Sullivan minimal models

Y. Félix

Let X be a 1-connected finite type CW complex. The Sullivan minimal models theory associates to X a finite type commutative differential graded algebra m_{X}. The differential algebra m_{X} is called the Sullivan minimal model of X, contains all the rational homotopy invariants of X and is all what we need to reconstruct the rationalization X_{0} of X ([6]).

To be more precise, Sullivan associates first to each space X a cdga (commutative differential graded algebra) consisting of simplicial forms on X, A(X). This algebra A(X) is connected to the usual singular cochains C^{∗}(X; ℚ) by a sequence of quasi-isomorphisms of differential graded algebras. For recall

(1) a quasi-isomorphism is a map that induces an isomorphism in cohomology;

(2) a graded algebra A is commutative if a.b = (−1)^{|b| |a|}a.b for a, b ∈ A.

Among the cdga’s there are special ones : Let V be a graded vector space, V = {V^{n}}_{n≥1}; the free cdga on V, ΛV, is T V/I, where I is the ideal generated by the elements xy − (−1)^{|x| |y|}yx, x, y ∈ V. The differential graded algebra ΛV satisfies the following universal property : Let φ : V → A be a linear map of degree zero and A be a commutative graded algebra, then φ extends in an unique way to a morphism of graded algebras Φ : ΛV → A.

We clearly have :

ΛV = Symmetric algebra (V^{even}) ⊗ Exterior algebra (V^{odd}).

The first key observation is contained in the following theorem :

Theorem. Let (A, d) be a cdga and suppose H^{0}(A, d) = ℚ, H^{1}(A, d) = 0 and dim H^{i}(A, d) < ∞, for all i, then there is a quasi-isomorphism

\xymatrix{(ΛV, d) \ar[r]^{φ} & (A, d)}

with ΛV free, V = {V^{i}}_{i≥2} , d(V) ⊂ Λ^{≥2}V and dim V^{i} < ∞ for i = 2, …

The cdga (ΛV, d) is defined up to isomorphism by this property and is called the minimal model of (A, d).

Definition. The minimal model of a 1-connected finite type CW complex X, m_{X}, is the minimal model of the Sullivan cdga A(X).

\underline{Examples}. (i) m_{S^{2n}} = (Λ(a, b), d), \begin{matrix}|a| = 2n, & |b| = 4n − 1 \\ da = 0, & db = a^{2} \end{matrix}

(ii) m_{S^{2n+1}} = (Λa, 0), |a| = 2n + 1,

(iii) m_{ℂP^{n}} = (Λ(a, b), d), \begin{matrix}|a| = 2, & |b| = 2n + 1 \\ da = 0, & db = a^{n+1}. \end{matrix}

Homotopy of morphisms. Two morphisms f, g : (ΛV, d) → (A, d) are homotopic, f ∼ g, if there is a morphism of cdga’s

H : (ΛV, d) ⟶ (A, d) ⊗ Λ(t, dt)

such that p_{0}H = f and p_{1}H = g. Here the map p_{i} : (A, d) ⊗ Λ(t, dt) → (A, d) is defined by

p_{i}(t) = i and p_{i}(dt) = 0.

Proposition. (i) Homotopy is an equivalence relation.

(ii) Let f, g, h be morphisms of cdga’s

\xymatrix{(ΛV, d) \ar[r]^{f} \ar[r]_{g} & (A, d) \ar[r]^{h} & (B, d)}

Suppose h is a quasi-isomorphism and hf ∼ hg, then f ∼ g.

Lifting lemma. Consider the following diagram

\xymatrix{ & (B, d) \ar[d]^{ψ} \\ (ΛV, d) \ar[r]_{φ} & (A, d)}

where ψ is a quasi-isomorphism and (ΛV, d) is minimal. Then there is a morphism Φ : (ΛV, d) → (B, d) such that ψΦ ∼ φ.

The lifting lemma is a very useful lemma. It helps in particular to the construction of the minimal model of a map between 1-connected finite type CW complexes f : X → Y. Indeed denote by m_{X} and m_{Y} minimal models for X and Y. We then have the following diagram

\xymatrix{A(X) & A(Y) \ar[l]_{A(f)} \\ m_{X} \ar[u]^{≃}_{α} & m_{Y} \ar[u]_{≃}^{β}}

The lifting lemma gives a map φ : m_{Y} → m_{X} such that αφ ∼ A(f)β. We then have :

Theorem. [m_{Y} , m_{X}] ≅ [X, Y]_{0}.

We want now to explain how to read the rational homotopy invariants of X from its minimal model m_{X} = (ΛV, d).

X  m_{X} = (ΛV, d)

H^{n}(X; ℚ)  H^{n}(m_{X})

π_{n}(X) ⊗ ℚ  (1)  Hom(V^{n}, ℚ)

The Hurewicz map ⟷ The Hurewicz map (2)

The Whitehead product The quadratic part of

on π_{∗}(X) ⊗ ℚ (3)  ⟷  the differential (4)
cat_{0}(X) (5)  cat_{0}(m_{X}) (6)

(1) A continuous map f : S^{n} → X induces a morphism

m_{X} → m_{S^{n}} ≅ (Λa/a^{2}, 0), |a| = n. This induces a bijection

π_{n}(X) = [S^{n}, X] ≅ [m_{X}, m_{S^{n}}] ≅ Hom(V^{n}, ℚa) ≅ Hom(V^{n}, ℚ).

(2) For a minimal cdga (ΛV, d), the projection

(ΛV, d) ⟶ (ΛV/Λ^{≥2}, d̄) = (ℚ ⊕ V, 0)

induces in homology a map

H^{n}(ΛV, d) ⟶ H^{n}(V, 0) = V^{n}

that is dual to the topological Hurewicz map.

(3) Let α ∈ π_{n}(X) and β ∈ π_{m}(X) be represented by maps f : (I^{n}, ∂I^{n}) → (X, x_{0}) and g : (I^{m}, ∂I^{m}) → (X, y_{0}). Then the Whitehead product w(α, β) is the homotopy class of the map

h : S^{n+m−1} = ∂I^{n+m} ⟶ X

where h(α, β) = f(α) if (α, β) ∈ I^{n}×∂I^{m} and g(β) if (α, β) ∈ ∂I^{n}×I^{m}. Of course π_{∗}(X) is isomorphic to π_{∗−1}(ΩX). By this isomorphism the Whitehead product corresponds up to the sign to the Samelson bracket [α′, β′] defined as follows. Let α′ = [f′] and β′ = [g′],

f′ : S^{n−1} → ΩX, g′ : S^{m−1} → ΩX.

We define a map k : S^{n−1} × S^{m−1} → ΩX by putting

k(a, b) = f′(a)g′(b)f′(a)^{−1}g′(b)^{−1}.

This map is homotopically trivial on S^{n−1}V S^{m−1} and factors thus up to homotopy through S^{n+m−2} = S^{n−1}ΛS^{m−1}. This gives [α′, β′].

(4) The differential d of the minimal model is decomposable

d = d_{2} + d_{3} + ... , d_{i} : V ⟶ Λ^{i}V.

Denote by A^{V} = Hom_{ℚ}(A, ℚ) for each graded vector space A. In our case d_{2} induces a map

\xymatrix{V^{V} ΛV^{V} \ar[r]^{φ}_{=} & (Λ^{2}V)^{V} \ar[r]^{(d_{2})^{V}} & V^{V}}

where

φ(f, g)(u ∧ v) = (−1)^{|u| |g|}f(u)g(v)
+ (−1)^{|u| |f|+|f| |g|}g(u)f(v).

This map induces a Lie bracket on V^{V}.

(5) The Lusternik-Schnirelmann category of a space X, cat X, is the least integer n such that X can be covered by (n + 1)-open sets each contractible in X. The rational category of X, cat_{0}(X), is the category of its rationalization X_{0} and we have

cat_{0}(X) ≤ cat(X) [7].

(6) Consider a diagram in which i_{n} is an injection, p_{n} the natural projection and φ_{n} a quasi-isomorphism

\xymatrix{(ΛV, d) \ar[rr]^{p_{n}} \ar[dr]_{i_{n}} &  & (ΛV/Λ^{>n}, d̄) \\  & (ΛV ⊗ ΛW_{n}, d) \ar[ur]_{φ_{n}}}

Then cat_{0}(ΛV, d) is the minimum n such that i_{n} admits a retraction. We have :

cat_{0}(X) = cat_{0}(m_{X}) [3].

The functor A of Sullivan induces an equivalence of categories between the homotopy category, H_{0} Top_{1}, of 1-connected rational spaces with finite Betti numbers and the homotopy category, H(cdga), of cdga’s (A, d) satisfying H^{0}(A, d) = Q, H^{1}(A, d) = 0 and dim H^{i}(A, d) < ∞, i ≥ 2.

[figure: H_{O}T op_{1}; A; <>; H(cdga)]

The spatial realization < − >: cdga → T op permits to realize each cdga as a topological space. In the homotopy category this functor is the inverse of A.

For recall a space X is called rational if its homotopy groups are rational vector spaces, or equivalently if its integral homology groups are rational vector spaces. Two spaces X and Y have the same rational homotopy type, X ∼_{0} Y, if there is a sequence of maps

X ⟶^{f_{1}} X_{1} ⟵^{f_{2}} X_{2} ⟶ … ⟶^{f_{n}} X_{n} ⟵ Y

inducing isomorphisms in homology with rational coefficients.

For instance each 1-connected rational space with finite Betti numbers has the rational homotopy type of a 1-connected finite type CW complex, and the rationalization of a 1-connected finite type CW complex X is given by < m_{X} >.

Therefore two spaces have the same rational homotopy type if and only if their Sullivan minimal models are isomorphic. By this way we can prove the following facts :

(i) Let X be a 1-connected finite type CW complex that is an H-space. Then X has the rational homotopy type of a product of odd dimensional spheres and loop spaces on odd dimensional spheres

X ∼_{0} ∏_{i} S^{2n_{i}+1} × ∏_{j} ΩS^{2n_{j}+1}.

(ii) Each 1-connected finite type CW complex that is a co-H-space has the rational homotopy type of a wedge of spheres

X ∼_{0} V S^{n_{i}}.

In particular, K(ℚ, 2n) ∼_{0} ΩS^{2n+1} and K(ℚ, 2n + 1) ∼_{0} S^{2n+1}. Each loop space has the rational homotopy type of a product of Eilenberg-Mac Lane spaces.

The first homotopy groups of a 1-connected space

The minimal model of a space X,

φ : m_{X} = (ΛV, d) → A(X)

is built inductively on V^{n} in order to make H^{n}(φ) surjective and H^{n+1}(φ) injective. This gives

V^{n} = W^{n} ⊕ R^{n}

where W^{n} is a vector space of cocycles making H^{n}(φ) surjective and where R^{n} is introduced to kill ker H^{n+1}(φ). The vector spaces have to be choosen minimal for those proposals.

In particular this gives :

V^{2} = H^{2}
V^{3} = H^{3} ⊕ Ker(Λ^{2}H^{2} → H^{4})
V^{4} = H^{4}/(H^{2}) ⊕ R^{4}.

This gives information on the rank of π_{2}, π_{3} and π_{4}. The determination of R^{4} depends on the space and not only on the cohomology of the space.

Examples

S^{2}V S^{2}V S^{5} (S_{a}^{2}V S_{b}^{2}) ∪_{[a,[a,b]]} e^{5}
rank π_{2}  2  2
rank π_{3}  3  3
rank π_{4}  2  1

The dichotomy theorem

The research of general results on the sequence rank π_{n}(X) when X is a 1-connected finite CW complex reduces to the study of the sequence dim V^{n} where (ΛV, d) is a minimal cdga and dim H^{∗}(ΛV, d) < ∞. Some works in that way have led to the following theorem

Theorem. ([3]). There are only two possibilities :

(1) either (elliptic case) Σ_{n} rank π_{n}(X) < ∞. In this case H^{∗}(X; ℚ) satisfies Poincaré duality, its Euler Poincaré characteristic is ≥ 0 and π_{n}(X) is a finite group for n ≥ 2· dim (X)

(2) or else (hyperbolic case) Σ_{n} rank π_{n}(X) = ∞. In that case, there is some A > 0 such that

∑_{p=0}^{n} rank π_{p}(X) ≥ A · e^{n} for n ≫ 0.

In particular for each integer N there is some p such that ℤ^{N} ⊂ π_{p}(X).

Fibrations

Let f : X → Y be a continuous map between 1-connected finite type CW complexes and let m_{f} : m_{Y} → m_{X} its minimal model.

Consider the following commutative diagram where i is the natural injection, φ is a quasi-isomorphism and where the quotient cdga (ΛW, d̄) is minimal.

\xymatrix{m_{Y} \ar[r]^{m_{f}} \ar[dr]_{i} & m_{X} \\  & (m_{Y} ⊗ ΛW, d) \ar[u]_{φ} \ar[r]^{p} & (ΛW, d̄)}

Then (ΛW, d̄) is the minimal model of the homotopy fibre F of f. Moreover all the rational homotopy invariants of the fibration can be deduced from the extension (called KS extension)

\xymatrix{m_{Y} \ar[r] & (m_{Y} ⊗ ΛW, d) \ar[r] & (ΛW, d̄)}

For instance the Serre spectral sequence of the fibration coincides up to the term E_{2} with the spectral sequence induced by the filtration

F_{p} = (m_{Y})^{≥p} ⊗ ΛW .

Write m_{Y} = (ΛV, d) and denote by d_{1} : W ⟶ V the linear part of the differential in m_{Y} ⊗ ΛV. Then the short exact sequence of complexes

O → (V, 0) → (V ⊕ W, d_{1}) → (W, 0) → 0

induces in homology a long exact sequence naturally isomorphic to the dual of the long homotopy exact sequence of the fibration F → X → Y.

The functor A and <> behaves well with respect to fibrations : the model of a fibration is a KS extension and the spatial realization of a KS extension is a fibration.

Exemple. Let (ΛV, d) be the minimal model of a space X, then the model of the Postnikov tower

X_{n} → … → X_{4} → X_{3} → X_{2}

is given by

(ΛV^{≤n}, d) … ← (ΛV^{≤4}, d) ← (ΛV^{≤3}, d) ← (ΛV^{2}, 0).

The homotopy fibre X⟨n⟩ of the projection X → X_{n} is called the n-connected cover of X. Its minimal model is (ΛV^{>n}, d) = (ΛV/ΛV^{≤n}, d).

Cofibration. Let m_{f} : m_{Y} → m_{X} the model of a map f. We consider the following diagram where φ is a quasi-isomorphism and ψ is surjective

\xymatrix{ &  & m_{Y} \ar[dr]^{m_{f}} \\ 0 \ar[r] & ker ψ \ar[r] & m_{Y} ⊗ E \ar[u]_{φ} \ar[r]_{ψ} & m_{X} \ar[r] & 0}

Then ℚ ⊕ ker φ is a model of the homotopy cofibre of F . This means dat the minimal model of ℚ ⊕ ker φ is the minimal model of the cofibre of F.

One again the functors A and <> behave well with respect to cofibrations. The spatial realization of a short exact sequence is a cofibration.

Self-equivalences.

Let aut(X) be the space of free homotopy self-equivalences of a 1-connected CW-complex X, and Aut (X) be the group of classes of self-equivalences.

By a theorem of Dror and Zabrodsky a subgroup G of Aut(X) is nilpotent provided that it acts trivially on the homology or on the homotopy groups of X [2].

Suppose X is a finite dimensional CW complex and N = dim X. We note Aut_{∗}^{N}(X) the subgroup of Aut(X) consisting of classes of self-equivalences φ such that π_{q}(φ) is the identity for q ⩽ N.

Then Aut_{∗}^{N}(X) is nilpotent and by a result of Maruyama ([5])

Aut_{∗}^{N}(X)_{0} = Aut_{∗}^{N}(X_{0})

Here G_{0} means the Malcev completion of the group G.

The Sullivan’s theory of minimal models implies an anti-isomorphism of groups

Aut_{∗}^{N}(X_{0}) ≅ Aut_{∗}^{N}(m_{X}), m_{X} ≅ (ΛV, d) ,

where Aut_{∗}^{n}(m_{X}) denotes the group of homotopy classes of self-homotopy equivalence φ of M_{X} inducing the identity on the indecomposables in degrees ≤ N, i.e. such that the composite

\xymatrix{V^{n} \ar[r]^{φ} & (Λ^{+}V)^{n} \ar[r] & (Λ^{+}V/Λ^{≥2}V)^{n} ≅ V^{n}}

is the identity for n ≤ N.

This leads to the following upper bound for the nilpotency.

Proposition.

Nil Aut_{∗}^{N}(X)_{0} < cat_{0}(X)

Denote by B the Dold functor which takes topological monoids to spaces. For any topological monoid G there is a weak equivalence ΩBG ∼ G. The space B(aut X) classifies fibrations over CW-complexes with fibre X.

\xymatrix{[Y, B(aut X)] \ar[r]^{≅} & \{classes of fibrations with bases Y and fibre X\}}

The subspace aut_{1}(X) is the path-component of aut(X) containing the identity of X. The space B(aut_{1}(X)) is the universal covering of B(aut(X)) and classifies fibrations with fibre X over simply connected CW complexes.

The rational homotopy groups of aut_{1}(X) and the Lie algebra structure of π_{∗}B aut(X) can be derived from the sullivan minimal model m_{X} = (ΛV, d) by the following process :

Denote by Der_{p}(ΛV) the vector space of derivations θ of ΛV of degree −p. The ℤ-graded vector space Der_{∗}(ΛV) is then a differential graded Lie algebra . The Lie bracket is defined by

[θ, θ′] = θ_{0}θ′ − (−1)^{|θ||θ′|}θ′_{0}θ

and the differential is defined by ∂θ = dθ − (−1)^{|θ|}θd. We then have :

Proposition. There is a graded lie algebra isomorphism

H_{q}(Der_{∗}(ΛV), ∂) = π_{q}(aut_{1}(X))_{0}, q ≥ 1

There are many problems connected to aut_{1}(X). We mention two of them :

Conjecture 1 (Halperin). Let X be a 1-connected finite CW complex whose rational cohomology H^{∗}(X; ℚ) is the quotient of a polynomial algebra by a regular sequence. Then

π_{odd}(B aut_{1} X) = 0

or equivalently :

aut_{1}(X) ∼_{0} ∏_{i} S^{2ni+1}

This conjecture has been proved for homogeneous spaces of rank zero and for other particular cases, but not in general.

An other problem concerns the homotopy nilpotency of aut_{1}X. For recall, if X is an associative H-space with strict unit and homotopy inverse, the commutator map

C_{2} : X × X → X is given by C_{2}(a, b) = aba^{−1}b^{−1}

and is well defined up to homotopy. The higher commutator maps C_{n} : X^{n} → X are defined by induction

C_{n}(a_{1}, …, a_{n}) = C_{2}(a_{1}, C_{n−1}(a_{2}, …, a_{n}))

The homotopy nilpotency of X, H nil(X), is the maximum integer n such that C_{n} is essential.

When X is 1-connected, a result of Arkowitz and Curjel shows that H nil ΩX = W L(X), where W L(X) is the maximum of the integers n such that there is non trivial n-iterated Whitehead product.

In particular, we have

H nil(aut_{1} X) = W L(B aut_{1} X)

Problem. If X is hyperbolic, is H nil(aut_{1} X) = ∞ ?

Adams-Hilton models.

Let X be a 2-reduced CW complex (i.e. X has only one 0-cell and no cell of dimension 1). Adams and Hilton ([1]) give for any ring k the construction of a quasi-isomorphism of differentiel graded algebras.

φ : (T (V), d) ⟶ C_{∗}(ΩX; k)

With the following properties :

- The construction is natural with respect to CW complexes

- V^{n} is a free k-module with a basis in bijective correspondance with the cells of X of dimension n + 1.

- The differential d is an algebrization of the attaching maps of the cells.

To be more precise, suppose Y = X ∪_{φ} e^{n} and

φ_{X} : (T (V), d) → C_{∗}(ΩX; k)

already defined. We put W = V ⊕ kx, with |x| = n − 1, and we denote by φ̃ : S^{n−2} → ΩX the adjoint of φ : S^{n−1} → X. We now consider

α = H_{n−2}(φ̃; k)(1) ∈ H_{n−2}(ΩX; k)

By induction α = φ_{X}[a], for some cycle or in T_{n}(V)_{n−2} ; We define now

φ_{Y} : (T (W), d) ⟶ C_{∗}(ΩY; k)

by putting : d(x) = a and φ_{Y} (x) = u with du = φ_{X}(a).

In particular, the Adams Hilton model of a wedge of spheres is a tensor algebra with differential (T (V), 0).

Suppose now k = ℚ. Then H_{∗}(ΩX; ℚ) is a graded Hopf algebra. Moreover the Hurewicz map

h : π_{∗}(ΩX) ⊗ ℚ ⟶ H_{∗}(ΩX; ℚ)

induces an isomorphism between π_{∗}(ΩX) ⊗ ℚ and the vector space of primitive elements in H_{∗}(ΩX; ℚ). In other words

H_{∗}(ΩX; ℚ) = ∪(π_{∗}(ΩX ⊗ ℚ)

When k = ℚ, the Adams-Hilton model (T (V), d) is in a natural way the envelopping algebra of a differential graded Lie algebra (𝕃(V), d) that satisfies the following properties

• H_{∗}(𝕃(V), d) ≅ π_{∗}(ΩX) ⊗ ℚ with the Samelson Bracket

• H_{∗}(V, d_{1}) ≅ H̃_{∗}(X; ℚ) where d_{1} denotes the linear part of the differential.

The relation between the differential Lie algebra (𝕃(V), d) and the Sullivan minimal model m_{X} is giuen by the Quillen functors C and L (this is the Baues-Lemaire conjecture proved by Majewski)

\xymatrix{cdga’s \ar[r]_{L} & differential graded Lie algebras \ar[l]_{C^{∗}}}

Let (L, d) be a differential graded Lie algebra, we define sL by (sL)_{n} = L_{n−1}. The chain coalgebra C_{∗}(L) is defined by :

C_{∗}(L) = (ΛsL, d_{0} + d_{1})
d_{0}(sx_{1} ∧ … ∧ sx_{k}) = − ∑_{i=1}^{k}(−1)^{n_{i}}sx_{1} ∧ … ∧ sdx_{i} ∧ … ∧ sx_{k}

d_{1}(sx_{1} ∧ … ∧ sx_{k}) = ∑_{1≤i≤j≤k}(−1)^{|sx_{i}|}(−1)^{n_{ij}}s[x_{i}x_{j}] ∧ sx_{1} ∧ … ŝx_{i} …
… ŝx_{j} … ∧ sx_{k}.

Here n_{i} = ∑_{j<i} |sx_{j}| and

sx_{1} ∧ … ∧ sx_{k} = (−1)^{n_{ij}}sx_{i} ∧ sx_{j} ∧ sx_{1} … ŝx_{i} … ŝx_{j} … sx_{k}

The diagonal △ is the unique morphism of graded algebras such that △(v) = v ⊗ 1 + 1 ⊗ v, for v ∈ sL .

The cochain algebra on L, C^{∗}(L), is defined as the dual algebra .

Theorem. Let (U𝕃(V), d) →^{≃} C_{∗}(ΩX; ℚ) be the Adams-Hilton model of a 1-connected finite type CW complex. Then m_{X} is quasi-isomorphic to the cdga C^{∗}(𝕃(V), d).

Spaces of category 2.

Over the rationals, spaces of category 2 coincide with 2-cones, i.e. cones of maps between wedges of spheres. Let X be a space of category 2, we thus have a cofibration with Σ_{1} and Σ_{2} wedges of spheres.

Σ_{1} ⟶^{f} Σ_{2} ⟶ X.

The rational homotopy Lie algebra of a wedge of spheres is a free Lie algebra. Therefore we have a map

π_{∗}Ωf ⊗ ℚ : 𝕃(U) → 𝕃(V)

Since the homotopy classes of Σ_{1} go to zero in π_{∗}(X), the map f induces a map of graded Lie algebras

φ : I_{X} = 𝕃(V)/(f(U)) → π_{∗}(ΩX) ⊗ ℚ.

where (f(U)) denotes the ideal generated by f(U). In fact φ is injective, admits a section and we have the following proposition.

Proposition. There is a short exact sequence of Lie algebras

0 → 𝕃(W) → π_{∗}(ΩX) ⊗ ℚ → I_{X} → 0.

In other words the new rational homotopy elements form a free graded Lie algebra.

More generally it should be interesting to have a structure theorem for the cokernel of the map π_{∗}(ΩX) ⊗ ℚ ⟶^{π_{∗}(i)⊗ℚ} π_{∗}(ΩY) ⊗ ℚ when i denotes the injection of X into Y = XU_{φ}e^{n}.

References

[1] J. F. Adams and P. J. Hilton, On the chain algebra of a loop space. Comment. Math. Helv. 30 (1956), 305-330.

[2] E. Dror and A. Zabrodsky, unipotency and nilpotency in homotopy equivalences. Topology 18 (1979), 187-197.

[3] Y. Félix and S. Halperin, Rational L.-S. category and its applications. Trans. Amer. Math. Soc. 273 (1982), 1-37.

[4] S. Halperin, Lecture on minimal models. Mémoire de la société mathématique de France, 9/10 (1983).

[5] K. Maruyama, Localization of a certain subgroup of self-homotopy equivalences. Pacific J. of Mathematics 136 (1989), 293-301.

[6] D. Sullivan, Infinitesimal computations in topology. Publ. I.H.E.S. 47 (1977), 269-331.

[7] G. H. Toomer, Topological localization, category and cocategory. Canadian Journal of Mathematics 27 (1975), 319-322