##
**Homotopie rationelle: modèles de Chen, Quillen, Sullivan.**
*(English)*
Zbl 0539.55001

Lecture Notes in Mathematics. 1025. Berlin etc.: Springer-Verlag. X, 211 p. DM 28.00; $ 10.90 (1983).

This book which has grown out of the author’s thesis emphasizes the differential Lie algebra viewpoint in rational homotopy theory and culminates in a corresponding classification theorem of fibrations. It also reviews and supplements the basic rational homotopy theory.

Let me mention some of the standard references for rational homotopy theory on which the book is based: S. Halperin, Lectures on minimal models, Publ. U. E. R., Math. Pures Appl. 3, No.4 (1981; Zbl 0505.55014); D. Quillen, Ann. Math., II Ser. 90, 205-295 (1969; Zbl 0191.537), referred to by [Qu] in this review; D. Sullivan, Publ. Math., Inst. Hautes Étud. Sci. 47 (1977), 269-331 (1978; Zbl 0374.57002). Let ADGC be the category of graded differential commutative algebras (A,d) \((A=\oplus_{i\geq 0}A^ i,\quad d: A^ i\to A^{i+1});\) let \(LDG_ 1\) be the category of graded differential Lie algebras \((L,\partial) (L=\oplus_{i\geq 0}L_ i\quad with\quad L_ 0=0,\quad \partial: L_ i\to L_{i-1}).\) Everything is over the rationals! (Note that the letters have been left under French order.) For the sake of simplicity the author mostly works with simply connected spaces and the analogues in the other categories.

Chapter I studies adjoint functors \(L_*\), \(C^*\) between the full subcategory of 1-connected algebras of finite type of ADGC and the full subcategory of Lie algebras of finite type of \(LDG_ 1\). The functor \(C^*\) applied to \((L,\partial)\) is the algebra of cochains on L used to define Lie algebra cohomology. In more detail: For any graded vector space V let \(\Lambda\) (V) be the free commutative algebra on V, let # denote the dual, then \(C^*(L,\partial)=(\Lambda(\#sL),d=d_ 1+d_ 2),\) where \((sL)_ q=L_{q-1}\) and where, up to the right signs, \((d_ 1| \#sL): \#sL\to \#sL\) is dual to \(\partial\), \((d_ 2| \#sL): \#sL\to \Lambda^ 2(\#sL)\) is dual to the Lie bracket.

Let A be 1-connected of finite type, then \(L_*(A,d)\) is the free Lie algebra \({\mathbb{L}}(s^{-1}\#A)\subset T(s^{-1}\#A)\) with the differential \(\partial =\partial_ 1+\partial_ 2\) given by \((\partial_ 1| s^{-1}\#A): s^{-1}\#A\to s^{-1}\#A\) dual to d and \((\partial_ 2| s^{-1}\#A): s^{-1}\#A\to {\mathbb{L}}^ 2(s^{- 1}\#A)\) dual to the product.

These functors are the duals of the functors C and L of [Qu]; it follows from [Qu] that the adjunctions are weak equivalences, i.e. homology isomorphisms.

Chapter II first reviews some of D. Quillen, Homotopical algebra [Lect. Notes Math. 43 (1967; Zbl 0168.209)]. In addition the homotopy theory \(ho\)-LD\(G_ 1\) of \(LDG_ 1\) is studied explicity. It is proved that \(L_*\) and \(C^*\) preserve homotopies.

We recall, that a Sullivan minimal model of a cohomologically 1-connected \(A\in ADGC\) is an algebra (\(\Lambda\) (V),d) with decomposable d, i.e. \(d(V)\subset \Lambda^{\geq 2} V,\) together with a weak equivalence \((\Lambda(V),d)\to(A,d_ a)\). Dually, a Quillen model of \((L,\partial_ L)\in LDG_ 1\) is a Lie algebra (\({\mathbb{L}}(V),\partial)\) together with a weak equivalence \(({\mathbb{L}}(V),\partial)\to(L,\partial_ L)\). The model is called minimal, if \(\partial\) is decomposable, i.e. \(\partial(V)\subset [{\mathbb{L}}(V),{\mathbb{L}}(V)].\) A Quillen (minimal) model of \((A,d_ a)\) with 1-connected cohomology of finite type is a Quillen (minimal) Lie algebra (\({\mathbb{L}}(V),\partial)\) together with a weak equivalence \(C^*({\mathbb{L}}(V),\partial)\to(A,d_ a).\)

As a first major application the functors \(C^*\), \(L_*\) are used to study the filtered model of S. Halperin and J. Stasheff [Adv. Math. 32, 233-279 (1979; Zbl 0408.55009)] as well as a variant introduced by Y. Félix [Can. J. Math. 33, 1448-1458 (1981; Zbl 0489.55008)], the so-called FHS-model. It is shown that the FHS-model of (A,d) with 1- connected cohomology of finite type is just \(C^*(L,\partial)\) where \((L,\partial)\) is a Quillen minimal model of (A,d).

Chapter III recalls the transition from topology to algebra as e.g. described in the papers cited above. The homotopy category of 1-connected ”nice” rational spaces of finite \({\mathbb{Q}}\)-type is equivalent to the full subcategory of ho-ADGC formed by the 1-connected algebras of finite type. The transition is given by forming the Sullivan minimal model of the algebra \(A_{PL}(X)\) of forms on a space X. Taking the Quillen minimal model of \(A_{PL}(X)\) one obtains an equivalence with the subcategory of \(ho\)-LD\(G_ 1\) of algebras of finite type. Note that it has been proved by J. M. Lemaire that the Quillen minimal model of \(A_{PL}(X)\) is homotopy equivalent to Quillen’s model of X as in [Qu]. Many examples are given as well as applications to the Eilenberg-Moore spectral sequence of a space and to a spectral sequence of Quillen.

In Chapter IV relations between the formal homological connections of K. T. Chen and minimal models are worked out. This question has also been studied in the paper reviewed below. Chapter V develops the duality between higher order Whitehead products and Massey products (over \({\mathbb{Q}}!)\). Departing from geometry higher order Whitehead products are defined in \((L,\partial)\in LDG\). On the other side a universal geometric object for Massey products is constructed displaying strongly Eckmann- Hilton duality. These higher order structures are then identified in the other models recovering results of P. Andrews and M. Arkowitz [Can. J. Math. 30, 961-982 (1978; Zbl 0441.55012)].

Chapter VI aims at identifying the Eilenberg-Moore spectral sequence of a fibration using a model in LDG. This was already done in chap. III in the absolute case. In Chapter VII a classification theorem for fibrations within \(LDG_ 1\) is proved. There are two variants, one for fibrations with a fixed fibre, one for fibrations with fibre of a fixed homotopy type. The author has also given a nice exposition of the topological classification of fibrations and of the results of chapters VI, VII in [Homotopie algébrique et algèbre locale, Journ. Luminy/France 1982, Astérisque 113-114, 132-147 (1984)].

Let me mention some of the standard references for rational homotopy theory on which the book is based: S. Halperin, Lectures on minimal models, Publ. U. E. R., Math. Pures Appl. 3, No.4 (1981; Zbl 0505.55014); D. Quillen, Ann. Math., II Ser. 90, 205-295 (1969; Zbl 0191.537), referred to by [Qu] in this review; D. Sullivan, Publ. Math., Inst. Hautes Étud. Sci. 47 (1977), 269-331 (1978; Zbl 0374.57002). Let ADGC be the category of graded differential commutative algebras (A,d) \((A=\oplus_{i\geq 0}A^ i,\quad d: A^ i\to A^{i+1});\) let \(LDG_ 1\) be the category of graded differential Lie algebras \((L,\partial) (L=\oplus_{i\geq 0}L_ i\quad with\quad L_ 0=0,\quad \partial: L_ i\to L_{i-1}).\) Everything is over the rationals! (Note that the letters have been left under French order.) For the sake of simplicity the author mostly works with simply connected spaces and the analogues in the other categories.

Chapter I studies adjoint functors \(L_*\), \(C^*\) between the full subcategory of 1-connected algebras of finite type of ADGC and the full subcategory of Lie algebras of finite type of \(LDG_ 1\). The functor \(C^*\) applied to \((L,\partial)\) is the algebra of cochains on L used to define Lie algebra cohomology. In more detail: For any graded vector space V let \(\Lambda\) (V) be the free commutative algebra on V, let # denote the dual, then \(C^*(L,\partial)=(\Lambda(\#sL),d=d_ 1+d_ 2),\) where \((sL)_ q=L_{q-1}\) and where, up to the right signs, \((d_ 1| \#sL): \#sL\to \#sL\) is dual to \(\partial\), \((d_ 2| \#sL): \#sL\to \Lambda^ 2(\#sL)\) is dual to the Lie bracket.

Let A be 1-connected of finite type, then \(L_*(A,d)\) is the free Lie algebra \({\mathbb{L}}(s^{-1}\#A)\subset T(s^{-1}\#A)\) with the differential \(\partial =\partial_ 1+\partial_ 2\) given by \((\partial_ 1| s^{-1}\#A): s^{-1}\#A\to s^{-1}\#A\) dual to d and \((\partial_ 2| s^{-1}\#A): s^{-1}\#A\to {\mathbb{L}}^ 2(s^{- 1}\#A)\) dual to the product.

These functors are the duals of the functors C and L of [Qu]; it follows from [Qu] that the adjunctions are weak equivalences, i.e. homology isomorphisms.

Chapter II first reviews some of D. Quillen, Homotopical algebra [Lect. Notes Math. 43 (1967; Zbl 0168.209)]. In addition the homotopy theory \(ho\)-LD\(G_ 1\) of \(LDG_ 1\) is studied explicity. It is proved that \(L_*\) and \(C^*\) preserve homotopies.

We recall, that a Sullivan minimal model of a cohomologically 1-connected \(A\in ADGC\) is an algebra (\(\Lambda\) (V),d) with decomposable d, i.e. \(d(V)\subset \Lambda^{\geq 2} V,\) together with a weak equivalence \((\Lambda(V),d)\to(A,d_ a)\). Dually, a Quillen model of \((L,\partial_ L)\in LDG_ 1\) is a Lie algebra (\({\mathbb{L}}(V),\partial)\) together with a weak equivalence \(({\mathbb{L}}(V),\partial)\to(L,\partial_ L)\). The model is called minimal, if \(\partial\) is decomposable, i.e. \(\partial(V)\subset [{\mathbb{L}}(V),{\mathbb{L}}(V)].\) A Quillen (minimal) model of \((A,d_ a)\) with 1-connected cohomology of finite type is a Quillen (minimal) Lie algebra (\({\mathbb{L}}(V),\partial)\) together with a weak equivalence \(C^*({\mathbb{L}}(V),\partial)\to(A,d_ a).\)

As a first major application the functors \(C^*\), \(L_*\) are used to study the filtered model of S. Halperin and J. Stasheff [Adv. Math. 32, 233-279 (1979; Zbl 0408.55009)] as well as a variant introduced by Y. Félix [Can. J. Math. 33, 1448-1458 (1981; Zbl 0489.55008)], the so-called FHS-model. It is shown that the FHS-model of (A,d) with 1- connected cohomology of finite type is just \(C^*(L,\partial)\) where \((L,\partial)\) is a Quillen minimal model of (A,d).

Chapter III recalls the transition from topology to algebra as e.g. described in the papers cited above. The homotopy category of 1-connected ”nice” rational spaces of finite \({\mathbb{Q}}\)-type is equivalent to the full subcategory of ho-ADGC formed by the 1-connected algebras of finite type. The transition is given by forming the Sullivan minimal model of the algebra \(A_{PL}(X)\) of forms on a space X. Taking the Quillen minimal model of \(A_{PL}(X)\) one obtains an equivalence with the subcategory of \(ho\)-LD\(G_ 1\) of algebras of finite type. Note that it has been proved by J. M. Lemaire that the Quillen minimal model of \(A_{PL}(X)\) is homotopy equivalent to Quillen’s model of X as in [Qu]. Many examples are given as well as applications to the Eilenberg-Moore spectral sequence of a space and to a spectral sequence of Quillen.

In Chapter IV relations between the formal homological connections of K. T. Chen and minimal models are worked out. This question has also been studied in the paper reviewed below. Chapter V develops the duality between higher order Whitehead products and Massey products (over \({\mathbb{Q}}!)\). Departing from geometry higher order Whitehead products are defined in \((L,\partial)\in LDG\). On the other side a universal geometric object for Massey products is constructed displaying strongly Eckmann- Hilton duality. These higher order structures are then identified in the other models recovering results of P. Andrews and M. Arkowitz [Can. J. Math. 30, 961-982 (1978; Zbl 0441.55012)].

Chapter VI aims at identifying the Eilenberg-Moore spectral sequence of a fibration using a model in LDG. This was already done in chap. III in the absolute case. In Chapter VII a classification theorem for fibrations within \(LDG_ 1\) is proved. There are two variants, one for fibrations with a fixed fibre, one for fibrations with fibre of a fixed homotopy type. The author has also given a nice exposition of the topological classification of fibrations and of the results of chapters VI, VII in [Homotopie algébrique et algèbre locale, Journ. Luminy/France 1982, Astérisque 113-114, 132-147 (1984)].

Reviewer: H.Scheerer

### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

55P62 | Rational homotopy theory |

55R15 | Classification of fiber spaces or bundles in algebraic topology |

57T35 | Applications of Eilenberg-Moore spectral sequences |

16W50 | Graded rings and modules (associative rings and algebras) |

17B70 | Graded Lie (super)algebras |

13N05 | Modules of differentials |