Table of Contents
Preface v
Introduction xi
1 Basic definitions and constructions 1
1.1 Graded algebra 1
1.2 Differential graded algebra 6
1.3 Simplicial sets 8
1.4 Polynomial differential forms 14
1.5 Sullivan algebras 18
1.6 The simplicial and spatial realizations of a A-algebra 24
1.7 Homotopy and based homotopy 30
1.8 The homotopy groups of a minimal Sullivan algebra 35
2 Homotopy Lie algebras and Sullivan Lie algebras 45
2.1 The homotopy Lie algebra of a. minimal Sullivan algebra 45
2.2 The fundamental Lie algebra of a Sullivan 1-algebra 48
2.3 Sullivan Lie algebras 54
2.1 Primitive Lie algebras and exponential groups 56
2.5 The lower central series of a group 63
2.6 The linear isomorphism (Λ sV)# ≅ $$$ 71
2.7 The fundamental group of a 1-ihiite minimal Sullivan algebra 75
2.8 The homology Hopf algebra of a 1-finite minimal Sullivan algebra 81
2.9 The action of GL on πn (|Λ V, d|,*) 84
2.10 Formal Sullivan 1-algebras 87
3 Fibrations and Λ-extensions 91
3.1 Fibrations, Serrc fibrations and homotopy fibrations 91
3.2 The classifying space fibration and Postnikov decompositions of a connected CW complex 93
3.3 Λ-extensions 95
3.4 Existence of minimal Sullivan models 101
3.5 Uniqueness of minimal Sullivan models 107
3.6 The acyclic closure of a minimal Sullivan algebra 111
3.7 Sullivan extensions and fibrations 114
4 Holonomy 117
4.1 Holonomy of a fibration 117
4.2 Holonomy of a Λ-extensions 125
4.3 Holonomy representations for a Λ-extensions 130
4.4 Nilpotent and locally nilpotent representations 133
4.5 Connecting topological and Sullivan holonomy 137
4.6 The holonomy action on the homotopy groups of a fibre 142
5 The model of the fibre is the fibre of the model 145
5.1 The main theorem 145
5.2 The holonomy action of π1 (Y, *) on π* (F) 160
5.3 The Sullivan model of a universal covering space 164
5.4 The Sullivan model of a spatial realization 166
6 Loop spaces and loop space actions 169
6.1 The loop cohomology coalgebra of (ΛV,d) 169
6.2 The transformation map ηL 176
6.3 The graded Hopf algebra, H* (|Λ U|;Q) 183
6.4 Connecting Sullivan algebras with topological spaces 187
7 Sullivan spaces 195
7.1 Sullivan spaces 195
7.2 The classifying space BG 199
7.3 The Sullivan 1-model of BG 205
7.4 Malcev completions 213
7.5 The morphism m|ΛV,d| : (ΛV, d) → APL (|Λ V, d|) 218
7.6 When BG is a Sullivan space 222
8 Examples 227
8.1 Nilpotent and rationally nilpotent groups 227
8.2 Nilpotent and rationally nilpotent spaces 227
8.3 The groups Z# … #Z 229
8.4 Semidirect products 231
8.5 Orientable Riemann surfaces 232
8.6 The classifying space of the pure braid group Pn is a Sullivan space 237
8.7 The Heisenberg group 238
8.8 Seifert manifolds 239
8.9 Arrangement of hypcrplanes 240
8.10 Connected sum of real projective spaces 241
8.11 A final example 242
9 Lusternik-Schnirelmann category 245
9.1 The LS category of topological spaces and commutative cochain algebras 245
9.2 The mapping theorem 248
9.3 Module category and the Toomer invariant 249
9.4 Cat = mcat 250
9.5 Cat = e (-)# 260
9.6 Jessup's Theorem 261
9.7 Example 265
10 Depth of a Sullivan algebra and of a Sullivan Lie algebra 267
10.1 Ext, Tor and the Hochschild-Serre spectral sequence 267
10.2 The depth of a minimal Sullivan algebra 272
10.3 The depth of a Sullivan Lie algebra 276
10.4 Sub Lie algebras and ideals of a Sullivan Lie algebra 279
10.5 Depth and relative depth 287
10.6 The radical of a Sullivan Lie algebra 295
10.7 Sullivan Lie algebras of finite type 298
11 Depth of a connected graded Lie algebra of finite type 301
11.1 Summary of previous results 301
11.2 Modules over an abclian Lie algebra 304
11.3 Weak depth 307
12 Trichotomy 313
12.1 Overview of results 313
12.2 The rationally elliptic case 317
12.3 The rationally hyperbolic case 317
12.4 The gap theorem 318
12.5 Rationally infinite spaces of finite category 319
12.6 Rationally infinite CW complexes of finite dimension 325
13 Exponential growth 329
13.1 The invariant log index 333
13.2 Growth of graded Lie algebras 333
13.3 Weak exponential growth and critical degree 337
13.4 Approximation of log index I 343
13.5 Moderate exponential growth 350
13.6 Exponential growth 358
14 Structure of a graded Lie algebra of finite depth 367
14.1 Introduction 367
14.2 The spectrum 368
14.3 Minimal sub Lie algebras 372
14.4 The weak complements of an ideal 377
14.5 L-equivalence 380
14.6 The odd part of a graded Lie algebra 387
15 Weight decompositions of a Sullivan Lie algebra 389
15.1 Weight decompositions 389
15.2 Exponential growth of L 393
15.3 The fundamental Lie algebra of 1-formal Sullivan algebra 395
16 Problems 401
Bibliography 405
Index 409
