Homology, Homotopy and Applications, vol.6(1), 2004, pp.341–350

NOTE ON THE RATIONAL COHOMOLOGY OF THE FUNCTION SPACE OF BASED MAPS

YASUSUKE KOTANI

(communicated by Hvedri Inassaridze)

Abstract In this paper, for a formal, path connected, finite-dimen-sional CW-complex X of finite type and a q-connected space Y of finite type with q ⩾ dim X, we determine the necessary and sufficient condition for the rational cohomology algebra H^{∗}(F_{∗}(X, Y); ℚ) of the function space F_{∗}(X, Y) of based maps to be free.

1. Introduction

Let F(X, Y) and F_{∗}(X, Y) be function spaces of free maps and based maps from a space X to a space Y respectively. Then F(X, Y) and F_{∗}(X, Y) are path connected if X is a path connected, finite-dimensional CW-complex of finite type and Y is a q-connected space with q ⩾ dim X.

A commutative graded algebra A = {A^{p}}_{p⩾0} satisfying A^{0} = ℚ is said to be free if A is isomorphic to a free commutative graded algebra ∧V on a graded vector space V.

A commutative cochain algebra (A, d) satisfying H^{0}(A) = ℚ is said to be formal if (A, d) and (H(A), 0) are connected by a chain of quasi-isomorphisms. A path connected space X is said to be formal if the commutative cochain algebra A_{PL}(X) of rational polynomial differential forms on X is formal.

It is known that, for an arbitrary n-connected space Y with n ⩾ 1, the rational cohomology algebra

H^{∗}(Ω^{n}Y; ℚ) = H^{∗}(F_{∗}(S^{n}, Y); ℚ)

of the n-fold loop space Ω^{n}Y of Y is free, and that spheres S^{n} are formal.

In this paper, for a formal, path connected, finite-dimensional CW-complex X of finite type and a q-connected space Y of finite type with q ⩾ dim X, we consider the condition for the rational cohomology algebra H^{∗}(F_{∗}(X, Y); ℚ) of the function space F_{∗}(X, Y) of based maps to be free.

Let H^{∗}(X; ℚ) = {H^{p}(X; ℚ)}_{p⩾0} be the rational cohomology algebra for a path connected space X with the cup product

∪: H^{∗}(X; ℚ) ⊗ H^{∗}(X; ℚ) → H^{∗}(X; ℚ).

Received October 23, 2003, revised July 12, 2004; published on August 6, 2004. 2000 Mathematics Subject Classification: Primary 55P62; Secondary 54C40. Key words and phrases: rational cohomology, function spaces, free commutative graded algebras, formal spaces. \begin{matrix}c \\ ◯ \end{matrix} 2004, Yasusuke KOTANI. Permission to copy for private use granted.

Recall that the rational cup length cup(X; ℚ) of X is defined by

sup{n ∈ ℤ | f_{1} ∪ ⋯ ∪ f_{n} ≠ 0 for f_{1}, …, f_{n} ∈ H^{+}(X; ℚ)},

where H^{+}(X; ℚ) = {H^{p}(X; ℚ)}_{p>0}.

Let (∧V, d) be a Sullivan algebra. Elements in ∧V of the form v_{1} ∧ ⋯ ∧ v_{k} for v_{1}, …, v_{k} ∈ V are said to have word length k. Then the differential d decomposes uniquely as the sum

d = d_{0} + d_{1} + d_{2} + ⋯

of derivations d_{i} raising the word length by i. (cf. [3, Section 12(a)]). Now, we define the differential length dl(∧V, d) of (∧V, d) by the least integer m such that d_{m−1} ≠ 0. If d_{i} = 0 for all i ⩾ 0, that is, d = 0, we define dl(∧V, 0) = ∞. We also define the differential length dl(Y) of a simply connected space Y of finite type by that of a minimal Sullivan model for Y. Then we can establish

Theorem 1.1. The differential length of a simply connected space of finite type is independent of a choice of minimal Sullivan models. Thus it is a rational homotopy invariant.

Our main theorem is as follows.

Theorem 1.2. Let X be a formal, path connected, finite-dimensional CW-complex of finite type and Y a q-connected space of finite type with q ⩾ dim X. Then H^{∗}(F_{∗}(X, Y); ℚ) is free if and only if cup(X; ℚ) < dl(Y).

This paper is organized as follows. In Section 2, we recall the construction of a minimal Sullivan model for F(X, Y) due to E. H. Brown, Jr. and R. H. Szczarba [2, Thoerem 1.9]. Moreover, we describe a minimal Sullivan model for F_{∗}(X, Y) is obtained by that for F(X, Y) using the evaluation fibration, which is established by K. Kuribayashi [4, Theorem 3.6]. The proofs of Theorems are given in Section 3 and 4 respectively. In Section 5, we give some examples.

The author would like to express his deepest gratitude to Professor K. Kuribayashi for permitting to use some results in [4, Section 3], and also to Professor T. Yamaguchi for helpful suggestions.

2. Minimal Sullivan models for F(X, Y) and F_{∗}(X, Y)

Let X and Y be as in Theorem 1.2. Then the construction of a minimal Sullivan model for F(X, Y) due to E. H. Brown, Jr. and R. H. Szczarba [2, Theorem 1.9] is described as follows.

Let m_{Y} : (∧V, d) ⟶^{≃} A_{PL}(Y) be a minimal Sullivan model for Y. Let H_{∗}(X; ℚ) = {H_{p}(X; ℚ)}_{p⩾0} be the rational homology coalgebra for X with the coproduct

Δ: H_{∗}(X; ℚ) → H_{∗}(X; ℚ) ⊗ H_{∗}(X; ℚ).

Let ∧V ⊗ H_{∗}(X; ℚ) be a graded vector space with grading |v ⊗ c| = |v| − |c| for v ∈ ∧V and c ∈ H_{∗}(X; ℚ). Let ∧(∧V ⊗ H_{∗}(X; ℚ)) be the free commutative graded algebra on ∧V ⊗ H_{∗}(X; ℚ) with the differential d ⊗ id, and let I be the ideal in ∧(∧V ⊗ H_{∗}(X; ℚ)) generated by 1 ⊗ 1 − 1 and all elements of the form

v′ ∧ v′′ ⊗ c − ∑(−1)^{|v′′||c′_{j}|}(v′ ⊗ c′_{j}) ∧ (v′′ ⊗ c′′_{j})  (2.1)

for v′, v′′ ∈ ∧V and c ∈ H_{∗}(X; ℚ) with Δc = ∑ c′_{j} ⊗ c′′_{j}. Then (d ⊗ id)(I) ∈ I ([2, Theorem 3.3]) and the composition map

ρ: ∧ (V ⊗ H_{∗}(X; ℚ)) ↪ ∧(∧V ⊗ H_{∗}(X; ℚ)) → ∧(∧V ⊗ H_{∗}(X; ℚ))/I

is an isomorphism of graded algebras ([2, Theorem 3.3]). Let δ be the differential on ∧(V ⊗H_{∗}(X; ℚ)) given by δ = ρ^{−1}(d⊗id)ρ. Then, by [2, Theorem 1.9], F(X, Y) has a minimal Sullivan model of the form

(∧(V ⊗ H_{∗}(X; ℚ)), δ).

Next, let us consider the evaluation fibration

F_{∗}(X, Y) → F(X, Y) ⟶^{ev_{∗}} Y,

where ev_{∗} is the evaluation map at the basepoint of X. Let i: (∧V, d) ↪ (∧(V ⊗ H_{∗}(X; ℚ)), δ) be the inclusion map defined by i(v) = v ⊗ 1 for v ∈ V. From the consideration in [4, Section 3], we have a commutative diagram

\xymatrix{A_{PL}(Y) \ar[r]^{A_{PL}(ev_{∗})} & A_{PL}(F(X, Y)) \\ (∧V, d) \ar@{^{(}->}[r]^{i} \ar[u]^{m_{Y}}_{≃} & (∧(V ⊗ H_{∗}(X; ℚ)), δ), \ar[u]^{m}_{≃}}

where m: (∧(V ⊗ H_{∗}(X; ℚ)), δ) ⟶^{≃} A_{PL}(F(X, Y)) is a minimal Sullivan model for F(X, Y) described above. Thus the inclusion map i is viewed as a model for the evaluation map ev_{∗}.

Let J be an ideal of ∧(V ⊗ H_{∗}(X; ℚ)) generated by v ⊗ 1 for v ∈ V. Let \overline{δ} be the differential on ∧(V ⊗ H_{∗}(X; ℚ))/J induced from δ on ∧(V ⊗ H_{∗}(X; ℚ)). Then, by [3, Proposition 15.5] and [4, Theorem 3.6], F_{∗}(X, Y) has a minimal Sullivan model of the form

(∧(V ⊗ H_{∗}(X; ℚ))/J, \overline{δ}) = (∧(V ⊗ H_{+}(X; ℚ)), \overline{δ}),

where H_{+}(X; ℚ) = {H_{p}(X; ℚ)}_{p>0}.

3. Proof of Theorem 1.1

It is known that minimal Sullivan models for a simply connected space of finite type are all isomorphic, and that the isomorphism class of a minimal Sullivan model for a simply connected space of finite type is a rational homotopy invariant. Hence, for the proof of Theorem 1.1, it is sufficient to prove the following.

Proposition 3.1. Let (∧V, d) and (∧V′, d′) be isomorphic Sullivan algebras. Then dl(∧V, d) = dl(∧V′, d′).

Proof. Let f: (∧V, d) ⟶^{≅} (∧V′, d′) be an isomorphism of differential graded algebras.

First, suppose that dl(∧V′, d′) = ∞, that is, d′ = 0. Then, since fd = d′f = 0 and f is an isomorphism, we have d = 0. Thus dl(∧V, d) = ∞.

Next, suppose that dl(∧V′, d′) = m < ∞, that is, d′_{i} = 0 for 0 ⩽ i < m − 1 and d′_{m−1} ≠ 0. Then, since f is an isomorphism, for an arbitrary element v ∈ V, there exists an element v′ ∈ V′ such that

f(v) = v′ + (higher terms).

Now, assume that dv has terms of the form v_{1} ∧ ⋯ ∧ v_{k} for v_{1}, …, v_{k} ∈ V and k ⩽ m − 1. Then f(dv) has terms of the form

f(v_{1} ∧ ⋯ ∧ v_{k}) = f(v_{1}) ∧ ⋯ ∧ f(v_{k}) = v′_{1} ∧ ⋯ ∧ v′_{k} + (higher terms)

for v′_{1}, …, v′_{k} ∈ V′ and k ⩽ m − 1. However, d′f(v) = f(dv) has no such terms because d′_{i} = 0 for 0 ⩽ i < m − 1. It is a contradiction. Hence we have d_{i} = 0 for 0 ⩽ i < m − 1 since d is a derivation. So we get the inequality dl(∧V, d) ⩾ dl(∧V′, d′). Since f^{−1} is also an isomorphism, we get the inverse inequality. Thus dl(∧V, d) = dl(∧V′, d′) = m.

4. Proof of Theorem 1.2

Let X and Y be as in Theorem 1.2. Let (∧V, d) be a minimal Sullivan model for Y and H_{∗}(X; ℚ) the rational homology coalgebra for X. Then, as described in Section 2, F_{∗}(X, Y) has a minimal Sullivan model of the form

(∧(V ⊗ H_{+}(X; ℚ)), \overline{δ}),

where \overline{δ} is induced from δ = ρ^{−1}(d⊗id)ρ on ∧(V ⊗H_{∗}(X; ℚ)) by reducing elements contained in the ideal J generated by v ⊗ 1 for v ∈ V.

It is easy to see that H^{∗}(F_{∗}(X, Y); ℚ) ≅ H(∧(V ⊗ H_{+}(X; ℚ)), \overline{δ}) is free if and only if \overline{δ} = 0, and that \overline{δ} = 0 if and only if δ(∧(V ⊗ H_{+}(X; ℚ))) ∈ J. Hence, for the proof of Theorem 1.2, it is sufficient to prove the following.

Proposition 4.1. (1). If cup(X; ℚ) < dl(Y), then δ(∧(V ⊗ H_{+}(X; ℚ))) ∈ J or

equivalently \overline{δ} = 0. (2). If cup(X; ℚ) ⩾ dl(Y), then δ(∧(V ⊗ H_{+}(X; ℚ))) ∉ J.

Thus we need to explain the differential δ in detail. Let Δ be the coproduct on H_{∗}(X; ℚ). Then the reduced coproduct

\overline{Δ}: H_{+}(X; ℚ) → H_{+}(X; ℚ) ⊗ H_{+}(X; ℚ)

is defined by \overline{Δ}c = Δc−c ⊗ 1−1 ⊗ c for c ∈ H_{+}(X; ℚ). Moreover, the k-th coproduct Δ^{(k)} and the k-th reduced coproduct \overline{Δ}^{(k)} are defined inductively by Δ^{(0)} = \overline{Δ}^{(0)} = id, Δ^{(1)} = Δ, \overline{Δ}^{(1)} = \overline{Δ} and

Δ^{(k)} = (Δ ⊗ id ⊗ ⋯ ⊗ id) ∘ Δ^{(k−1)} : H_{∗}(X; ℚ) → H_{∗}(X; ℚ)^{⊗k+1},
\overline{Δ}^{(k)} = (\overline{Δ} ⊗ id ⊗ ⋯ ⊗ id) ∘ \overline{Δ}^{(k−1)} : H_{+}(X; ℚ) → H_{+}(X; ℚ)^{⊗k+1},

where H^{⊗k+1} denotes the (k + 1)-times tensor product of H.

Let H^{∗}(X; ℚ) be the rational cohomology algebra for X with the cup product ∪. Since X is of finite type, H^{∗}(X; ℚ) with ∪ and H_{∗}(X; ℚ) with Δ are dual each other. Hence we have immediately

Lemma 4.2. If cup(X; ℚ) = n, then \overline{Δ}^{(k−1)} ≠ 0 for 0 < k ⩽ n and \overline{Δ}^{(k−1)} = 0 for all k > n.

Let B_{H_{∗}} = {c_{0} = 1, c_{1}, c_{2}, … } be a basis for H_{∗}(X; ℚ) with 0 < |c_{1}| ⩽ |c_{2}| ⩽ · · ·. Then, for an arbitrary element c_{j} ∈ B_{H_{∗}} and k ⩾ 2, we may denote

Δ^{(k−1)}c_{j} = ∑ μ_{j_{1},...,j_{k}}c_{j_{1}} ⊗ ⋯ ⊗ c_{j_{k}},

where 0 ≠ μ_{j_{1},...,j_{k}} ∈ ℚ and c_{j_{1}}, …, c_{j_{k}} ∈ B_{H_{∗}}. By the definition of the reduced coproduct, we have immediately

Lemma 4.3. \overline{Δ}^{(k−1)}c_{j} = 0 if and only if there exists an integer s such that c_{j_{s}} = 1 in each term of Δ^{(k−1)}c_{j}.

Moreover, since the cup product ∪ is associative and commutative, so is the coproduct Δ, that is, (Δ ⊗ id)Δ = (id ⊗Δ)Δ and τΔ = Δ, where τ is defined by τ(c ⊗ c′) = (−1)^{|c||c′|}c′ ⊗ c. Hence we have immediately

Lemma 4.4. μ_{j_{1},...,j_{s},j_{s+1},...,j_{k}} = (−1)^{|c_{js} ||c_{j_{s+1}} |}μ_{j_{1},...,j_{s+1},j_{s},...,j_{k}}.

Let B_{V} = {v_{1}, v_{2}, … } be a basis for V with 0 < |v_{1}| ⩽ |v_{2}| ⩽ · · ·. Then, if dv_{i} = v_{i_{1}} ∧ ⋯ ∧ v_{i_{k}} for v_{i} ∈ B_{V} and Δ^{(k−1)}c_{j} = ∑ μ_{j_{1},...,j_{k}}c_{j_{1}} ⊗ ⋯ ⊗ c_{j_{k}} for c_{j} ∈ B_{H_{∗}}, we have

δ(v_{i} ⊗ c_{j}) = ∑(−1)^{ε(i_{1},j_{1};...;i_{k},j_{k})}μ_{j_{1},...,j_{k}}(v_{i_{1}} ⊗ c_{j_{1}}) ∧ ⋯ ∧ (v_{i_{k}} ⊗ c_{j_{k}}), (4.1)

where the sign (−1)^{ε(i_{1},j_{1};...;i_{k},j_{k})} is determined by (2.1), that is,

v_{i_{1}} ∧ ⋯ ∧ v_{i_{k}} ⊗ c_{j} = ∑(−1)^{ε(i_{1},j_{1};...;i_{k},j_{k})}μ_{j_{1},...,j_{k}}(v_{i_{1}} ⊗ c_{j_{1}}) ∧ ⋯ ∧ (v_{i_{k}} ⊗ c_{j_{k}})

in the graded algebra ∧(∧V ⊗ H_{∗}(X; ℚ))/I. More precisely, ε(i_{1}, j_{1}; …; i_{k}, j_{k}) is given by

Lemma 4.5. ε(i_{1}, j_{1}; …; i_{k}, j_{k}) = ∑_{l=1}^{k−1}(|v_{i_{l+1}}| + ⋯ + |v_{i_{k}}|)|c_{j_{l}}|

Proof. We prove by induction on k. Let k = 2. Then, if Δc_{j} = ∑ μ_{j_{1},j_{2}}c_{j_{1}} ⊗ c_{j_{2}} for c_{j} ∈ B_{H_{∗}}, we have

v_{i_{1}} ∧ v_{i_{2}} ⊗ c_{j} = ∑(−1)^{|v_{i_{2}} ||c_{j_{1}} |}μ_{j_{1},j_{2}}(v_{i_{1}} ⊗ c_{j_{1}}) ∧ (v_{i_{2}} ⊗ c_{j_{2}}),

and so ε(i_{1}, j_{1}; i_{2}, j_{2}) = |v_{i_{2}}||c_{j_{1}}|.

Let k ⩾ 3 and assume that the formula is true until k − 1. Since Δ^{(k−1)} = (Δ ⊗ id ⊗ ⋯ ⊗ id) ∘ Δ^{(k−2)}, if Δ^{(k−1)}c_{j} = ∑ μ_{j_{1},...,j_{k}}c_{j_{1}} ⊗ ⋯ ⊗ c_{j_{k}} for c_{j} ∈ B_{H_{∗}}, we can denote

Δ^{(k−2)}c_{j} = ∑ μ_{j′_{1},j_{3},...,j_{k}}c_{j′_{1}} ⊗ c_{j_{3}} ⊗ ⋯ ⊗ c_{j_{k}}

with Δc_{j′_{1}} = ∑ μ′_{j_{1},j_{2}}c_{j_{1}} ⊗ c_{j_{2}} and μ_{j_{1},...,j_{k}} = μ′_{j_{1},j_{2}}μ_{j′_{1},j_{3},...,j_{k}}. Then, by putting v_{i′_{1}} = v_{i_{1}} ∧ v_{i_{2}}, we have

v_{i_{1}} ∧ ⋯ ∧ v_{i_{k}} ⊗ c_{j} = v_{i′_{1}} ∧ v_{i_{3}} ∧ ⋯ ∧ v_{i_{k}} ⊗ c_{j}

= ∑(−1)^{ε(i′_{1},j′_{1};i_{3},j_{3};...;i_{k},j_{k})}μ_{j′_{1},j_{3},...,j_{k}}(v_{i′_{1}} ⊗ c_{j′_{1}}) ∧ (v_{i_{3}} ⊗ c_{j_{3}}) ∧ ⋯ ∧ (v_{i_{k}} ⊗ c_{j_{k}}).

Furthermore, since

v_{i′_{1}} ⊗ c_{j′_{1}} = v_{i_{1}} ∧ v_{i_{2}} ⊗ c_{j′_{1}} = ∑(−1)^{|v_{i_{2}} ||c_{j_{1}} |}μ′_{j_{1},j_{2}}(v_{i_{1}} ⊗ c_{j_{1}}) ∧ (v_{i_{2}} ⊗ c_{j_{2}}),

we have

v_{i_{1}} ∧ ⋯ ∧ v_{i_{k}} ⊗ c_{j}
= ∑(−1)^{ε(i′_{1},j′_{1};i_{3},j_{3};...;i_{k},j_{k})+|v_{i_{2}} ||c_{j_{1}} |}μ_{j_{1},...,j_{k}}(v_{i_{1}} ⊗ c_{j_{1}}) ∧ ⋯ ∧ (v_{i_{k}} ⊗ c_{j_{k}}),

and so

ε(i_{1}, j_{1}; …; i_{k}, j_{k})

= ε(i′_{1}, j′_{1}; i_{3}, j_{3}; …; i_{k}, j_{k}) + |v_{i_{2}}||c_{j_{1}}|

= (|v_{i_{3}}| + ⋯ + |v_{i_{k}}|)|c_{j′_{1}}| + ∑_{l=3}^{k−1}(|v_{i_{l+1}}| + ⋯ + |v_{i_{k}}|)|c_{j_{l}}| + |v_{i_{2}}||c_{j_{1}}|

= ∑_{l=1}^{k−1}(|v_{i_{l+1}}| + ⋯ + |v_{i_{k}}|)|c_{j_{l}}|

because |c_{j′_{1}}| = |c_{j_{1}}| + |c_{j_{2}}|.

Now we can prove Proposition 4.1.

Proof of Proposition 4.1. Notice that cup(X; ℚ) < ∞ since X is finite-dimensional.

First, suppose that dl(Y) = ∞. Then, since d = 0, we have δ = ρ^{−1}(d ⊗ id)ρ = 0, and so \overline{δ} = 0.

Next, suppose that dl(Y) = m < ∞. Fix a basis B_{H_{∗}} = {c_{0} = 1, c_{1}, c_{2}, … } for H_{∗}(X; ℚ) with 0 < |c_{1}| ⩽ |c_{2}| ⩽ ⋯ and a basis B_{V} = {v_{1}, v_{2}, … } for V with 0 < |v_{1}| ⩽ |v_{2}| ⩽ · · ·. Then, for an arbitrary element v_{i} ∈ B_{V}, we may denote

dv_{i} = ∑_{k⩾m} λ_{i_{1},...,i_{k}}v_{i_{1}} ∧ ⋯ ∧ v_{i_{k}},

where 0 ≠ λ_{i_{1},...,i_{k}} ∈ ℚ and v_{i_{1}}, …, v_{i_{k}} ∈ B_{V} with i_{1} ⩽ ⋯ ⩽ i_{k}.

(1). For an arbitrary element c_{j} ∈ B_{H_{∗}} with c_{j} ≠ 1 and k ⩾ m, we may denote

Δ^{(k−1)}c_{j} = ∑ μ_{j_{1},...,j_{k}}c_{j_{1}} ⊗ ⋯ ⊗ c_{j_{k}},

where 0 ≠ μ_{j_{1},...,j_{k}} ∈ ℚ and c_{j_{1}}, …, c_{j_{k}} ∈ B_{H_{∗}}. Since cup(X; ℚ) < dl(Y) = m, by Lemma 4.2, \overline{Δ}^{(k−1)}c_{j} = 0 for k ⩾ m, and so, by Lemma 4.3, there exists an integer s such that c_{j_{s}} = 1 in each term of Δ^{(k−1)}c_{j} for k ⩾ m. Hence we have

δ(v_{i} ⊗ c_{j})

= ∑_{k⩾m}(−1)^{ε(i_{1},j_{1};...;i_{k},j_{k})}λ_{i_{1},...,i_{k}}μ_{j_{1},...,j_{k}}(v_{i_{1}} ⊗ c_{j_{1}}) ∧ ⋯ ∧ (v_{i_{k}} ⊗ c_{j_{k}}) ∈ J

for an arbitrary element v_{i} ∈ B_{V}. Thus δ(∧(V ⊗ H_{+}(X; ℚ)) ∈ J since δ is a derivation.

(2). Since cup(X; ℚ) ⩾ dl(Y) = m, by Lemma 4.2, there exists an element c_{j} ∈ B_{H_{∗}} such that c_{j} ≠ 1 and

\overline{Δ}^{(m−1)}c_{j} = ∑ μ_{j_{1},...,j_{m}}c_{j_{1}} ⊗ ⋯ ⊗ c_{j_{m}} ≠ 0.

Since dl(Y) = m, there exists an element v_{i} ∈ B_{V} such that dv_{i} has a term of the form λ_{i_{1},...,i_{m}}v_{i_{1}} ∧ ⋯ ∧ v_{i_{m}} with λ_{i_{1},...,i_{m}} ≠ 0 and i_{1} ⩽ ⋯ ⩽ i_{m}. Then δ(v_{i} ⊗ c_{j}) has terms of the form

∑(−1)^{ε(i_{1},j_{1};...;i_{m},j_{m})}λ_{i_{1},...,i_{m}}μ_{j_{1},...,j_{m}}(v_{i_{1}} ⊗ c_{j_{1}}) ∧ ⋯ ∧ (v_{i_{m}} ⊗ c_{j_{m}})

with c_{j_{s}} ≠ 1 for 1 ⩽ s ⩽ m.

If i_{1} < ⋯ < i_{m}, we see that each term (v_{i_{1}} ⊗ c_{j_{1}}) ∧ ⋯ ∧ (v_{i_{m}} ⊗ c_{j_{m}}) cannot be canceled by other terms.

If i_{s} = i_{s+1} for some s, |v_{i_{s}}| must be even. Then we have

(v_{i_{s}} ⊗ c_{j_{s}}) ∧ (v_{i_{s}} ⊗ c_{j_{s+1}})
= (−1)^{(|v_{is} |−|c_{js} |)(|v_{is} |−|c_{j_{s+1}} |)}(v_{i_{s}} ⊗ c_{j_{s+1}}) ∧ (v_{i_{s}} ⊗ c_{j_{s}})
= (−1)^{|c_{js} ||c_{j_{s+1}} |}(v_{i_{s}} ⊗ c_{j_{s+1}}) ∧ (v_{i_{s}} ⊗ c_{j_{s}})

and, by Lemma 4.5,

ε(i_{1}, j_{1}; …; i_{s}, j_{s}; i_{s}, j_{s+1}; …; i_{m}, j_{m})
− ε(i_{1}, j_{1}; …; i_{s}, j_{s+1}; i_{s}, j_{s}; …; i_{m}, j_{m})
= (|v_{i_{s}}| + |v_{i_{s+2}}| + ⋯ + |v_{i_{k}}|)|c_{j_{s}}| + (|v_{i_{s+2}}| + ⋯ + |v_{i_{k}}|)|c_{j_{s+1}}|
− (|v_{i_{s}}| + |v_{i_{s+2}}| + ⋯ + |v_{i_{k}}|)|c_{j_{s+1}}| − (|v_{i_{s+2}}| + ⋯ + |v_{i_{k}}|)|c_{j_{s}}|
= |v_{i_{s}}|(|c_{j_{s}}| − |c_{j_{s+1}}|) ≡ 0 mod 2.

Hence, by considering the coefficients with Lemma 4.4, we see that each term (v_{i_{1}} ⊗ c_{j_{1}})∧⋯∧(v_{i_{m}}⊗c_{j_{m}}) cannot be canceled by other terms. (For example, see Example 3 in Section 5).

Thus there exists an element v_{i} ⊗ c_{j} ∈ ∧(V ⊗ H_{+}(X; ℚ)) such that δ(v_{i} ⊗ c_{j}) ∉ J.

5. Some examples

Since cup(X; ℚ) < ∞ if X is finite-dimensional and dl(Y) > 1 for any simply connected space Y of finite type, we have

Proposition 5.1. Let X be a formal, path connected, finite-dimensional CWcomplex of finite type and Y a q-connected space of finite type with q ⩾ dim X.

Then, if Y has a minimal Sullivan model of the form (∧V, 0) or all cup products on H^{+}(X; ℚ) are trivial, H^{∗}(F_{∗}(X, Y); ℚ) is always free.

Example 1. The following spaces have a minimal Sullivan model with a trivial differential:

• odd dimensional spheres,

• path connected H-spaces of finite type (cf. [3, Section 12(a), Example 3]),

• classifying spaces of path connected topological groups of finite type (cf. [3, Proposition 15.15]),

• Eilenberg-MacLane spaces of type (π, n) with n ⩾ 1, π is Abelian and π ⊗_{ℤ} ℚ is finite dimensional (cf. [3, Section 15(b), Example 2]),

• a product of above spaces.

Example 2. The following spaces are formal and all cup products on the positive dimensional rational cohomology algebra are trivial:

• spheres,

• suspensions of spaces (cf. [3, Proposition 13.9]),

• co-H-spaces,

• a wedge of above spaces.

Note that a co-H-space is rationally homotopy equivalent to a wedge of spheres (cf. [1, Section 7]), and a wedge of formal spaces is also formal.

A product of spheres S^{i_{1}} ×⋯×S^{i_{n}} is an (i_{1} +⋯+i_{n})-dimensional CW-complex and a formal space with cup(S^{i_{1}} × ⋯ × S^{i_{n}}; ℚ) = n.

It is known that the n-th James reduced product space J_{n}(S^{2i}) of a 2i-dimen-sional sphere S^{2i} is a 2ni-dimensional CW-complex which has the rational cohomology

H^{∗}(J_{n}(S^{2i}); ℚ) = ℚ[c]/(c^{n+1})

with |c| = 2i, and has a minimal Sullivan model of the form

(∧(v, θ), dθ = v^{n+1})

with |v| = 2i. Hence we have

Proposition 5.2. (1). Let Y be a q-connected space of finite type with q ⩾ i_{1} + ⋯ + i_{n}. Then

H^{∗}(F_{∗}(S^{i_{1}} × ⋯ × S^{i_{n}}, Y); ℚ)

is free if and only if dl(Y) > n.

(2). Let Y be a 2ni-connected space of finite type. Then

H^{∗}(F_{∗}(J_{n}(S^{2i}), Y); ℚ)

is free if and only if dl(Y) > n.

(3). Let X be a formal, path connected, p-dimensional CW-complex of finite type

with p < 2i. Then

H^{∗}(F_{∗}(X, J_{n}(S^{2i})); ℚ)

is free if and only if cup(X; ℚ) < n + 1.

Example 3. H^{∗}(F_{∗}(S^{1} × S^{3}, S^{6}); ℚ) is not free.

Notice that dl(S^{6}) = 2 = cup(S^{1} × S^{3}; ℚ). A basis for H_{∗}(X; ℚ) is given by \{1, c_{1}, c_{3}, c_{4}\} with |c_{j}| = j, \overline{Δ}c_{1} = \overline{Δ}c_{3} = 0 and

\overline{Δ}c_{4} = μ_{1,3}c_{1} ⊗ c_{3} + μ_{3,1}c_{3} ⊗ c_{1},

where μ_{1,3} = (−1)^{1·3}μ_{3,1} = −μ_{3,1}. A minimal Sullivan model for S^{6} is given by (∧(v_{6}, v_{11}), d) with |v_{i}| = i, dv_{6} = 0 and dv_{11} = v_{6}^{2}. By applying the construction described in Section 2, F_{∗}(S^{1} × S^{3}, S^{6}) has a minimal Sullivan model of the form

(∧(\{v_{6}, v_{11}\} ⊗ \{c_{1}, c_{3}, c_{4}\}), \overline{δ}).

Then, by the formula (4.1) and Lemmas 4.4 and 4.5, we have

δ(v_{11} ⊗ c_{4})
= (−1)^{6·4}(v_{6} ⊗ c_{4}) ∧ (v_{6} ⊗ 1) + (−1)^{6·0}(v_{6} ⊗ 1) ∧ (v_{6} ⊗ c_{4})
+ (−1)^{6·1}μ_{1,3}(v_{6} ⊗ c_{1}) ∧ (v_{6} ⊗ c_{3}) + (−1)^{6·3}μ_{3,1}(v_{6} ⊗ c_{3}) ∧ (v_{6} ⊗ c_{1})
= (v_{6} ⊗ c_{4}) ∧ (v_{6} ⊗ 1) + (−1)^{(6−0)(6−4)}(v_{6} ⊗ c_{4}) ∧ (v_{6} ⊗ 1)
+ μ_{1,3}(v_{6} ⊗ c_{1}) ∧ (v_{6} ⊗ c_{3}) + (−1)^{(6−3)(6−1)+1}μ_{1,3}(v_{6} ⊗ c_{1}) ∧ (v_{6} ⊗ c_{3})
= 2(v_{6} ⊗ c_{4}) ∧ (v_{6} ⊗ 1) + 2μ_{1,3}(v_{6} ⊗ c_{1}) ∧ (v_{6} ⊗ c_{3}),

and so \overline{δ}(v_{11} ⊗ c_{4}) = 2μ_{1,3}(v_{6} ⊗ c_{1}) ∧ (v_{6} ⊗ c_{3}) ≠ 0.

References

[1] M. Arkowitz, Co-H-spaces, Handbook of algebraic topology, 1143–1173, North-Holland, 1995.

[2] E. H. Brown, Jr. and R. H. Szczarba, On the rational homotopy type of function spaces, Trans. Amer. Math. Soc. 349 (1997), 4931–4951.

[3] Y. Félix, S. Halperin and J.-C. Thomas, Rational homotopy theory, Graduate Texts in Mathematics 205, Springer-Verlag, 2001.

[4] K. Kuribayashi, Rational model for the evaluation map and iterated cyclic homology, preprint.

This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2004/n1a18/v6n1a18.(dvi,ps,pdf)

Yasusuke KOTANI kotani@math.kochi-u.ac.jp

Department of Mathematics Faculty of Science Kochi University Kochi 780-8520, Japan