Homology, Homotopy and Applications, vol.1, No.10, 1999, pp.205–217 ISSN 1512–0139

ON SPACES OF THE SAME STRONG n-TYPE

YVES FÈLIX and JEAN-CLAUDE THOMAS

(communicated by Lionel Schwartz)

Abstract

Let X be a connected CW complex and [X] be its homotopy type. As usual, SNT(X) denotes the pointed set of homotopy types of CW complexes Y such that their n^{th}-Postnikov approximations X^{(n)} and Y^{(n)} are homotopy equivalent for all n. In this paper we study a particularly interesting subset of SNT(X), denoted SNT_{π}(X), of strong n type; the n^{th}-Postnikov approximations X^{(n)} and Y^{(n)} are homotopy equivalent by homotopy equivalences satisfying an extra condition at the level of homotopy groups. First, we construct a CW complex X such that SNT_{π}(X) ≠ {[X]} and we establishe a connection between the pointed set SNT_{π}(X) and sub-groups of homotopy classes of selfequivalences via a certain lim_{←}^{1} set. Secondly, we prove a conjecture of Arkowitz and Maruyama concerning subgroups of the group of self equivalences of a finite CW complex and we use this result to establish a characterization of simply connected CW complexes with finite dimensional rational cohomology such that SNT_{π}(X) = {[X]}.

1. Introduction

Let X be a connected CW complex and [X] be its homotopy type. As usual, SNT(X) denotes the pointed set of homotopy types of CW complexes Y such that their n^{th}-Postnikov approximations X^{(n)} and Y^{(n)} are homotopy equivalent (then X and Y have the same n type for all n ⩾ 1).

The first example of a space X with SNT(X) ≠ {[X]} has been given by J.F. Adams in 1957 ([1]). In 1966 Brayton Gray, ([7]), found one with finite type. In general, the determination of spaces X such that SNT(X) is a singleton remains an open problem. For instance, in ([12]), C.A. McGibbon and J. Möller conjecture that SNT(ΩX) = {[ΩX]} for a simply connected finite CW complex X and prove this conjecture when X is an H_{0}-space, and in fact using, [13], their proof works as well in order to prove the conjecture for any rational elliptic space, [6].

In this paper we consider a particular subset of SNT(X) denoted SNT_{π}(X) that we now describe. Denote by P_{X} the set of pairs (Y, (f_{n})_{n⩾1}), where f_{n} : Y^{(n)} → X^{(n)} is a homotopy equivalence and such that the following diagrams commute for n ⩾ 2.

π_{∗}(Y^{(n)}) ⟶^{π_{∗}(f_{n})} π_{∗}(X^{(n)})
↓π_{∗}(p_{Y}^{(n)})  ↓π_{∗}(p_{X}^{(n)})
π_{∗}(Y^{(n−1)}) ⟶^{π_{∗}(f_{n−1})} π_{∗}(X^{(n−1)})

The first author was partially supported by a CNRS research position held in Angers and the second by UPRESA CNRS 60093. Both authors were partially supported by a CNRS-FNRS-CGRI Tournesol agreement Received 28 September 1999, revised 18 December 1999; published on 27 December 1999. 1991 Mathematics Subject Classification: 55P10, 55P15, 18G55 Key words and phrases: Derived functor, Postnikov tower, Self-equivalences, lim^{1}, Phantom map \begin{matrix}c \\ ◯ \end{matrix} 1999, Yves Fèlix and Jean-Claude Thomas. Permission to copy for private use granted.

We put SNT_{π}(X) = { [Y] | (Y, (f_{n})) ∈ P_{X} }.

When [Y] belongs to SNT_{π}(X), we say that X and Y have the same strong n type.

It follows from the works of Adams ([1]), Wilkerson ([18]), McGibbon and Möller ([12], [10]) that

SNT (X) = SNT_{π}(X) = {[X]}

in each of the following cases (a) if π_{i}(X) is a finite group for all i > 0;

(b) if X is a simply connected finite type rational space;

(c) if X is a simply connected, ℚ-finite type, H_{0}-space such that the natural map Aut(X) → Aut H^{⩽n}(X; ℤ) has a finite cokernel for n ⩾ 1. Here Aut H^{⩽n}(X; ℤ) denotes the group of ring homomorphisms.

Example 1. SNT_{π}(X) ≠ {[X]}) when X is the 0-localization of the CW complex

Z = (S_{a}^{2} × S_{b}^{2} × ⋁_{p⩾1} S^{6p}) ∨ ( ⋁_{n⩾1} S_{n}^{4}) ∪_{(γ_{n,p})} ⋃_{\begin{matrix}p⩾1 \\ n>p \end{matrix}} e_{n}^{4+6p} ,

where γ_{n,p} is the Whitehead bracket [S_{n}^{4}, S^{6p}]. Observe that X is not of finite type (see (b) above !). More precisely we prove the following equality (Theorem 4 below)

SNT_{π}(X) ≅ ∏_{n⩾1} ℚ/ ⨁_{n⩾1} ℚ .

Our main result establishes a connection between the pointed set SNT_{π}(X) and sub-groups of homotopy classes of self-equivalences via a certain lim_{←}^{1} set. We will state the precise results with its first consequences after some definitions.

If X is a based topological space, Aut (X) denotes the group of homotopy classes of self homotopy equivalences of X, and Aut_{π}(X) the subgroup of homotopy classes which induce the identity on the homotopy groups of X. A tower of groups

⋯ → Aut X^{(n)} →^{q_{n}} Aut X^{(n−1)} → ⋯ → Aut X^{(1)} ,

is defined by the homomorphisms q_{n}([f]) = [f^{(n−1)}] where f^{(n−1)} denotes the map induced up to homotopy by f :

X^{(n)}  ⟶^{f}  X^{(n)}
p_{Y}^{(n)} ↓  ↓ p_{X}^{(n)}
X^{(n−1)}  ⟶^{f^{n−1}} X^{(n−1)} .

In 1975, C. Wilkerson ([18]) establishes the existence of a natural pointed set bijection

θ^{X} : SNT(X) → lim_{←}^{1} Aut X^{(n)} .

where lim_{←}^{1}G_{n} is defined for a tower of groups

⋯ → G_{n} →^{q_{n}} G_{n−1} → ⋯ →^{q_{2}} G_{1} ,

as the orbit set for the action of the group ∏_{n} G_{n} acting on the set ∏_{n} G_{n} by

(γ_{n}) · (α_{n}) = (γ_{n}α_{n}q_{n+1}(γ_{n+1})^{−1}) .

The sequence of injections Aut_{π}X^{(n)} → Aut X^{(n)} induces a natural map

j_{X} : lim_{←}^{1} Aut_{π}X^{(n)} → lim_{←}^{1} Aut X^{(n)} .

Theorem 1. Let X be a simply connected CW complex. Then there is a pointed set bijection SNT_{π}(X) → Image j_{X}.

Example 2. Assume X is a simply connected finite type CW-complex that has the rational homotopy type of a bouquet of spheres. By ([12], Lemma 1), the groups Aut_{π}X^{(n)} are finite, so that lim_{←}^{1} Aut_{π}X^{(n)} = {∗} and SNT_{π}(X) = {[X]}, see [12].

Example 3. Denote by f : ℂP^{∞} → S^{3} a phantom map and by H : S^{3} → S^{2} the Hopf map. The homotopy cofibre Z, of H ∘f, and the space S^{2} ∨ΣℂP^{∞} have the same n type for all n. D. Stanley ([16]) has proved that the Lusternik-Schnirelmann category of Z is two whereas the category of S^{2} ∨ ΣℂP^{∞} is one. Therefore the spaces Z and S^{2} ∨ ΣℂP^{∞} do not have the same homotopy type and thus not the same strong n type because they are rational suspensions.

Theorem 2. Let X be a simply connected finite type CW complex with finite dimensional rational cohomology, then SNT_{π}(X) = {[X]} if and only if for some integer N, the morphism Aut_{π}(X) → Aut_{π}(X^{(N)}) has a finite cokernel.

2. The six term exact sequence

Let X be a simply connected finite type CW complex. Denote by G_{n} the image of the natural morphism AutX^{(n)} → Autπ_{⩽n}(X). Then the short exact sequence of towers

1 → Aut_{π}X^{(n)} → AutX^{(n)} → G_{n} → 1

induces a 6-term exact sequence, [8]:

1 → lim_{←} Aut_{π}X^{(n)} → lim_{←} AutX^{(n)} → lim_{←} G_{n} →^{δ} lim_{←}^{1} Aut_{π}X^{(n)} →^{j_{X}}

→^{j_{X}} lim_{←}^{1}AutX^{(n)} → lim_{←}^{1}G_{n} → ∗

The image of j_{X} is, by Theorem 1, in bijection with the subset SNT_{π}(X) of SNT (X) so that we obtain a 6-term exact sequence

1 → Aut_{π}X → AutX → lim_{←} G_{n} →^{δ} lim_{←}^{1} Aut_{π}X^{(n)} →^{j_{X}} SNT (X) → lim_{←}^{1}G_{n} → ∗ .

In particular, in the case SNT_{π}(X) = {[X]} we have a bijection

SNT (X) ⟶^{≅} lim_{←}^{1}G_{n} .

This 6-term exact sequence provides us with anaother description of the set SNT_{π}X. The group lim_{←} G_{n} is the subgroup of Aut π_{∗}(X) consisting of those automorphisms φ such that, for each n ⩾ 1, the restriction of φ to π_{⩽n} can be realized by some automorphism α_{n} ∈ AutX^{(n)}. The connecting map δ associates to φ the class of the element (α_{n}(α_{n+1})^{−1})_{n⩾1} in

lim_{←}^{1} Aut_{π}X^{(n)}.

The group lim_{←} G_{n} acts on the set lim_{←}^{1} Aut_{π}X^{(n)} in the following way : Let φ be an element

in lim_{←} G_{n} whose restriction to π_{⩽n} is realized by an automorphism α_{n} ∈ AutX^{(n)}, and let

β = [(β_{n})] ∈ lim_{←}^{1} Aut_{π}X^{(n)}, then

(φ · β)_{n} = α_{n}β_{n}(α_{n+1})^{−1} .

Therefore we deduce:

Proposition 1. The set SNT_{π}(X) is the orbit space of lim_{←}^{1} Aut_{π}X^{(n)} for this action.

Let us precise here another interesting fact concerning the above 6-term exact sequence. Let aut_{π}X denote the monoid of self equivalences which induce the identity map at the level of homotopy groups and recall that π_{0}(aut_{π}X) = Aut_{π}X. From the work of Bousfield and Kan ([4]), we have a diagram of short exact sequences

1 → lim_{←}^{1}π_{1}( aut _{π}X^{(n)}) → aut _{π}X →^{ρ_{1}}  lim_{←} Aut _{π}X^{(n)} → 1
∥  ↓  ↓
1 → lim_{←}^{1}π_{1}( aut _{π}X^{(n)}) → AutX →^{ρ}  lim_{←} Aut X^{(n)} → 1

Moreover the kernel of ρ is the subgroup of automorphisms φ whose restrictions to X^{(n)} are homotopic to the identity for each n ⩾ 1. This subgroup of Aut_{π}X is called the group of weak identities of X. Observe that if X is an H-space, the set of homotopy classes of phantom maps from X into X, P h(X, X), is a group, and that the map θ : P h(X, X) → Ker ρ defined by θ(f) = id + f is a group isomorphism ([15]).

There is another interesting subgroup of Aut_{π}X^{(n)}, the subgroup Aut_{Ω}X^{(n)} formed by the automorphisms φ such that Ωφ is homotopic to the identity. By ([5]), the maps Aut_{Ω}X^{(n)} → Aut_{π}X^{(n)} are injections of finitely generated nilpotent groups. Since these injections become isomorphisms after Malcev completion, the quotients H_{n} = Aut_{π}X^{(n)}/ Aut_{Ω}X^{(n)} are finite groups. Therefore lim_{←}^{1}H_{n} = 0, and we have a surjection

lim_{←}^{1} Aut_{Ω}X^{(n)} → lim_{←}^{1} Aut_{π}X^{(n)} .

This shows that

Proposition 2.

SNT_{π}(X) = Image ( lim_{←}^{1} Aut_{Ω}X^{(n)} → lim_{←}^{1}AutX(n) ) .

In other words, the set SNT_{π}(X) can also be described as the subset of SNT (X) formed by the spaces Y for which there exist maps f_{n} : Y^{(n)} → X^{(n)} such that the following diagrams commute for n ⩾ 2.

ΩY^{(n)}  ⟶^{Ωf_{n}} ΩX^{(n)}
↓Ωp_{Y}^{(n)}  ↓Ωp_{X}^{(n)}
ΩY^{(n−1)} ⟶^{Ωf_{n−1}} ΩX^{(n−1)}

3. Proof of Theorem 1

Recall that P_{X} denotes the set of pairs (Y, (f_{n})_{n⩾1}), where f_{n} : Y^{(n)} → X^{(n)} is an homotopy equivalence and such that the following diagrams commute

π_{∗}(Y^{(n)}) ⟶^{π_{∗}(f_{n})} π_{∗}(X^{(n)})
↓π_{∗}(p_{Y}^{(n)})  ↓π_{∗}(p_{X}^{(n)})
π_{∗}(Y^{(n−1)}) ⟶^{π_{∗}(f_{n−1})} π_{∗}(X^{(n−1)})

We introduce an equivalence relation on the set P_{X}. Two pairs (Y, (f_{n})) and (Z, (g_{n})) in P_{X} are equivalent if there is an homotopy equivalence φ : Z → Y such that the maps f_{n} ∘ φ ∘ g_{n}^{−1} belongs to Aut_{π}(X^{(n)}) for all n. For instance, for any homotopy equivalence m : Y → Z and each element (Z, (g_{n})), the elements (Y, (g_{n} ∘ m)) and (Z, (g_{n})) are equivalent. We denote by E_{X} the quotient set P_{X}/ ∼. To each pair (Y, (f_{n})) ∈ P_{X} we associate the homotopy type of Y into SNT_{π}(X). We obtain in this way a well defined map ρ : E_{X} → SNT_{π}(X).

The following proposition implies then clearly Theorem 1.

Proposition 3. Let X be a simply connected CW complex. There is a natural pointed set bijection θ_{1}^{X} : E_{X} → lim_{←}^{1} Aut_{π}X^{(n)}. Moreover, the bijections θ^{X} and θ_{1}^{X} fit together in a commutative diagram

lim_{←}^{1} Aut_{π}X^{(n)} ⟶ Image j_{X} ↪ lim_{←}^{1}AutX^{(n)}
↑ θ_{1}^{X}  ↑ ≅  ↑ θ^{X}
E_{X}  ⟶^{ρ} SNT_{π}(X) ↪ SNT (X) .

Proof. Since the simplicial Postnikov decomposition is functorial, we consider Kan complexes instead of CW complexes. The n^{th}-Postnikov complex of a simplicial complex X is the quotient X^{(n)} = X/ ∼_{n} where x ∼_{n} y if the corresponding faces of x and y of dimensions less than or equal to n are equal. The quotient X^{(n)} is a Kan complex, and the natural projections

p_{X}^{(n)} : X^{(n)} → X^{(n−1)}

are Kan fibrations with Eilenberg-MacLane complexes K(π_{n}(X), n) as fibres.

We suppose that X is a minimal Kan complex. Therefore each X^{(n)} is also minimal. As usual, AutX^{(n)} denotes the group of simplicial isomorphisms and Aut_{π}X^{(n)} denotes the subgroup of automorphisms inducing the identity map on the homotopy groups.

If α_{n} ∈ AutX^{(n)}, the composite α_{n}∘p^{(n+1)} is again a Kan fibration and a Postnikov section. Let ᾱ = (α_{0}, ⋯) denotes the sequence of the α_{n} in ∏_{n} Aut_{π}(X^{(n)}), and define

Xᾱ = lim_{←}(X^{(n)}, α_{n−1} ∘ p^{(n)}) .

Then Xᾱ is again a minimal Kan complex and the commutativity of the square

X^{(n)}  = X^{(n)}
p_{X}^{(n)} ↓  ↓ ^{α_{n−1}∘p^{(n)}}
X^{(n−1)} ⟶^{α_{n−1}} X^{(n−1)}

shows that for each n there are homotopy equivalences f ᾱ ; n : X^{(n)} → X(n) ; ᾱ such that the following diagram commutes

π_{∗}(X^{(n)}) ⟶^{π_{∗}(f ᾱ ; n)}  π_{∗}(X(n) ; ᾱ)
π_{∗}(p_{X}^{(n)}) ↓  ↓ π_{∗}(α_{n−1}∘p^{(n)})
π_{∗}(X^{(n−1)}) ⟶^{π_{∗}(α_{n−1})} π_{∗}(X^{(n−1)}) .

In particular, (Xᾱ, (f ᾱ ; n)) defines an element in P_{X} and in E_{X}.

Suppose that ᾱ and β̄ define the same element in lim_{←}^{1} Aut_{π}X^{(n)}. Then there is a sequence

γ̄ = (γ_{n}) ∈ ∏_{n} Aut_{π}X^{(n)} such that α_{n} = γ_{n} ∘ β_{n} ∘ (q_{n+1}(γ_{n+1}))^{−1}. We then have maps γ_{n} : X(n) ; β̄ ⟶ X(n) ; ᾱ such that the following diagrams commute up to homotopy

X^{(n)} ⟶^{γ_{n}} X^{(n)}
β_{n−1}∘p_{X}^{n} ↓  ↓ _{α_{n−1}∘p_{X}^{n}}
X^{(n−1)} ⟶^{γ_{n−1}} X^{(n−1})

and thus the homotopy class [(f ᾱ ; n)^{−1} ∘ γ_{n} ∘ f β̄ ; n] belongs to Aut_{π}X^{(n)}. Here, if f is a homotopy equivalence, then f^{−1} means any representative for the class [f]. The correspondance ᾱ → (Xᾱ, (f ᾱ ; n)) induces therefore a well defined map

Φ : lim_{←}^{1} Aut_{π}X^{(n)} → E_{X} ,

Which respects base points.

In order to show that Φ is a bijection we construct the inverse map. The map

θ : P_{X} → lim_{←}^{1} Aut_{π}X^{(n)} , θ(Y, (f_{n})) = ([f_{n} ∘ (f_{n+1}^{(n)})^{−1}])

factors to give a map θ_{1}^{X} : E_{X} → lim_{←}^{1} Aut_{π}X^{(n)}. Indeed, suppose that (Y, (f_{n})) and (Z, (g_{n})) are equivalent, then there exists a map φ such that, for each integer n ⩾ 1, the class α_{n} = [g_{n} ∘ φ^{(n)} ∘ f_{n}^{−1}] ∈ Aut_{π}X^{(n)}. This shows that

(α_{n}) · [f_{n} ∘ (f_{n+1}^{(n)})^{−1}] = [g_{n} ∘ (g_{n+1}^{(n)})^{−1}] .

The relation Φ ∘ θ_{1}^{X} = id results from the commutativity, up to homotopy, of the diagram

Y^{(n)}  ⟶^{f_{n}} X^{(n)}
p_{Y}^{n} ↓  ↓ _{f_{n−1}∘(f_{n}^{n−1})^{−1}p_{X}^{n}}
Y^{(n−1)} ⟶^{f_{n}^{(n−1)}} X^{(n−1)}

The relation θ_{1}^{X} ∘ Φ = id is trivially satified. □

4. A conjecture of Arkowitz and Maruyama

For a simply connected CW complex Z we denote by Aut_{#}^{n}(Z), n ⩾ 2, the kernel of the natural morphism Aut(Z) → Aut π_{⩽n}(Z). In ([3]) Arkowitz and Maruyama conjecture that for a finite simply connected CW complex there is an integer N such that the restriction map Aut_{π} → Aut_{#}^{N}Z is an isomorphism. Here we prove the conjecture as a corollary of its rational version.

Theorem 3. Let Z be a simply connected finite type CW complex and let Z_{0} its rationalization. Suppose there H^{>M}(Z; ℚ) = 0 for some M, then there is some integer N such that the restriction map

Aut_{π}Z_{0} → Aut_{#}^{N}Z_{0} ,

is an isomorphism.

Moreover if H^{>M}(Z; ℤ) = 0, then there is some integer N such that Aut_{π}Z ≅ Aut_{#}^{N}Z .

Example 3. Suppose W is the fat wedge of the three spheres S^{2}, S^{2} and S^{4}, ie. S^{2}×S^{2}×S^{4} = W ∪_{ω} D^{8} where ω ∈ π_{7}(W) denotes a triple Whitehead product. Then the rational homotopy Lie algebra of W is generated by elements of degrees less than or equal to 6 :

π_{∗}(ΩW) ⊗ ℚ ≅ Ab(x_{1}, y_{1}, z_{3}) ∐ 𝕃(t_{6}) .

The cokernel of the morphism Aut_{#}^{7}(W) → Aut_{#}^{6}(W) is not finite, but for n ⩾ 8, the cokernel of the correponding injection Aut_{#}^{n}(W) → Aut_{#}^{n−1}(W) is finite.

Example 4. The integer N depends on the space and not only on the dimension of the space. Consider for instance the 6-dimensional rational space X whose minimal Lie model is given by the following differential graded Lie algebra

L = (𝕃(x_{i}, y_{i}, z_{i}, a_{i,j}, b_{i,j}, u, v, w)_{i,j=1,...,n; i⩽j}, d)

|x_{i}| = |y_{i}| = |z_{i}| = 2, |u| = 1, |v| = 2, |a_{i,j}|, |b_{i,j}|, |w| = 5, d(u) = d(v) = d(x_{i}) = d(y_{i}) = d(z_{i}) = 0, d(a_{i,j}) = [x_{i}, z_{j}], d(b_{i,j}) = [y_{i}, z_{j}], and

d(w) = [x_{1}, y_{1}] + … + [x_{n}, y_{n}] .

The autoomorphism φ equal to the identity on all generators except w and defined on w by φ(w) = w + [v, [v, u]] belongs to Aut_{#}^{5+2n}(X) but not to Aut_{π}(X).

Proof of Theorem 3. Denote by (∧V, d) the Sullivan minimal model of Z ([17]). We denote by Aut(∧V, d) the group of homotopy classes of automorphisms of (∧V, d). Each automorphism of (∧V, d) induces an isomorphism on the vector space of indecomposable elements Q(∧V) = ∧V/ ∧^{⩾2} V ≅ V. We denote by Aut_{π}(∧V, d) the kernel of the representation morphism Aut(∧V, d) → Aut(Q(V)) and by Aut_{#}^{N}(∧V, d) the kernel of the morphism Aut(∧V, d) → Aut(Q(V)^{⩽N}). We denote also by Iso(∧V, d) the group of automorphisms of (∧V, d), and by Iso_{1}(∧V, d) the subgroup of autoomorphisms of the form id + φ, with φ : V → ∧^{⩾2}V.

From the Sullivan theory of minimal models one deduce the following group isomorphisms

AutZ_{0} = Aut(∧V, d)
Aut_{#}^{N}Z_{0} = Aut_{#}^{N}(∧V, d)
Aut_{π}Z_{0} = Aut_{π}(∧V, d)

where Z_{0} denote the 0-localization of Z.

Since H^{>M}(∧V, d) = 0, each automorphism of (∧V^{⩽M}, d) extends to an automorphism of (∧V, d). This means that the restriction map

ρ : Aut(∧V, d) → Aut(∧V^{⩽M}, d)

is onto. Let θ : (∧V, d) → (A, d) be a quasi-isomorphism with A^{>M} = 0. Suppose now that f, g ∈ Aut(∧V). If ρ(f) ∼ ρ(g), ( ∼ means homotopy in the category of c.d.g.a.’s ), then θ ∘ f ∼ θ ∘ g, and since θ is a quasi-isomorphism, f ∼ g. This shows that ρ is also injective.

We form the vector space Hom^{0}(V^{⩽M}, ∧^{⩾2}V), consisting of the degree zero linear maps. We choose an homogeneous basis, (x_{i})_{i=1,...,N_{1}}, of V^{⩽M}, and an homogeneous basis, (y_{j})_{j=1,...,N_{2}}, for (∧^{⩾2}V)^{⩽M}. Then for any φ ∈ Hom^{0}(V^{⩽M}, ∧^{⩾2}V) we write

φ(x_{i}) = ∑_{j} α_{i}^{j}y_{j} .

Of course id + φ is an automorphism of (∧V^{⩽M}, d) if and only if it commutes with the differential, i.e. if and only if, for i = 1, …, N_{1}, we have

(id + φ)d(x_{i}) = d(x_{i} + φ(x_{i})) .

By expressing these conditions in terms of the α_{i}^{j}, we obtain a bijective correspondance between Iso_{1}(∧V^{⩽M}, d) and some sub-algebraic variety W in ℚ^{N_{1}N_{2}}.

We choose a linear section, σ : (d(∧^{⩾2}V))^{>M} → (∧V)^{⩾M}, of the differential d. Then each ψ = id + φ in Iso_{1}(∧V^{⩽M}, d) extends in a natural way to an automorphism of (∧V, d) by the following rule : Let z ∈ V^{M+k}, k ⩾ 1, then ψdz is a coboundary so that we put ψ(z) = σψdz.

Performing this construction on a homogeneous basis of each V^{n+k} we construct the required homomorphism. Moreover, ψ is an isomorphism since it is a quasi-isomorphism and (∧V, d) is minimal ([17]). We put Γ_{M} = Iso_{1}(∧V^{⩽M}, d), and for m ⩾ M,

Γ_{m} = { ψ ∈ Γ , ψ(z) − z ∈ ∧^{⩾2}V , for z ∈ V^{⩽m} } .

the choice of the section isomorphisms of (∧V, d). We ψ such that in their extension Now the sequence

… ⊂ Γ_{M+2} ⊂ Γ_{M+1} ⊂ Γ_{M}

is a sequence of inclusions of algebraic varieties. Since ℚ[α_{i}^{j}] is noetherian, this sequence stabilizes : there is an integer N such that

Γ_{N} = Γ_{N+r} ∀r ⩾ 1 .

Denote by Γ_{id} the normal subgroup of Γ consisting of the automorphisms homotopic to the identity. Then f ∼ g if and only if fg^{−1} ∈ Γ_{id}. Thus Aut(∧V^{⩽M}, d) is isomorphic to the quotient Iso_{1}(∧V^{⩽M}, d))/Γ_{id}. This yields the equality

Aut_{#}^{N}(∧V, d) = Aut_{#}^{N+r}(∧V, d) ∀r ⩾ 1 .

Since Aut_{#}^{k}(∧V, d) ≅ Aut_{#}^{k}(Z_{0}), ([17]), we obtain finally

Aut_{#}^{N}Z_{0} = Aut_{#}^{N+r}Z_{0} ∀r ⩾ 1

Suppose now that Z is a simply connected finite CW complex. Then Aut_{π}(Z) ⊂ Aut_{#}^{N}(Z) is an normal inclusion of finitely generated nilpotent groups ([9]) with same Malcev completions. It is then easy to see by induction on the nilpotency index that the quotient is a finite group. Therefore there is an integer P ⩾ N such that Aut_{π}(Z) ≅ Aut_{#}^{P}(Z). □

5. Proof of Theorem 2

The proof of Theorem 2 is based on a sequence of 4 lemmas.

Lemma 1. There is a rational homotopy equivalence K : W → X where W is a finite CW complex.

Henceforth we consider K : W → X as in lemma 1, and by Theorem 3, we fix an integer N such that Aut_{#}^{N}X_{0} = Aut_{π}X_{0} and H^{⩾N}(X; ℚ) = 0.

Lemma 2. The morphism Aut_{π}W^{(n+1)} ⟶^{f→f^{(n)}} Aut_{π}W^{(n)} is injective and has a finite cokernel for each n > N.

Lemma 3. The morphism Aut_{π}X^{(n+1)} ⟶^{f→f^{(n)}} Aut_{π}X^{(n)} has a finite kernel and a finite cokernel for n ⩾ N.

Theorem 2 follows from previous lemmas and the following lemma due to McGibbon and Möller ([12], Lemma 3).

Lemma 4. Let G_{n} ⟶^{q_{n}} G_{n−1} be a tower of groups. Suppose that q_{n}(G_{n}) has finite index into G_{n−1}, for n > N. Then lim_{←}^{1}G_{n} = {∗} if and only if lim_{←} G_{n} → G_{N} has a finite cokernel.

Proof of Lemma 1.

Suppose that H^{>M}(X; ℚ) = 0. We consider the cellular chain complex of X,

C_{M+1}(X; ℤ) →^{∂} C_{M}(X; ℤ) → C_{M−1}(X; ℤ) → ⋯

where each C_{k}(X; ℤ) is freely generated by the k-cells of X as a ℤ-module. In particular, C_{M+1}(X; ℤ) = ⊕_{i=1}^{r}ℤe_{i}. By an elementary argument in linear algebra, there exists elements i_{1}, ⋯, i_{s} ∈ { 1, ⋯, r } such that ∂ ⊗ ℚ sends isomorphically ⊕_{j=1}^{s}ℚe_{i_{j}} to the image of ∂ ⊗ ℚ in C_{M}(X; ℚ).

Denote by W the subcomplex of X obtained by adding to the M-skeleton of X the (M +1)- cells e_{i_{1}}, ⋯, e_{i_{s}}. The injection K : W → X induces an isomorphism in the homology of the rational cellular chain complexes and is therefore a rational homotopy equivalence. □

Proof of Lemma 2. Since n > N > dim W, and since W^{(n)} = W ∪_{φ_{α}} e^{n_{α}}, n_{α} > n + 2, the restriction maps

Aut_{#}^{n}W^{(n+1)} → Aut_{#}^{n}W^{(n)}
AutW → AutW^{(n+1)} → AutW^{(n)}

are isomorphisms. This gives directly the first part of the lemma.

Consider the following diagram obtained by restriction and rationalization

Aut_{#}^{n+1}W^{(n+1)} ↪^{ρ} Aut_{#}^{n}W^{(n+1)} ⟶^{rep} Autπ_{n+1}(W)
r′ ↓  ↓ ^{r}
Aut_{#}^{n+1}W_{0}^{(n+1)} = Aut_{#}^{n}W_{0}^{(n+1)} ⟶^{rep_{0}} Autπ_{n+1}(W) ⊗ ℚ

By hypothesis, (cf. Theorem 3), the representation map rep_{0} is trivial, the image of the representation rep : Aut_{#}^{n}W^{(n+1)} → Aut π_{n+1}(W) is finite and thus its kernel Aut_{#}^{n+1}W^{(n+1)} is a subgroup of finite index.

Therefore the composite

Aut_{π}W^{(n+1)} = Aut_{#}^{n+1}W^{(n+1)} ↪ Aut_{#}^{n}W^{(n+1)} → Aut_{#}^{n}W^{(n)} = Aut_{π}W^{(n)}

has a finite cokernel. □

Proof of Lemma 3. We use the same argument as in ([12]). We define

Δ(f^{(n)}) = {(α, β) ∈ Aut_{π}W^{(n)} × Aut_{π}X^{(n)} | f^{(n)}α = βf^{(n)} }

Then Δ(f^{(n)}) is a group and since f : W → X is a rational equivalence, by a result of Wilkerson ([19]) the projections from Δ(f^{(n)}) to Aut_{π}W^{(n)} and to Aut_{π}X^{(n)} have finite kernels and cokernels.

Now, using Lemma 2, a simple diagram chasing yields the result.

Aut_{π}W^{(n+1)} ← Δ(f^{(n+1)}) → Aut_{π}X^{(n+1)}
↓  ↓  ↓
Aut_{π}W^{(n)} ← Δ(f^{(n)}) → Aut_{π}X^{(n)}
□

6. Example of a space X with SNT_{π}(X) ≠ {[X]}

Consider the CW complex

Z = (S_{a}^{2} × S_{b}^{2} × ⋁_{p⩾1} S^{6p}) ∨ ( ⋁_{n⩾1} S_{n}^{4}) ∪_{(γ_{n,p})} ⋃_{\begin{matrix}p⩾1 \\ n>p \end{matrix}} e_{n}^{4+6p} ,

where γ_{n,p} is the Whitehead bracket [S_{n}^{4}, S^{6p}].

The space T is constructed as a deformation of the space Z.

T = ((S_{a}^{2} ∨ S_{b}^{2}) × (∨_{p⩾1}S^{6p})) ∪_{[S_{a}^{2},S_{b}^{2}]} e^{4} ∨ ( ⋁_{n⩾1} S_{n}^{4}) ∪_{(γ_{n,p})} ⋃_{\begin{matrix}p⩾1 \\ n>p \end{matrix}} e_{n}^{4+6p} ∪_{\overline{ω}_{p}} ⋃_{p⩾1} e^{4+6p} ,

where \overline{ω}_{p} = ω_{p} + ∑_{j=1}^{p}[S^{6p}, S_{j}^{4}]. Here the element ω_{p} is a triple Whitehead product in the π_{6p+3}

of the fat wedge T (S_{a}^{2}, S_{b}^{2}, S^{6p}).

The spaces X and Y are respectively the rationalizations of Z and T. We show that X and Y do not have the same homotopy type, and that there exists a sequence of homotopy equivalences between the n^{th} skeleton

f_{n} : X_{n} → Y_{n}

such that

π_{r}(f_{n}) = π_{r}(f_{m}) ,

when r ⩽ n ⩽ m. In particular f_{n} induces an homotopy equivalence f_{n}^{(n−1)} : X^{(n−1)} → Y^{(n−1)}, so that SNT_{π}(X) ≠ {X}.

By Quillen rational homotopy theory ([14]), there is an equivalence of homotopy category between the homotopy category of simply connected CW complexes and the homotopy category of connected differential graded Lie algebras. It thus suffices to prove the corresponding results into the category of differential graded Lie algebras.

The Quillen models L_{X} and L_{Y} of X and Y are respectively given by :

L_{X} = (𝕃(a, b, c, x_{i}, y_{i}, v_{i}, w_{i}, t_{i}, z_{i,j} ; i ⩾ 1, j > i), d) ,

with |a| = |b| = 1, |c| = 3, |x_{i}| = 6i − 1, |y_{i}| = 3, |z_{i,j}| = 6i + 3, |v_{i}| = 6i + 1, |w_{i}| = 6i + 1, |t_{i}| = 6i + 3, d(c) = [a, b], d(x_{i}) = 0, d(y_{i}) = 0, d(z_{i,j}) = [x_{i}, y_{j}], d(v_{i}) = [a, x_{i}], d(w_{i}) = [b, x_{i}], d(t_{i}) = ω_{i} = [v_{i}, b] + [c, x_{i}] + [w_{i}, a].

L_{Y} = (𝕃(a, b, c, x_{i}, y_{i}, v_{i}, w_{i}, t_{i}, z_{i,j} ; i ⩾ 1, j > i), d) ,

with same degrees and differentials, except that

d(t_{i}) = ω_{i} + ∑_{j=1}^{i}[x_{i}, y_{j}] .

Suppose that φ : L_{X} → L_{Y} is a quasi-isomorphism and denote by 𝕃^{p} the ideal of L_{X} or L_{Y} generated by iterated brackets of lenght ⩾ p.

By composition with an automorphism of π_{2}(X) ⊗ ℚ = aℚ ⊕ bℚ we can suppose that φ(a) = a and φ(b) = b. Then we have

φ(c) = c + ∑_{k⩾1} γ_{k}y_{k}
φ(x_{i}) = μx_{i} mod𝕃^{2}, μ ≠ 0
φ(v_{i}) = μv_{i} mod𝕃^{2}
φ(w_{i}) = μw_{i} mod𝕃^{2}
φ(t_{i}) = δt_{i} + ∑_{j>i} β_{i,j}z_{i,j}

and,

(dφ − φd)(t_{i}) = (δ − μ)ω_{i} + ∑_{j=1}^{i} δ[x_{i}, y_{j}] + ∑_{j>i} β_{i,j}[x_{i}, y_{j}] − ∑_{k⩾1} γ_{k}μ[y_{k}, x_{i}] mod𝕃^{3} .

Therefore γ_{k} ≠ 0 for k ⩽ i. Since this has to be true for any i ⩾ 1, we have

φ(c) = c + ∑_{k⩾1} γ_{k}y_{k}

with all γ_{k} ≠ 0, which is impossible.

On the other hand, by taking μ = δ = γ_{k} = 1 for k ⩽ p, and γ_{k} = 0 for k > p, we obtain an homotopy equivalence f_{n} : X^{n} → Y^{n} with n = 4 + 6p.

Remark now that the composition of injections

((S^{2} × S^{2} × (∨_{p⩾1}S^{6p})) ∨ (∨_{n⩾1}S_{n}^{4}))_{0} →^{i} X → ([(S^{2} × S^{2}) ∨ (∨_{n⩾1}S_{n}^{4})] × [∨_{p⩾1}S^{6p}])_{0}

induces a surjective map on homotopy groups. It results then from results of Anick ([2]) that π_{∗}(i) is a surjective map, and using this it is clear that π_{r}(f_{m}) = π_{r}(f_{n}) when r ⩽ m ⩽ n.

Theorem 4. Let X be the rationalizations of the CW complex

Z = (S_{a}^{2} × S_{b}^{2} × ⋁_{p⩾1} S^{6p}) ∨ ( ⋁_{n⩾1} S_{n}^{4}) ∪_{(γ_{n,p})} ⋃_{\begin{matrix}p⩾1 \\ n>p \end{matrix}} e_{n}^{4+6p} ,

where γ_{n,p} is the Whitehead bracket [S_{n}^{4}, S^{6p}]. Then

SNT_{π}(X) ≅ lim_{←}^{1}Aut_{π}X^{(n)} ≅ ∏_{n⩾1} ℚ/ ⨁_{n⩾1} ℚ .

Proof. Since π_{∗}(ΩX) ⊗ ℚ is generated as an algebra by the elements a, b, x_{i} and y_{i}, i ⩾ 1, with the notations of section 4, the restriction map Aut X → lim_{←} G_{n} is surjective. Therefore

SNT_{π}(X) = lim_{←}^{1}Aut_{π}X^{(n)}.

Denote by L_{X}^{(n)} the quotient of L_{X} by the ideal J = (L_{X})^{>n+1} ⊕ S, where S = d(L_{X}^{n+2}). Clearly Aut _{π}X^{(n)} can be identified with the group of homotopy classes of automorphisms φ of L_{X}^{(n}) such that φ is the identity on the vector space generated by the elements a, b, x_{i}, y_{i}, i ⩾ 1.

To each φ ∈ Aut _{π}X^{(n)}, n ⩾ 4, we can associate the element φ(c) − c ∈ ⊕_{i⩾1}ℚy_{i}. We define in this way a linear map Aut _{π}X^{(n)} → π_{3}(ΩX) whose kernel and cokernel are respectively denoted by K_{n} and I_{n};

0 → K_{n} → Aut _{π}X^{(n)} → I_{n} → 0 .

For φ ∈ K_{n} and n ⩾ 6i + 3, the elements φ(v_{i}) − v_{i} and φ(w_{i}) − w_{i} belong to the sub Lie algebra L generated by a, b and the x_{j}. This define a map

ψ : K_{n} → ⨁_{i | 6i+3⩽n} (L_{6i+3} ⊕ L_{6i+3}) ,

and this gives a short exact sequence

0 → K′_{n} = Kerψ → K_{n} ⟶^{ψ} ⨁_{i | 6i+3⩽n} (L_{6i+3} ⊕ L_{6i+3}) → 0 .

Since the induced map ψ : lim_{←} K_{n} → ⨁_{i⩾1}(L_{6i+3} ⊕ L_{6i+3}) is surjective, we have

lim_{←}^{1}K_{n} ≅ lim_{←}^{1}K′_{n} .

Now remark that the sequence K′_{n} → K′_{n−1} is a sequence of surjective maps. It follows that lim_{←}^{1}K′_{n} = 0. This gives the isomorphism

lim_{←}^{1}Aut _{π}X^{(n)} ≅ lim_{←}^{1}I_{n} .

Now, clearly, I_{6i+4} = ⨁_{j>i} ℚy_{j}. We consider then the short exact sequence of towers

0 → ⨁_{j⩽i} ℚy_{j} → ⨁_{j⩾1} ℚy_{j} → ⨁_{j>i} ℚy_{j} → 0 .

The associated six-terms exact sequence reduces to the exact sequence

0 → ⨁_{j⩾1} ℚy_{j} → ∏_{j⩾1} ℚy_{j} → lim_{←}^{1}I_{n} → 0 .

□

References

1. J.F. Adams. An example in homotopy theory. Proc. Cambridge Philosophical Society 53 (1957), 922-923.

2. D. Anick. Non commutative graded algebras and their Hilbert series. J. of Algebra 78 (1982), 120-140.

3. M. Arkowitz and K.-I. Maruyama. Self homotopy equivalences which induce the identity on homology, cohomology or homotopy groups. Preprint.

4. A.K. Bousfield and D. Kan. Homotopy limits, completions and localizations. Lecture Notes in Mathematics 304, Springer-Verlag (1972).

5. E. Dror and A. Zabrodsky. Unipotency and nilpotency in homotopy equivalences. Topology 18 (1979), 187-197.

6. Y. Félix, S. Halperin and J.-C. Thomas. Elliptic spaces Bulletin of the Amer. Math. Soc. 25, (1991) 69-73.

7. B. Gray. Spaces of the same n-type, for all n. Topology 5 (1966), 241-243.

8. C. Jensen. Les foncteurs dérivés de lim_{←} et leurs applications en théorie des modules. Lecture Notes in Mathematics 254, Springer-Verlag, 1972.

9. K.-I. Maruyama. Finitely presented subgroups of self-homotopy equivalences group. Math. Z. 221 (1996), 537-548.

10. C .A. McGibbon and J.M. Moller. On spaces of the same n-type for all n. Topology 31 (1992), 177-201.

11. C .A. McGibbon and J.M. Moller. How can you tell two spaces apart when they have the same n-type for all n ? Adams Memorial Symposium on Algebraic Topology, London Math. Soc. Lecture Note Series 175 (1992), 131-144.

12. C.A. McGibbon and J.M. Moller. On infinite dimensional spaces that are rationally equivalent to a bouquet of spheres. Proceeding of the Barcelona conferences (1998).

13. C.A. McGibbon and C. Wilkerson. Loop space of finite complexes at large primes. Proc. Amer. Math. Soc. 96 (1986), 698-702

14. D. Quillen. Rational homotopy theory. Annals of Math. 90 (1969), 205-295.

15. J. Roitberg. Weak identities, phantom maps and H-spaces. Israel J. Math. 66 (1989), 319-329.

16. D. Stanley. On the Lusternik-Schnirelmann category of maps, Preprint (1998).

17. D. Sullivan. Infinitesimal computations in topology. Publ. IHES 47 (1977), 269-331.

18. C. Wilkerson. Classification of spaces of the same n-type for all n. Proceedings of the American Mathematical Society 80 (1976), 279-285.

19. C. Wilkerson. Applications of minimal simplicial groups. Topology. 15 (1976), 111-130.

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/1999/n10/n10.(dvi,ps,dvi.gz,ps.gz)

Yves Fèlix felix@agel.ucl.ac.be

Départment de Mathématiques Chemin du Cyclotron 2 1348 Louvain-la-Neuve Belgique

Jean-Claude Thomas jean-claude.thomas@univ-angers.fr

Université d’Angers Faculté des Sciences 2bd Lavoisier, Cedex 01 France