾ 2). Then, for m ⩽ n − 1,

dim Im (ρ_{p} ∘ h ∘ ad^{m} ∘ [ , ]) ⩾ dim Ker ξ.

Moreover, if ξ ≡ 0, then

dim Im (ρ_{p} ∘ h ∘ ad^{m} ∘ [ , ]) = \frac{1}{2}s(s + 1) − εs,

where dim H_{n}(X; 𝔽_{p}) = s and ε = 0 or 1 as n is even or odd.

In the case p = 2, the above (in)equality does not hold in general. Indeed, the result [8, Lemma 4.4] due to Kallel and Sjerve implies that ρ_{p} ∘h∘ad^{m} ∘[ι_{n}, ι_{n}] = 0 for the generator ι_{n} of π_{n}(S^{n}) although n is even. The Whitehead products are trivial for an H-space. So one can see that the condition ξ ≡ 0, which deduces the equality, can not be relaxed even though p is odd. (We can give ΩSU(m) as such an example.) It is important to mention that Theorem 1.1 recovers an inequality concerning the Whitehead product due to Chen [2, Theorem 2]. The same argument as in the proof of Theorem 1.1 enables us to obtain an estimate of the dimension of the image by the composition ρ_{p} ∘ h ∘ ad^{m} ∘ [ , ] : π_{n}(X) ⊗ π_{n+1}(X) → H_{2n−m}(Ω^{m}X; 𝔽_{p}).

Theorem 1.2. Let p be a prime number or zero and X an (n − 1)-connected space (n ⩾ 2). Then, for m ⩽ n − 1,

dim Im {ρ_{p} ∘ h ∘ ad^{m} ∘ [,] : π_{n}(X) ⊗ π_{n+1}(X) → H_{2n−m}(Ω^{m}X; 𝔽_{p})} ⩾ dim Ker ζ,

where ζ : H^{n}(X; 𝔽_{p}) ⊗ H^{n+1}(X; 𝔽_{p}) → H^{2n+1}(X; 𝔽_{p}) denotes the cup product.

When p = 0, we have the following theorem by analyzing the minimal model for X.

Theorem 1.3. (1) The inequality in Theorem 1.1 becomes an equality if p = 0.

(2) The inequality in Theorem 1.2 becomes an equality if p = 0 and the multiplication m_{2} : H^{2}(X; ℚ) ⊗ H^{2}(X; ℚ) → H^{4}(X; ℚ) is a monomorphism.

This paper is organized as follows. In Section 2, after recalling briefly the Koszul resolution and the module derivation, we describe our main theorem and its applications. Section 3 is devoted to proving the main theorem. In Section 4, we prove Theorems 1.1, 1.2 and 1.3.

2. Main theorem and its applications

In order to describe our main theorem, we need algebraic notation and terminology. For any non-negatively graded vector space V of finite type over 𝔽_{p}, we denote by S(V) the symmetric algebra generated by V. Let Γ(V) be the divided power algebra generated by V. The desuspension s^{−1}V is the graded vector space defined by (s^{−1}V)^{i} = V^{i+1} and we denote by s^{−1}v ∈ s^{−1}V the element which corresponds to v ∈ V.

We here recall results on the torsion products.

Lemma 2.1. ([15, Proposition 3.5], [9, Propositions 1.1, 1.5] ) Let A be a symmetric algebra S(V) over 𝔽_{p}.

(i) There exists a projective resolution K^{•} ⟶^{φ} A → 0 of A as a left A ⊗ A-module such that K^{•} = A ⊗ A ⊗ Γ(s^{−1}V), d(s^{−1}u) = u ⊗ 1 − 1 ⊗ u for u ∈ V and φ is the multiplication of A, where bideg s^{−1}u = (−1, deg u). Hence, as a bigraded algebra,

Tor_{A⊗A}(A, A) ≅ S(V) ⊗ Γ(s^{−1}V).

(ii) Let B^{•}(A ⊗ A, A) → A → 0 be the bar resolution of A as a left A ⊗ A-module. Then there exists an isomorphism of algebras

Ψ : Tor_{A⊗A}(A, A)_{bar}⟶^{≅}Tor_{A⊗A}(A, A)_{KT}

such that Ψ(1[u ⊗ 1 − 1 ⊗ u]1) = s^{−1}u for u ∈ V. Here Tor_{A⊗A}(A, A)_{bar} and Tor_{A⊗A}(A, A)_{KT} denote the torsion products obtained from the bar resolution and the Koszul-Tate resolution which is defined in (i), respectively.

Following [10], we define the module derivation 𝔇 : A → Tor_{A⊗A}^{∗,∗}(A, A) by 𝔇(a) = 1[a ⊗ 1 − 1 ⊗ a]1 for any graded commutative algebra A over 𝔽_{p}. The map 𝔇 enjoys the following property:

𝔇(ab) = (−1)^{(deg a+1) deg b}b𝔇(a) + (−1)^{deg a}a𝔇(b)

for any a, b ∈ A. In particular, if A is taken to be the symmetric algebra S(V), then 𝔇(v) = s^{−1}v for any v ∈ V up to the isomorphism Ψ. Observe that the image of the module derivation 𝔇 is in Tor_{A⊗A}^{−1,∗}(A, A).

Let H^{∗} be a graded simply connected commutative algebra over 𝔽_{p} and QH^{∗} denote the vector space of indecomposable elements. Choosing a section s : QH^{∗} → H^{∗} of the projection H^{∗} → QH^{∗} = H^{∗}/H̄^{∗}·H̄^{∗}, we define a surjective algebra map q : A = S(QH^{∗}) → H^{∗} by q(v) = s(v) for v ∈ QH^{∗}. Put A = S(QH^{∗}). It follows from Lemma 2.1 that Tor_{A⊗A}(A, A) ≅ S(QH^{∗}) ⊗ Γ(s^{−1}QH^{∗}) as an algebra. In particular, Tor_{A⊗A}^{−1,∗}(A, A) ≅ S(QH^{∗}) ⊗ s^{−1}QH^{∗} as an A-module. Therefore we can define a morphism of A-modules

η : Tor_{A⊗A}^{−1,∗}(A, A) → H^{∗} ⊗ S(s^{−1}QH^{∗})

by η(a ⊗ s^{−1}v) = q(a) ⊗ s^{−1}v.

For any simply connected space Y, let σ^{∗} : H^{∗}(Y; 𝔽_{p}) → H^{∗−1}(ΩY; 𝔽_{p}) denote the cohomology suspension. We are now ready to describe our main theorem.

Theorem 2.2. Let X be an n-connected space and m an integer which is less than or equal to n. Suppose that there exist a subspace V ⊂ QH^{∗}(X; 𝔽_{p}) and an element ρ ∈ Ker q ∩ S(V) ⊂ S(QH^{∗}(X; 𝔽_{p})) such that

\underbrace{σ^{∗} ∘ ⋯ ∘ σ^{∗}}_{m−times} |_{V} : V → H^{∗}(Ω^{m}X; 𝔽_{p})

is a monomorphism and the image of ρ under the composition map

η𝔇 : A = S(QH^{∗}(X; 𝔽_{p})) → Tor_{A⊗A}^{−1,∗}(A, A) → H^{∗}(X; 𝔽_{p}) ⊗ S(s^{−1}QH^{∗}(X; 𝔽_{p}))

is non zero. Then the evaluation fibration Ω^{m}X → Λ^{m}X → X is not TNCZ with respect to 𝔽_{p}.

Remark 2.3. If X is n-connected, then the cohomology suspension σ^{∗} : H^{i}(X; 𝔽_{p}) → H^{i−1}(ΩX; 𝔽_{p}) is a monomorphism for i ⩽ 2n + 1. (see [17, (6.5)Corollary]). Therefore it follows that the m-fold cohomology suspension σ^{∗} ∘ ⋯ ∘ σ^{∗} : H^{i}(X; 𝔽_{p}) → H^{i−m}(Ω^{m}X; 𝔽_{p}) is a monomorphism for i ⩽ 2n − m + 2. Moreover, the map η𝔇 is computable if algebra generators in H^{∗}(X; 𝔽_{p}) and relations between the elements are clarified. From these facts, one can expect that the sufficient condition in Theorem 2.2 is reasonable.

In order to prove Theorem 2.2, we will rely on the Eilenberg-Moore spectral sequences which are obtained from m fibre squares. The construction of the fibre square is as follows: Let us consider the fibration Ω^{m}X ⟶ X^{I^{m}} ⟶^{res} X^{∂I^{m}}, where res is defined by res(γ) = γ|_{∂I^{m}}. We define the map c_{m−1} : X → X^{∂I^{m}} by c_{m−1}(x)(t) = x for x ∈ X and t ∈ ∂I^{m}. The pullback of the fibration by the map c_{m−1} : X → X^{∂I^{m}} is regarded as the m-fold free loop space Λ^{m}X = map(S^{m}, X). Moreover the map res : X^{I^{m}} → X^{∂I^{m}} can be replaced by the map c_{m−1} : X → Λ^{m−1}X with the homotopy equivalence c̃ : X → X^{I^{m}} defined by c̃(x)(t) = x for x ∈ X and t ∈ I^{m}. Since the space X^{∂I^{m}} can be viewed as the (m − 1)-fold free loop space Λ^{m−1}X, we have a fibre square FS_{m}(X):

Λ^{m}X ⟶  X
↓_{ev_{m}} ↓^{c_{m−1}}
X ⟶_{c_{m−1}} Λ^{m−1}X .

Observe that the fibration on the left hand side in FS_{m}(X) is the evaluation fibration Ω^{m}X → Λ^{m}X → X and that FS_{1}(X) is the fibre square which Smith has introduced and studied in [15]. Suppose that X is an m-connected space. Then the fibre square FS_{m}(X) gives rise to the Eilenberg-Moore spectral sequence {_{m}E_{r}^{∗,∗}, d_{r}} converging to H^{∗}(Λ^{m}X; 𝔽_{p}) with

mE_{2}^{∗,∗} ≅ Tor_{H^{∗}(Λ^{m−1}X;𝔽_{p})}^{∗,∗}(H^{∗}(X; 𝔽_{p}), H^{∗}(X; 𝔽_{p}))

as a bigraded algebra.

An important point in proving Theorem 2.2 is what we can translate information of some relation on elements in H^{∗}(X; 𝔽_{p}) to that in H^{∗}(Λ^{m}X; 𝔽_{p}) via the module derivation 𝔇 : H^{∗}(X; 𝔽_{p}) →_{1}E_{2}^{−1,∗} and H^{∗}(X; 𝔽_{p})-module maps with degree −1 which are defined below.

Let A and M be simply connected graded commutative algebras and ϕ : A → M an algebra map. We regard M as an A-bimodule via ϕ. The map 𝔇̃ : Ker ϕ → Tor_{A}(M, M) defined by 𝔇̃(a) = 1[a]1 is an A-module map.

Lemma 2.4. 𝔇̃(ab) = (−1)^{deg a}a[b]1 for a ∈ A and b ∈ Ker ϕ.

Proof. Let ∂ : B^{∗}(M, A, M) → B^{∗−1}(M, A, M) be the external differential of the bar complex which induces the torsion product Tor_{A}(M, M). Then, for a ∈ A and b ∈ Ker ϕ, we see that ∂(1[a|b]1) = a[b]1 + (−1)^{deg a+1}[ab] − (−1)^{deg a+1}1[a]b = a[b]1 + (−1)^{deg a+1}[ab]. This completes the proof.

We conclude this section with examples.

Example 2.5. Let X be an (n − 1)-connected space (n ⩾ 1) and put r_{X} = inf{i ⩾ 2|H^{i}(X; 𝔽_{p}) ≠ 0}. Assume that r_{ΣX} is even and p ≠ 2 or dim H^{r_{ΣX}} (ΣX; 𝔽_{p}) ⩾ 2. Then, for m ⩽ n, the evaluation fibration Ω^{m}ΣX → Λ^{m}ΣX → ΣX is not TNCZ with respect to 𝔽_{p}.

Proof. Let {x_{i}} be a basis of V = H^{r_{ΣX}} H^{∗}(ΣX; 𝔽_{p}). Assume that r_{ΣX} is even and p ≠ 2. Then, for any element x_{i}, we see that x_{i}^{2} ∈ Ker q ∩ S(V) and η𝔇(x_{i}^{2}) = 2x_{i}(s^{−1}x_{i}) ≠ 0 in H^{∗}(X; 𝔽_{p}) ⊗ S(s^{−1}QH^{∗}(X; 𝔽_{p})). Suppose that dim H^{r_{ΣX}} (ΣX; 𝔽_{p}) ⩾ 2. It is readily seen that x_{i}x_{j} ∈ Ker q ∩ S(V) (i ≠ j) and that

η𝔇(x_{i}x_{j}) = (−1)^{r_{ΣX}} (x_{j}s^{−1}x_{i}) + (−1)^{r_{ΣX}} (x_{i}s^{−1}x_{j}) ≠ 0

in H^{∗}(X; 𝔽_{p}) ⊗ S(s^{−1}QH^{∗}(X; 𝔽_{p})). Since the map (σ^{∗})^{(m)} : H^{r_{ΣX}} (ΣX; 𝔽_{p}) → H^{r_{ΣX}−m}(Ω^{m}ΣX; 𝔽_{p}) is isomorphism, the result follows from Theorem 2.2

In the case m = 1, we can obtain a characterization for the evaluation fibration ΩΣX → ΛΣX → ΣX to be TNCZ. Indeed, the cohomology of the free loop space ΛY of a simply connected space Y is isomorphic to the Hochschild homology of the singular cochain complex of Y as a vector space([7]). Moreover since suspension spaces are 𝕂-formal in the sense of Anick (or El haouari) for any field 𝕂 ([1]), it follows that H^{∗}(ΛΣX; 𝔽_{p}) is isomorphic to the Hochschild homology HH(H^{∗}(ΣX; 𝔽_{p})). Therefore direct computation of the Hochschild homology of spheres leads us to the following result.

Example 2.6. Suppose that X is a connected space. Then the evaluation fibration ΩΣX → ΛΣX → ΣX is TNCZ with respect to 𝔽_{p} if and only if H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(S^{2k}; 𝔽_{p}) for some k and p is odd or H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(S^{m}; 𝔽_{p}) for some m and p = 2 .

One may expect that the evaluation fibration F_{m}(X) : Ω^{m}X → Λ^{m}X → X is TNCZ with respect to 𝔽_{p} if and only if the cohomology algebra H^{∗}(X; 𝔽_{p}) is free because the assertion is true if p = 0. However it is not true in general when p > 0. In fact the evaluation fibration F_{1}(ℂP (n)) over the complex project space is TNCZ with respect to 𝔽_{p} if and only if n + 1 ≡ 0 mod p (see [9, Theorem 2]).

Since it is difficult to determine the cohomology of the m-fold free loop space in general, we can not deduce easily that a give evaluation fibration is TNCZ. Theorem 2.2 is applicable to the case where X is the Stiefel manifold.

Example 2.7. For any 1 ⩽ m < n, the evaluation fibration

Ω^{m}SO(n + k)/SO(n) ⟶ Λ^{m}SO(n + k)/SO(n) ⟶ SO(n + k)/SO(n)

is not TNCZ with respect to 𝔽_{p} if n is even and p is odd.

Proof. Put X = SO(n + k)/SO(n). As is known [11], H^{∗}(X; 𝔽_{p}) ≅

∧(e_{2n+3}, ..., e_{2n+2k−3}) ⊗ 𝔽_{p}[x_{n}]/(x_{n}^{2}) if n is even and k is odd
∧(e_{2n+3}, ..., e_{2n+2k−5}, e′_{n+m−1}) ⊗ 𝔽_{p}[x_{n}]/(x_{n}^{2}) if n and k are even .

We choose a 1-dimensional subvector space V of QH^{∗}(X; 𝔽_{p}) so that V = 𝔽_{p}{x_{n}}. Since the element x_{n} is of the least degree, it follows that σ^{∗}∘⋯ σ^{∗}|_{V} is a monomorphism (see Remark 2.3). Moreover we see that η𝔇(x_{n}^{2}) = 2x_{n}s^{−1}x_{n} ≠ 0 in H^{∗}(x; 𝔽_{p})⊗S(s^{−1}QH^{∗}(X; 𝔽_{p})). By virtue of Theorem 2.2, we have the result.

3. Proof of the main theorem

For based spaces Y and X, let map_{∗}(Y, X) denote the space of all based maps from Y to X. Throughout this section, we assume that X is an n-connected space. For any integer m ⩽ n, let us consider a morphism of fibre squares

[figure: Ω^{m}X =; map_{∗}(S^{m}, X); j_{m}; map_{∗}(I^{m}, X); res; Λ^{m}X =; map(S^{m}, X); ev_{m}; X^{I^{m}}; res; ∗; map_{∗}(∂I^{m}, X); j_{m−1}; = Ω^{m−1}X; X; c_{m−1}; X^{∂I^{m}}; = Λ^{m−1}X.]

Observe that the back square gives rise to the Eilenberg-Moore spectral sequence {_{m}Ê_{r}^{∗,∗}, d̂_{r}} converging to H^{∗}(Ω^{m}X; 𝔽_{p}) with

mÊ_{2}^{∗,∗} ≅ Tor_{H^{∗}(Ω^{m−1}X;𝔽_{p})}^{∗,∗}(𝔽_{p}, 𝔽_{p})

as a bigraded algebra. For the rest of this section, the cohomology algebra H^{∗}(X; 𝔽_{p}) will be denoted by H^{∗} and the coefficient fields of the cohomologies will be omitted when no confusion results. Let {_{m}f_{r}} : {_{m}E_{r}^{∗,∗}, d_{r}} → {_{m}Ê_{r}^{∗,∗}, d̂_{r}} be the morphism of spectral sequences induced from the above morphism of fibre squares. Let {_{m}F^{−i}}_{i⩾0} and {_{m}F̂^{−j}}_{j⩾0} denote the filtrations of the Eilenberg-Moore spectral sequences {_{m}E_{r}^{∗,∗}, d_{r}} and {_{m}Ê_{r}^{∗,∗}, d̂_{r}}, respectively. The naturality of the morphism of spectral sequences allows us to obtain the following lemma.

Lemma 3.1. The following two diagrams consist of commutative squares:

H^{∗} ⟶^{𝔇} Tor_{H^{∗}⊗H^{∗}}^{−1,∗}(H^{∗}, H^{∗}) ≅ _{1}E_{2}^{−1,∗} → _{1}E_{∞}^{−1,∗} ← _{1}F^{−1} ↪ H^{∗}(ΛX)
↓_{=}  ↓^{Tor_{id⊗ε}(ε,ε)}  ↓^{_{1}f_{2}}  ↓^{_{1}f_{∞}}  ↓^{j_{1}^{∗}}
H^{∗} \underrightarrow{𝔇̃_{Ω}} Tor_{H^{∗}}^{−1,∗}(𝔽_{p}, 𝔽_{p}) ≅ _{1}Ê_{2}^{−1,∗} → _{1}Ê_{∞}^{−1,∗} = _{1}F̂^{−1} ↪ H^{∗}(ΩX),
Ker c_{m−1}^{∗}  \underrightarrow{𝔇̃} Tor_{H^{∗}(Λ^{m−1}X)}^{−1,∗}(H^{∗}, H^{∗}) ≅ _{m}E_{2}^{−1,∗} → ⋯
↓_{j_{m−1}^{∗}}  ↓^{Tor_{j{∗ \atop m−1}} (ε,ε)}  ↓^{_{m}f_{2}}
H^{∗}(Ω^{m−1}X) \underrightarrow{D̃_{Ω}} Tor_{H^{∗}(Ω^{m−1}X)}^{−1,∗}(𝔽_{p}, 𝔽_{p}) ≅ _{m}Ê_{2}^{−1,∗} → ⋯
⋯ → _{m}E_{∞}^{−1,∗} ← _{m}F^{−1} ↪ H^{∗}(Λ^{m}X)
↓^{_{m}f_{∞}}  ↓^{j_{m}^{∗}}
⋯ → _{m}Ê_{∞}^{−1,∗} = _{m}F̂^{−1} ↪ H^{∗}(Ω^{m}X).

Here 𝔇̃_{Ω} is the map defined by mapping x to [x], 𝔇 and 𝔇̃ are the module derivation and the H^{∗}-module map defined in §2, respectively, and ε : H^{∗} → 𝔽_{p} is the augmentation.

The upper sequences in Lemma 3.1 are covered by more algebraic sequences.

Lemma 3.2. (i) There exist an H^{∗}-module map α̃_{1} : H^{∗} ⊗ s^{−1}QH^{∗} →_{1}F^{−1} = F^{−1}H^{∗}(ΛX; 𝔽_{p}) and an H^{∗}-algebra map α_{1} : H^{∗} ⊗ S(s^{−1}QH^{∗}) → H^{∗}(ΛX; 𝔽_{p}) such that the following two diagrams are commutative:

A = S(QH^{∗}) ⟶^{D}  Tor_{A⊗A}^{−1,∗}(A, A)
↓_{q}  ↓^{Tor_{q⊗q}(q,q)}
H^{∗}  ⟶^{D} Tor_{H^{∗}⊗H^{∗}}^{−1,∗}(H^{∗}, H^{∗})

≅^{(I)} S(QH^{∗}) ⊗ s^{−1}QH^{∗} ⟶^{q⊗1} H^{∗} ⊗ s^{−1}QH^{∗} ↪ H^{∗} ⊗ S(s^{−1}QH^{∗})
↓ ; α̃_{1}  ↓^{α_{1}}
≅ _{1}E_{2}^{−1,∗} → _{1}E_{∞}^{−1,∗} ⟵  _{1}F^{−1}  ↪  H^{∗}(ΛX),

A_{1} := H^{∗} ⊗ S(s^{−1}QH^{∗}) ⟶^{ϕ_{1}} H^{∗}
↓_{α_{1}}  ↓^{=}
H^{∗}(ΛX)  ⟶^{c_{1}^{∗}} H^{∗},

where ϕ_{1} : H^{∗}⊗S(s^{−1}QH^{∗}) → H^{∗} is defined by ϕ_{1}(h⊗s^{−1}v) = 0 and ϕ_{1}(h⊗1) = h. (ii) There exist an H^{∗}-module map α̃_{m} : H^{∗} ⊗ (s^{−1})^{(m)}QH^{∗} → F^{−1}H^{∗}(Λ^{m}X; 𝔽_{p}) = _{m}F^{−1} and an H^{∗}-algebra map α_{m} : H^{∗} ⊗ S((s^{−1})^{(m)}QH^{∗}) → H^{∗}(Λ^{m}X; 𝔽_{p}) such that the following two diagrams are commutative:

Ker ϕ_{m−1} \underrightarrow{D̃}  Tor_{A_{m−1}}^{−1,∗}(A, A)
↓_{α_{m−1}}  ↓^{Tor_{α_{m−1}} (q,q)}
Ker c_{m−1}^{∗} \underrightarrow{D̃} Tor_{H^{∗}(Λ^{m−1}X)}^{−1,∗}(H^{∗}, H^{∗})
≅ H^{∗} ⊗ (s^{−1})^{(m)}QH^{∗} ⟶^{η} H^{∗} ⊗ S((s^{−1})^{(m)}QH^{∗})
↓ ; α̃_{m}  ↓^{α_{m}}
≅ _{m}E_{2}^{−1,∗} → _{m}E_{∞}^{−1,∗} ←_{m}F^{−1}  ↪  H^{∗}(Λ^{m}X),

A_{m} := H^{∗} ⊗ S((s^{−1})^{(m)}QH^{∗}) ⟶^{ϕ_{m}} H^{∗}
↓_{α_{m}}  ↓^{=}
H^{∗}(Λ^{m}X)  ⟶^{c_{m}^{∗}} H^{∗}

where (s^{−1})^{(m)}QH^{∗} denotes the m-fold desuspension of the vector space QH^{∗} and ϕ_{m} : H^{∗} ⊗ S((s^{−1})^{(m)}QH^{∗}) → H^{∗} is defined by ϕ_{m}(h ⊗ (s^{−1})^{(m)}v) = 0 and ϕ_{m}(h ⊗ 1) = h.

By virtue of Lemma 2.1, we can obtain the isomorphism (I). Observe that the

composition map S(QH^{∗})⊗s^{−1}QH^{∗}⟶^{q⊗1}H^{∗}⊗s^{−1}QH^{∗} ↪ H^{∗}⊗S(s^{−1}QH^{∗}) = A_{1}

coincides with the map η defined before Theorem 2.2.

Proof of Lemma 3.2. We define H^{∗}-module map α̃_{1} : H^{∗} ⊗s^{−1}QH^{∗} → F^{−1}H^{∗}(ΛX) by α̃_{1}(s^{−1}v) = 1[q(v) ⊗ 1 − 1 ⊗ q(v)]1, where 1[q(v) ⊗ 1 − 1 ⊗ q(v)]1 denotes a representative element of {𝔇(q(v))} such that c_{1}^{∗}(1[q(v) ⊗ 1 − 1 ⊗ q(v)]1) = 0. Moreover, choosing 1[α̃_{m−1} (s^{−1})^{(m−1)}v]1 as a representative element of {𝔇̃((s^{−1})^{(m−1)}v)} ∈ mE_{∞}^{−1,∗}, we define H^{∗}-module maps α̃_{m} : H^{∗} ⊗ (s^{−1})^{(m)}QH^{∗} → F^{−1}H^{∗}(Λ^{m}X) by α̃_{m}((s^{−1})^{(m)}v) = 1[α̃_{m−1}(s^{−1})^{(m−1)}v]1 inductively. Since the evaluation fibration ev_{m} : Λ^{m}X → X has a section c_{m} : X → Λ^{m}X, there is no loss in generality in supposing that c_{m}^{∗}(1[α̃_{m−1}((s^{−1})^{(m−1)}v)]1) = 0. We have the required maps α̃_{1} and α̃_{m}.

The module map 𝔇̃ defined in Ker ϕ_{m−1} or Ker c_{m−1}^{∗} will be denoted below by 𝔇̃_{m}. From the definitions of 𝔇 and 𝔇̃_{m}, we see that Im η𝔇 ⊂ Ker ϕ_{1} and that Im η𝔇̃_{i−1} ⊂ Ker ϕ_{i}. This fact allows us to compose the maps 𝔇, 𝔇̃_{m} and η’s as η𝔇̃_{m−1} ⋯ η𝔇̃_{1}η𝔇.

Lemma 3.3. The following diagram is commutative:

A = S(QH^{∗}) ⟶^{η𝔇̃_{m−1}···η𝔇̃_{1}η𝔇} H^{∗} ⊗ S((s^{−1})^{(m)}QH^{∗})
↓_{q} ↓^{α_{m}}
H^{∗}(X; 𝔽_{p})  H^{∗}(Λ^{m}X; 𝔽_{p})
↓_{π}  ↓^{j_{m}^{∗}}
QH^{∗}(X; 𝔽_{p})  ⟶_{(σ^{∗})^{(m)}}  H^{∗}(Ω^{m}X; 𝔽_{p}),

where (σ^{∗})^{(m)} is the m-fold suspension map.

Proof. It follows from [13, Proposition 4.5] that the composition map

H^{∗}(Ω^{i−1}X)⟶^{𝔇_{Ω}}T or_{H^{∗}(Ω^{i−1}X)}^{−1,∗}(𝔽_{p}, 𝔽_{p}) ≅ _{i}Ê_{2}^{−1,∗} → _{i}Ê_{∞}^{−1,∗} = F^{−1} → H^{∗−1}(Ω^{i}X)

coincides with the cohomology suspension. Lemmas 3.1 and 3.2 yield the result.

Lemma 3.4. α_{m}η𝔇̃_{m−1} ⋯ η𝔇̃_{1}η𝔇(Ker q) = 0.

Proof. Let π_{1} : F^{−1}H^{∗}(ΛX; 𝔽_{p}) → _{1}E_{∞}^{−1,∗} be the natural projection. From the commutativity of the diagram (i) of Lemma 3.2, it follows that π_{1}α̃_{1}(q ⊗1)𝔇(x) = 0 for any x ∈ Ker q. Therefore we see that α̃_{1}(q ⊗ 1)𝔇(x) + ev_{1}^{∗}y = 0 for some y ∈ H^{∗}(X; 𝔽_{p}). Since c_{1}^{∗}ev_{1}^{∗} = id, it follows from the definition of α̃_{1} that y = 0. Thus α̃_{1}(q ⊗ 1)𝔇(x) = 0 and hence α_{1}η𝔇(x) = 0. The same argument still works well on the diagram (ii) of Lemma 3.2. In consequence we have α_{i}η𝔇̃_{i−1}(z) = 0 for z ∈ Ker α_{i−1}. This completes the proof.

Proof of Theorem 2.2. We can choose a basis S for H^{∗}(Ω^{m}X; 𝔽_{p}) extending that of (σ^{∗})^{(m)}(V), say {(σ^{∗})^{(m)}x_{1}, ..., (σ^{∗})^{(m)}x_{s}, ..} ∪ {b_{1}, b_{2}, ..}, where {x_{1}, .., x_{s}, ..} is a basis for V with deg x_{1} ⩽ deg x_{2} ⩽ · · ·. Suppose that the evaluation fibration Ω^{m}X → Λ^{m}X → X is TNCZ with respect to 𝔽_{p}. Then there exists an isomorphism Φ : H^{∗}(Λ^{m}X; 𝔽_{p}) → H^{∗}(X; 𝔽_{p}) ⊗ H^{∗}(Ω^{m}X; 𝔽_{p}) of an H^{∗}(X; 𝔽_{p})-module such that in_{2}^{∗}Φ = j_{m}^{∗}, where in_{2} : Ω^{m}X → X × Ω^{m}X is the inclusion into the second factor. We here consider the image of (s^{−1})^{(m)}x_{i} by in_{2}^{∗}Φα_{m}. From Lemma 3.3, it follows that in_{2}^{∗}Φα_{m}((s^{−1})^{(m)}x_{i}) = j^{∗}α_{m}((s^{−1})^{(m)}x_{i}) = (σ^{∗})^{(m)}x_{i}. Therefore we can write

Φα_{m}((s^{−1})^{(m)}x_{i}) = (σ^{∗})^{(m)}x_{i} + ∑_{z∈S,deg z<deg x_{i}−m} Q_{(z,i)}z

where Q_{(z,i)} are appropriate elements of H^{⩾1}(X; 𝔽_{p}). Since η𝔇ρ ≠ 0 by the assumption, we see that the element η𝔇(ρ) is expressed as ∑_{i=1}^{k} P_{i}s^{−1}x_{i} with elements P_{i} of H^{∗}(X; 𝔽_{p}) in which P_{k} ≠ 0. Letting η̃ = η𝔇̃_{m−1} ⋯ η𝔇̃_{1}η𝔇, then Lemma 2.4 implies that η̃(ρ) = ∑_{i=1}^{k} P_{i}(s^{−1})^{(m)}x_{i}. From Lemma 3.4, it turns out that

0 = Ψα_{m}η(ρ) = ∑_{i=1}^{k} P_{i}((σ^{∗})^{(m)}x_{i} + ∑_{z∈S,deg z<deg x_{i}−m} Q_{(z,i)}z)
= P_{k}((σ^{∗})^{(m)}x_{k} + ∑_{z∈S,deg z<deg x_{k}−m} Q_{(z,k)}z)

+ ∑_{i=1}^{k−1} P_{i}((σ^{∗})^{(m)}x_{i} + ∑_{z∈S,deg z<deg x_{i}−m} Q_{(z,i)}z).

The fact that H^{∗}(X; 𝔽_{p}) ⊗ H^{∗}(Ω^{m}X; 𝔽_{p}) is a free H^{∗}(X; 𝔽_{p})-module enables us to deduce that P_{k} = 0, which is a contradiction.

4. Proofs of Theorems 1.1, 1.2 and 1.3

Proof of Theorem 1.1.. Let {E^{r}, d^{r}} and {Ẽ^{r}, d̃^{r}} be the integral homology Leray-Serre spectral sequence for the evaluation fibration F_{m} and its mod p reduction, respectively. From [8, Theorem 3.5] and a result due to Hansen [5], we can obtain the following commutative diagram (4.1):

π_{n}(X) ⊗ π_{n}(X)  ⟶^{[,]}  π_{2n−1}(X)
↓_{1⊗ad^{m}}  ↓_{ad^{m}}
π_{n}(X) ⊗ π_{n−m}(Ω^{m}X) ⟶^{{,}} π_{2n−m−1}(Ω^{m}X)
↓_{h⊗h} ↓^{h}
H_{n}(X) ⊗ H_{n−m}(Ω^{m}X) = E_{n,n−m}^{2}  E_{0,2n−m−1}^{2} = H_{2n−m−1}(Ω^{m}X)
↓_{≅}  ↓
E_{n,n−m}^{n}  ⟶^{d^{n}}  E_{0,2n−m−1}^{n}
↓_{ρ_{p}}  ↓^{ρ_{p}}
Ẽ_{n,n−m}^{n}  ⟶^{d̃^{n}}  Ẽ_{0,2n−m−1}^{n},

where {, } denotes the brace product in the sense of James associated with the fibration F_{m}. Thus in order to prove Theorem 1.1, it suffices to show that dim(Im d̃^{n}) ⩾ dim Ker ξ and that dim(Im d̃^{n}) = 1/2 · s(s + 1) − εs if ξ′ ≡ 0.

Let {ι_{i}}_{1⩽i⩽s} be a basis for H_{n}(X; 𝔽_{p}) and ι_{i}(m) the iterated transgression image of the element ι_{i} in H_{n−m}(Ω^{m}X; 𝔽_{p}). We denote by x^{∗} the dual element to an element x of H_{∗}(X; 𝔽_{p}) or H_{∗}(Ω^{m}X; 𝔽_{p}). Let ξ′ : 𝔽_{p}{ι_{i}^{∗}⊗ι_{j}^{∗}; i−j ⩾ ε} → H^{2n}(X; 𝔽_{p}) be the restriction of the cup product to the subspace of H^{n}(X; 𝔽_{p}) ⊗ H^{n}(X; 𝔽_{p}) generated by the elements ι_{i}^{∗} ⊗ ι_{j}^{∗}. Observe that dim Ker ξ′ = dim Ker ξ. For a basis {b_{l}} of Ker ξ′, we write b_{l} = ∑_{i−j⩾ε} α_{l}^{ij}ι_{i}^{∗} ⊗ ι_{j}^{∗}. Since ∑ α_{l}^{ij}ι_{i}^{∗} · ι_{j}^{∗} = ξ′(b_{l}) = 0, it follows from Lemmas 2.4 and 3.4 that (4.2):

0 = α_{m}η𝔇̃_{m−1} ⋯ η𝔇̃_{1}η𝔇(ξ′(b_{l}))
= (−1)^{n(m−1)} ∑ α_{l}^{ij}\{ι_{j}^{∗} · \overline{ι_{i}(m)^{∗}} + (−i)^{n}ι_{i}^{∗} · \overline{ι_{j}(m)^{∗}}\},

where \overline{ι_{i}(m)^{∗}} is the element of H^{∗}(Λ^{m}X; 𝔽_{p}) such that j_{m}^{∗}(\overline{ι_{i}(m)^{∗}}) = ι_{i}(m)^{∗}. We look at the mod p cohomology Leray-Serre spectral sequence {E_{r}, d_{r}} for F_{m}. Let V be the subspace of E_{2}^{n,n−m} generated by the elements ∑ α_{l}^{ij}{ι_{j}^{∗} ⊗ι_{i}(m)^{∗} +(−i)^{n}ι_{i}^{∗} ⊗ ι_{j}(m)^{∗}}. Since ev^{∗} is a monomorphism, every element of V is a permanent cycle. The equality (4.2) implies that all elements in V are in the image by the differential d_{n} : E_{n}^{0,2n−m−1} → E_{n}^{n,n−m}. Therefore, letting dim H^{2n−m−1}(Ω^{m}X; 𝔽_{p}) = k and dim H^{2n−m−1}(X; 𝔽_{p}) = l, we have dim H^{2n−m−1}(Λ^{m}X; 𝔽_{p}) ⩽ k − dim Ker ξ′ + l because dim V = dim Ker ξ′. We again turn to the mod p homology spectral sequence {Ẽ^{r}, d̃^{r}}. The map ev_{∗} is an epimorphism so that every element in Ẽ_{∗,0}^{2} = H_{∗}(X; 𝔽_{p}) is a permanent cycle. Thus we have dim H_{2n−m−1}(Λ^{m}X; 𝔽_{p}) = k − dim Im d̃_{n,n−m}^{n}+l and hence (4.3): dim Im d̃_{n,n−m}^{n} ⩾ dim Ker ξ′. We have the former half of Theorem 1.1. Suppose that ξ ≡ 0. Then dim Ker ξ′ = 1/2·s(s+1)−εs. Since the Whitehead product is skew commutative, it follows from the diagram (4.1) that d̃^{n}(ι_{i} ⊗ ι_{j}(m)) = d̃^{n}(ι_{j} ⊗ ι_{i}(m)), further d̃^{n}(ι_{i} ⊗ ι_{j}(m)) = d̃^{n}(ι_{j} ⊗ ι_{i}(m)) = 0 if n is odd. Thus we see that dim Im d̃_{n,n−m}^{n} ⩽ 1/2·s(s+1)−εs. Combining this inequality with (4.3), we can conclude that dim Im d̃_{n,n−m}^{n} = 1/2 · s(s + 1) − ε.

If X is (n − 1)-connected, then the Hurewicz homomorphism h : π_{n+1}(X) → H_{n+1}(X) is surjective. Therefore, in the homology Leray-Serre spectral sequence {E^{r}, d^{r}} in the above proof, every element of E_{n,n+1−m}^{2} and E_{n+1,n−m}^{2} is spherical, that is; the element is in the image of the the Hurewicz homomorphism. With this in mind, we shall prove Theorem 1.2. The proof is similar to that of Theorem 1.1 although a little complicated argument is needed. In fact, we have to consider two differentials d̃^{n+1} : Ẽ_{n+1,n−m}^{n+1} → Ẽ_{0,2n−m}^{n+1} and d̃^{n} : Ẽ_{n,n+1−m}^{n} → Ẽ_{0,2n−m}^{n} in the mod p homology Leray-Serre spectral sequence {Ẽ^{r}, d̃^{r}}.

Proof of Theorem 1.2.. Under the notation in the proof of Theorem 1.1, let us consider the commutative diagram (4.2):

π_{i}(X) ⊗ π_{j}(X)  ⟶^{[,]}  π_{i+j−1}(X)
↓_{1⊗ad^{m}}  ↓_{ad^{m}}
π_{i}(X) ⊗ π_{j−m}(Ω^{m}X) ⟶^{{,}} π_{i+j−1−m}(Ω^{m}X)
↓_{h⊗h} ↓^{h}
H_{i}(X) ⊗ H_{j−m}(Ω^{m}X) = E_{i,j−m}^{2}  E_{0,i+j−1−m}^{2} = H_{i+j−1−m}(Ω^{m}X)
↓  ↓
E_{i,j−m}^{i}  ⟶^{d^{i}}  E_{0,i+j−1−m}^{i}
↓_{ρ_{p}}  ↓^{ρ_{p}}
Ẽ_{i,j−m}^{i}  ⟶^{d̃^{i}}  Ẽ_{0,i+j−1−m}^{i},

for i = n and j = n + 1. Observe that the composition of left vertical arrows is surjective. As in the proof of Theorem 1.1, it suffices to show that dim(Im d̃^{n}) ⩾ Ker ζ. By comparing the dimension of the E^{∞}-term of {Ẽ^{r}, d̃^{r}} with that of the E_{∞}-term of {E_{r}, d_{r}} on the total degree 2n − 1 − m, we see that

dim Im {d̃^{n} : Ẽ_{n,n−m}^{n} → Ẽ_{0,2n−m−1}^{n}} = dim Im {d_{n} : E_{n}^{0,2n−m−1} → E_{n}^{n,n−m}}.

Put k = dim H^{2n−m}(Ω^{m}X; 𝔽_{p}) and l = dim H^{2n−m}(X; 𝔽_{p}). Then it follows that

dim H^{2n−m}(Λ^{m}X; 𝔽_{p}) = k − dim d_{n}^{0,2n−m} − dim d_{n+1}^{0,2n−m}
+ dim cokerd_{n}^{0,2n−m−1} + l
⩽ k − dim d_{n}^{0,2n−m} + dim E_{2}^{n,n−m}
− dim Im d_{n}^{0,2n−m−1} + l.

On the other hand, dim H_{2n−m}(Λ^{m}X; 𝔽_{p}) = k − dim d̃_{n,n+1−m}^{n} − dim d̃_{n+1,n−m}^{n+1} + dim E_{n,n−m}^{2} − dim Im d̃_{n,n−m}^{n} + l. We here consider the commutative diagram (4.2) for i = n + 1 and j = n. Since the composition of the left vertical arrows is also surjective in this case, the skew commutativity of the Whitehead product enables us to deduce that Im d̃_{n+1,n−m}^{n+1} = 0. It turns out that dim Im d_{n}^{0,2n−m} ⩽ dim Im d̃_{n,n+1−m}^{n}. We are left to show that dim Ker ζ ⩽ dim Im d_{n}^{0,2n−m}. Since the Hurewicz map h : π_{i}(X) → H_{i}(X) is an isomorphism for i = n and an epimorphism for i = n + 1, it follows that the iterated cohomology suspension (σ^{∗})^{(m)} : H^{i}(X; 𝔽_{p}) → H^{i−m}(Ω^{m}X; 𝔽_{p}) is a isomorphism for i = n and a monomorphism for i = n + 1. Lemma 3.3 implies that every element x in the image of the map (σ^{∗})^{(m)} : H^{n+1}(X; 𝔽_{p}) → H^{n+1−m}(Ω^{m}X; 𝔽_{p}) is also in Im j_{m}^{∗}. Therefore the element x is a permanent cycle. As in the proof of Theorem 1.1, applying the map α_{m}η𝔇̃_{m−1} ⋯ η𝔇̃_{1}η𝔇 to appropriate elements of Ker ζ, we have non-trivial relations between indecomposable elements of H^{∗}(Λ^{m}X; 𝔽_{p}). In the cohomology Leray-Serre spectral sequence, the relations are caused by the differential d_{n}^{0,2n−m} : E_{n}^{0,2n−m} → E_{n}^{n,n+1−m}. Thus the same argument as in the proof of Theorem 1.1 enables us to obtain the required inequality.

Remark 4.1. The proof of [2, Theorem 2] relies on the bijectivity of the Hurewicz homomorphism [2, page 259, line 6]. So it does not seem to be easy to improve the proof due to Chen for showing Theorem 1.2.

The rest of this section is devoted to proving Theorem 1.3. We first recall an equality related to the differential of the minimal model for X and the Whitehead product. Let (∧V, d) be the minimal model for the given (n−1)-connected space X. It is known that there exists a non-degenerate ℚ-bilinear map ⟨ ; ⟩ : V^{s} × π_{s}(X) ⊗_{ℤ} ℚ → ℚ for any s ⩾ 2. We define another bilinear map ⟨ ; ⟩_{2} : ∧^{2}V ×(π_{∗}(X)⊗_{ℤ} ℚ)⊗ (π_{∗}(X)⊗_{ℤ}ℚ) → ℚ by ⟨v∧w; γ_{0}⊗γ_{1}⟩_{2} = ⟨v; γ_{1}⟩⟨w; γ_{0}⟩+(−1)^{deg w deg γ_{0}}⟨v; γ_{1}⟩⟨w; γ_{0}⟩. The linear map [ , ]_{ℚ} : π_{∗}(X) ⊗_{ℤ} ℚ ⊗ π_{∗}(X) ⊗_{ℤ} ℚ → π_{∗}(X) ⊗_{ℤ} ℚ is defined by [γ_{0} ⊗ q_{0}, γ_{1} ⊗ q_{1}]_{ℚ} = [γ_{0}, γ_{1}] ⊗ q_{0}q_{1}, where [ , ] is the Whitehead product, q_{i} ∈ ℚ and γ_{i} ∈ π_{∗}(X). Observe that (Im [ , ]) ⊗_{ℤ} ℚ = Im [ , ]_{ℚ}. We then obtain the equality (4.3):

⟨d_{1}v; γ_{0} ⊗ γ_{1}⟩_{2} = (−1)^{k+s−1}⟨v; [γ_{0}, γ_{1}]_{ℚ}⟩,

where v ∈ V, γ_{0} ∈ π_{k}(X) ⊗_{ℤ} ℚ, γ_{1} ∈ π_{s}(X) ⊗_{ℤ} ℚ and d_{1} is the quadratic part of the differential d (see [4, Proposition 13.16]). To simplify, the composition V^{n} ⊗V^{n+1} ×

(π_{n}(X)⊗_{ℤ}ℚ)⊗(π_{n+1}(X)⊗_{ℤ}ℚ)⟶^{≅}(∧^{2}V)^{2n+1}×π_{∗}(X)⊗_{ℤ}ℚ⊗π_{∗}(X)⊗_{ℤ}ℚ⟶^{⟨ ; ⟩_{2}}ℚ

is denoted by ⟨ ; ⟩′ below. It is immediate that ⟨v ⊗ w; γ_{0} ⊗ γ_{1}⟩′ = ⟨v; γ_{1}⟩⟨w; γ_{0}⟩ for v ⊗ w ∈ V^{n} ⊗ V^{n+1}. Therefore, the map ⟨ ; ⟩′ is also non-degenerate. Thus the equality (4.3) yields that dim Im d_{1} = dim(Im [ , ] ⊗_{ℤ} ℚ). We note that V^{n} ≅ H^{n}(X; ℚ) and V^{n+1} = V̂^{n+1} ⊕ ℚ{v_{α}} for which V̂^{n+1} ≅ H^{n+1}(X; ℚ), ℚ{v_{α}} = 0 if n ≠ 2 and ℚ{v_{α}} ≅ Ker m_{2} if n = 2. Moreover, we see that the map ζ : H^{n}(X; ℚ)⊗H^{n+1}(X; ℚ) = V^{n}⊗V^{n+1} → H^{2n+1}(X; ℚ) factors through the injection \overline{ζ} : V^{n} ⊗V^{n+1} → (Ker d)^{2n+1}. For dimensional reasons, we can write (Ker d)^{2n+1} = ℚ{v_{i}^{n} · v_{j}^{n+1}} ⊕ ℚ{w_{β}} ⊕ ℚ{∑ a_{l}^{iα}v_{i}^{n}v_{α}}. Here {v_{i}^{n}} and {v_{j}^{n+1}} are bases for V^{n} and V^{n+1} respectively and w_{β} are appropriate indecomposable elements. Observe that d^{2n} = d_{1}^{2n}. In general, \overline{ζ}(Ker ζ) ⊂ Im d so that dim(Im [ , ] ⊗_{ℤ} ℚ)^{2n} ⩾ Ker ζ. Thus we have the following proposition.

Proposition 4.2. dim(Im [ , ] ⊗_{ℤ} ℚ)^{2n} = Ker ζ if and only if \overline{ζ}(Ker ζ) = Im d.

Proof of Theorem 1.3.. We first prove (2). From the injectivity of the map m_{2}, we see that \overline{ζ}(Ker ζ) = Im d. Moreover, since dim(Im ρ_{0} ∘ h ∘ ad^{m} ∘ [ , ]) = dim(h ⊗ 1 ∘ ad^{m}⊗1(Im [ , ]⊗_{ℤ}ℚ)), it follows from Proposition 4.2 that dim(Im ρ_{0}∘h∘ad^{m}∘[ , ]) ⩽ dim(Im [ , ] ⊗_{ℤ} ℚ) = dim Ker ζ. The result follows from Theorem 1.2. The same argument as above, in which V^{n+1} is replaced with V^{n}, enables us to deduce that the inequality in Theorem 1.1 becomes an equality if p = 0.

As mentioned above, the inequality dim(Im [ , ]⊗_{ℤ} ℚ)^{2n} ⩾ Ker ζ holds in general for (n−1)-connected space (n ⩾ 2). We conclude this section with an example which asserts that the inequality can be strict.

Example 4.3. We define a minimal differential graded algebra (∧V, d) over ℚ as follows: let V be the vector space generated by the elements x_{α}, x_{β}, x_{γ} with degree 2, v, w_{αβ}, w_{γα} with degree 3 and z with degree 4. Define the differential d by d(x_{i}) = 0 = d(v), d(w_{αβ}) = x_{α}x_{β}, d(w_{γα}) = x_{γ}x_{α} and d(z) = x_{γ}w_{αβ} − w_{γα}x_{β}. Then it follows that dim Ker ζ = 0 and dim(Im d)^{5} = 1. From Proposition 4.2, we can conclude that dim(Im [ , ] ⊗_{ℤ} ℚ)^{2n} > Ker ζ.

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Katsuhiko Kuribayashi kuri@geom.xmath.ous.ac.jp

Department of Applied Mathematics, Okayama University of Science, Okayama 700-0005, Japan