Rational Homotopy Theory II

Yves F´elix,Steve Halperin and Jean-Claude Thomas

World Scientic Book, 412 pages, to appear in March 2012.

Abstract Sullivan’s seminal paper, Infinitesimal Computations in Topology, includes the ap- plication of his techniques to non-simply connected spaces, and these ideas have been used frequently by other authors. Our objective in this sequel to our “Rational Homo- topy Theory I, published by Springer-Verlag in 2001, is to provide a complete descrip- tion with detailed proofs of this material. This then provides the basis for new results, also included, and which we complement with recent advances for simply-connected spaces. There do remain many interesting unanswered questions in the field, which hopefully this text will make it easier for others to resolve.

1 Introduction

Rational homotopy theory assigns to topological spaces invariants which are preserved by continuous maps f for which H∗(f; Q) is an isomorphism. The two standard approaches of the theory are due respectively to Quillen [58] and Sullivan [61], and [62]. Each constructs from a class of CW complexes X an algebraic model MX , and then constructs from MX a CW complex XQ, together with a map ϕX : X → XQ. Both H∗(XQ; Z) and πn(XQ) are rational vector spaces, and with appropriate hypotheses

H∗(ϕX ): H∗(X) ⊗ Q → H∗(XQ; Z) , and

πn(ϕX ): πn(X) ⊗ Q → πn(XQ), n ≥ 2, are isomorphisms. In each case the model MX belongs to an algebraic homotopy category, and a homo- topy class of maps f : X → Y induces a homotopy class of morphisms Mf : MX → MY (in Quillen approach) and a homotopy class of morphisms Mf : MY → MX (in Sullivan’s approach). These are referred to as representatives of f. In Quillen’s approach, X is required to be simply connected and MX is a rational differential graded Lie algebra which is free as a graded Lie algebra. In this case H∗(ϕX ) and π≥2(ϕX ) are always isomorphisms. Here, as in [18], we adopt Sullivan’s approach, and in this Introduction provide an overview of the material in the monograph, together with brief summaries of the individual Chapters. Sullivan’s approach associates to each path connected space X a cochain algebra MX of the form (∧V, d) in which the free commutative graded algebra ∧V is generated by ≥1 m m V = V , and V = ⊕m ∧ V with ∧ V = V ∧ ··· ∧V (m factors). Additionally, each ∧V ≤k is preserved by d, and d also satisfies a ”nilpotence” condition: (∧V, d) is called a minimal Sullivan algebra. A minimal Sullivan algebra determines a simplicial set h∧V, di

1 with spatial realization |∧V, d|, and when (∧V, d) is the model of the CW complex X then this determines (up to homotopy) the map

ϕX : X → XQ = | ∧ V, d| . This approach makes non-simply connected spaces accessible to rational homotopy the- ory. For example, if H1(X; Q) is finite dimensional then π1(XQ) is the Malcev completion of π1(X): π1(ϕX ) induces an isomorphism

∼ n = lim π1(X)/π (X) ⊗ / π1(X ) , ←− n 1 Q Q

n where (π1 (X)) denotes the lower central series of π1(X). On the other hand, this approach also comes at the cost of a finiteness condition: If X is simply connected then H∗(ϕX ) and π≥2(ϕX ) are isomorphisms if and only if H∗(X; Q) is a graded vector space of finite type. In the case of non-simply connected CW complexes X, two additional ingredients are required for rational homotopy theory: first, the action by covering transformations ∗ of π1(X) on the cohomology H (Xe; Q) of the universal covering space of X; second, a Sullivan representative ψ for a classifying map mapping X to the classifying space for ∗ π1(X) and inducing an isomorphism of fundamental groups. If H (Xe; Q) has finite type, then the groups π≥2(X) ⊗ Q can be computed from a minimal Sullivan model of Xe, and if ∗ the action of π1(X) on H (Xe; Q) is nilpotent, then this Sullivan model can be computed from ψ. Minimal Sullivan algebras (∧V, d) are equipped with a homotopy theory and a range of invariants analogous to those which arise in topology. Key among these are the graded 1 homotopy Lie algebra L = {Lp}p≥0 and, when dim H (∧V, d) < ∞, the group GL. Here p+1 2 Lp = Hom(V , Q), the Lie bracket is dual to the component d1 : V → ∧ V of d, and an exponential map converts L0 to GL. The group GL acts by conjugation in L and also in H(∧V ≥2, d) where (∧V ≥2, d) is obtained by dividing by V 1∧ ∧ V . A third key invariant is the Lusternik-Schnirelmann category, cat (∧V, d), defined as the least m for which (∧V, d) is a homotopy retract of (∧V/ ∧>m V, d). 1 When (∧V, d) is the minimal Sullivan model of a CW complex X such that dim H (X; Q) < ≥2 ∞, then GL = π1(XQ), L≥1 = π≥1(ΩXQ), and there is a natural map H(∧V , d) → H(Xe; Q) equivariant via π1(ϕX ) with respect to the actions of GL and the action by cover- ing transformations of π1(X). Finally, as in the simply connected case ([18]) cat (∧V, d) ≤ cat X. Sullivan models (∧V, d) are constructed via a functor, APL (inspired by the differen- tial forms on a manifold) from spaces to rational cochain algebras, for which H(APL(X)) ∗ and H (X; Q) are naturally isomorphic algebras. Then (∧V, d) is the unique (up to iso- morphism) minimal Sullivan algebra admitting a morphism m :(∧V, d) → APL(X) for which H(m) is an isomorphism. Moreover any morphism ϕ :(∧W, d) → APL(X) from an arbitrary minimal Sullivan algebra determines by adjunction a homotopy class of maps |ϕ| : X → | ∧ V, d|; in particular, |m| is homotopic to the map ϕX above. This second step can be applied to construct a minimal Sullivan model (∧V, d) → (A, d) for any commutative cochain algebra satisfying H0(A, d) = lk. While these may not be Sullivan models of a topological space, the homotopy machinery of minimal Sullivan algebras is established independently of topology, and so can be applied in this more general context.

2 In particular, minimal Sullivan algebras become a valuable tool in the study of graded n Lie algebras E = E≥0 with lower central series denoted by (E ), provided that E0 acts nilpotently by the adjoint representation in each Ei, i ≥ 1, and that

n dim E0/[E0,E0] < ∞ , dim Ei < ∞ , i ≥ 1 , and ∩n E0 = 0 .

Such Lie algebras are called Sullivan Lie algebras. Note that we may have dim E0 = ∞. For a Sullivan Lie algebra, E, lim C∗(E/En) is a minimal Sullivan algebra, called −→ n the associated Sullivan algebra for E, and lim E/En is its homotopy Lie algebra. (Here ←− n C∗(−) is the classical Cartan-Chevalley-Eilenberg cochain algebra construction.) In the reverse direction, if (∧V, d) is any minimal Sullivan algebra for which dim H1(∧V, d) < ∞ then its homotopy Lie algebra is a Sullivan Lie algebra whose associated Sullivan algebra 2 is (∧V, d1) with d1 the component of d mapping V to ∧ V . In summary, the interplay between spaces, minimal Sullivan algebras and graded Lie algebras is illustrated by the diagram

Graded Lie algebras O L lim C∗(L/Ln) −→ n

APL Minimal Sullivan | . | Spaces / / Spaces algebras

A crucial technical tool in the theory of minimal Sullivan algebras is the conversion of cochain algebra morphisms to Λ-extensions (∧V, d) → (∧V ⊗ ∧Z, d) in which (∧V, d) is a minimal Sullivan algebra and d : Zp → ∧V ⊗ ∧Z≤p satisfies a ”nilpotence” condition; when V 1 6= 0 it may happen that Z0 6= 0. Division by ∧+V ⊗ ∧Z gives a quotient cochain algebra (∧Z, d) and the Λ-extension determines holonomy representations of L and, if 1 dim H (∧V, d) < ∞, of GL in H(∧Z, d). These Λ-extensions are the Sullivan analogues of fibrations, and (∧Z, d) is the Sullivan analogue of the fibre. The analogy is not merely abstract: suppose

(∧V, d) / (∧V ⊗ ∧Z, d)

α β

 APL(p)  APL(Y ) / APL(X) is a commutative diagram in which Y is a CW complex and p is the projection of a fibration with fibre F . Then β factors to give a morphism γ :(∧Z, d) → APL(F ), and, ∗ in this setting H(γ): H(∧Z, d) → H (F ; Q) is equivariant via π1(|ψ|) with respect to the homolony representation of GL in H(∧Z, d) and π1(Y ) in H(F ; Q). There are two important examples of Λ-extensions. First, if (∧V, d) is a minimal Sullivan algebra then (∧V 1, d) → (∧V, d) is a Λ-extension, the Sullivan analogue of a classifying map for a CW complex X. If this morphism is a Sullivan representative for the ∗ classifying map, and if H (X; Q) has finite type and the covering space action of π1(X) is nilpotent, then X is a Sullivan space and the quotient (∧V ≥2, d) is a minimal Sullivan model for Xe. Many classical examples are Sullivan spaces, including all closed orientable Riemann surfaces.

3 Second, if (∧V, d) is any minimal Sullivan algebra, converting the augmentation (∧V, d) → lk yields a Λ-extension (∧V, d) → (∧V ⊗∧U, d) with H(∧V ⊗∧U, d) = lk; this is the acyclic closure of (∧V, d). Here the differential in the quotient ∧U is zero and so the holonomy representation is a representation of the homotopy Lie algebra L of ∧V in ∧U. The Λ-extension also determines a diagonal

∆ : ∧U → ∧U ⊗ ∧U which makes ∧U into a commutative graded Hopf algebra, and there is a natural homo- morphism ηL : UL → Hom (∧U, lk) of graded algebras which converts right multiplication by L to the dual of the holonomy representation. In particular, in the case of the acyclic closure of the associated Sullivan algebra of a Sullivan Lie algebra, E, this yields a morphism UE → Hom(∧U, lk) which identifies ∧U as a sort of ”predual” of UE. In this setting we define

UE depth E = least p (or ∞) such that Torp (lk, ∧U) 6= 0 .

This generalizes the definition in [18] for Lie algebras E = E≥1 of finite type, because in # ∗ UE this case UE = (∧U) and ExtUE(lk, UE) is the dual of Tor∗ (lk, ∧U). The invariant depth E plays an important role in the growth and structure theorems for the homotopy Lie algebra of a simply connected space of finite LS category. These were established after [18] appeared, and so are included here. The extent to which they may be generalized to non-simply connected spaces remains an open question. Although the present volume is a sequel to [18] it can be read independently, since all the definitions, conventions and results are stated here, whether or not they appear in [18], although we do quote proofs from [18] whenever this is possible. As in [18], we work where possible over an arbitrary field lk of characteristic zero, and with rare exceptions, definitions and notation are unchanged from [18]; in particular, V # denotes the dual of a graded vector space V . Also, for simplicity, the cohomology algebra H∗(X; lk) of a space X is denoted by H(X). That said, by and large the material in this monograph either is a non-trivial extension of, or is in addition to, the content of [18]. In particular, it includes: • the extension of Sullivan models from simply connected spaces to path connected spaces with general (not necessarily nilpotent) fundamental group G.

• an analysis of L0, the fundamental Lie algebra of (∧V, d).

∗ • a description of the holonomy action of π1(B) on H (F ) in terms of Sullivan models. • a complete proof that under the most general possible hypotheses the Sullivan fibre associated with a fibration B ←− E is the Sullivan model of the fibre F of p, even when B is not simply connected.

• an analysis of the minimal Sullivan model of a classifying space and the introduction of Sullivan spaces.

• the definition of the depth of L for any Sullivan algebra and a homological analysis of its properties extending those provided in [18] when L0 = 0. • complete proofs of the growth and structure theorems for the higher rational homo- topy groups of a connected CW complex.

4 2 Contents

1. Basic definitions and constructions

1.1 Graded algebra 1.2 Differential graded algebra 1.3 Simplicial sets 1.4 Polynomial differential forms 1.5 Sullivan algebras 1.6 The simplicial and spacial realizations of a Λ-algebra 1.7 Homotopy and based homotopy 1.8 The homotopy groups of a minimal algebra

2. 2.1 The homotopy Lie algebra of a minimal Sullivan algebra 2.2 The fundamental Lie algebra of a Sullivan 1-algebra 2.3 Sullivan Lie algebras 2.4 Primitive Lie algebras and exponential groups 2.5 The lower central series of a group 2.6 The linear isomorphism (∧sV )# =∼ ULd 2.7 The fundamental group of a 1-finite minimal Sullivan algebra 2.8 The homology Hopf algebra of a 1-finite minimal Sullivan algebra

2.9 The action of GL on πn(| ∧ V, d|) 2.10 Formal Sullivan 1-algebras

3. Fibrations et Λ-extensions

3.1 Fibrations, Serre fibrations and homotopy fibrations 3.2 The classifying space fibration and Postnikov decompsition of a CW complex 3.3 λ-extensions 3.4 Existence of minimal Sullivan models 3.5 Uniqueness of the minimal model 3.6 The acyclic closure of a minimal Sullivan algebra 3.7 Sullivan extension and fibrations

4. Holonomy

4.1 Holonomy of a fibration 4.2 Holonomy of a λ-extension 4.3 Holonomy representation for a λ-extension 4.4 Nilpotent and locally nilpotent representations 4.5 Connecting topology and Sullivan homotopy 4.6 The holonomy action on the homotopy groups of the fibre

5 5. The model of the fibre is the fibre of the model

5.1 The main theorem

5.2 The holonomy action of π1(Y, ∗) on π∗(F ) 5.3 The Sullivan model of a universal covering space 5.4 The Sullivan model of a spacial realization

6. Loop spaces and loop spaces actions

6.1 The loop cohomology coalgebra of (λV, d)

6.2 The transformation map ηL

6.3 The graded Hopf algebra, H∗(|λU|; Q) 6.4 Connecting Sullivan algebras with topological spaces

7. Sullivan spaces

7.1 Sullivan spaces 7.2 The classifying space BG 7.3 The Sullivan 1-model of BG 7.4 Malcev completions

7.5 The morphism mλV,d| :(λV, d) → APL(λV, d|) 7.6 When BG is a Sullivan space

8. Examples

8.1 Nilpotent and rationally nilpotent groups 8.2 Nilpotent and rationally nilpotent spaces 8.3 The group Z# ··· #Zs 8.4 Semidirect product 8.5 Orientable Riemann surfaces 8.6 The classifying space of the pure braid group is a Sullivan space 8.7 The Heisenberg group 8.8 Siefert manifolds 8.9 Arrangement of hyperplanes 8.10 Connected sum of real projective spaces 8.11 A final example

9. Lusternik-Schnirelmann category

9.1 The LS category of topological spaces and commutative cochain algebras 9.2 The mapping theorem 9.3 Module category and the Toomer invariant 9.4 cat =mcat 9.5 cat = e(−)∗

6 9.7 Jessup’s Theorem 9.8 Example

10. Depth of a Sullivan algebra and of a Sullivan Lie algebra

10.1 Ext, Tor and the Hochschild-Serre spectral sequence 10.2 The depth of a minimal Sullivan algebra 10.3 The depth of a Sullivan Lie algebra 10.4 Sub Lie algebras and ideals of Sullivan Lie algebra 10.5 Depth and relative depth 10.6 The radical of a Sullivan Lie algebra 10.7 Sullivan Lie algebras of finite type

11. Depth of a connected Sullivan Lie algebra of finite type

11.1 Summary of previous results 11.2 Modules over an abelian Lie algebra 11.3 Weak depth

12. Trichotomy

12.1 Overview of the results 12.2 The rationally elliptic case 12.3 The rationally hyperbolic case 12.4 The gap theorem 12.5 Rationally infinite spaces of finite category 12.6 Rationally infinite spaces of finite dimension

13. Exponential growth

13.1 The invariant lod index 13.2 Growth of graded Lie algebra 13.3 Weak exponential growth and critical degree 13.4 Approximation of log index 13.5 Moderate exponential growth 13.6 Exponentiam growth

14. Structure of graded Lie algebras of finite depth

14.1 Introduction 14.2 The spectrum 14.3 Minimal sub Lie algebras 14.5 L-equivalence 14.6 The odd part of a graded Lie algebra

15. Weight decomposition of a Sullivan algebra

7 15.1 Weight decomposition 15.2 Exponential growth of L 15.3 The fundamental Lie algebra of a 1-formal Sullivan algebra

16. Problems

AMS Classification : 55P35, 55P62, 17B70

Key words : Homotopy groups, graded Lie algebra, exponential growth, LS category.

3 Introduction

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