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Ⅰ Homotopy Theory，Resolutions for Fibrations，and P-local Spaces1

0 Topological spaces1

1 CW complexes，homotopy groups and cofibrations4

（a）CW complexes4

（b）Homotopy groups10

（c）Weak homotopy type12

（d）Cofibrations and NDR pairs15

（e）Adjunction spaces18

（f）Cones，suspensions，joins and smashes20

2 Fibrations and topological monoids23

（a）Fibrations24

（b）Topological monoids and G-fibrations28

（c）The homotopy fibre and the holonomy action30

（d）Fibre bundles and principal bundles32

（e）Associated bundles，classifying spaces，the Borel construction and the holonomy fibration36

3 Graded（differential）algebra40

（a）Graded modules and complexes40

（b）Graded algebras43

（c）Differential graded algebras46

（d）Graded coalgebras47

（e）When k is a field48

4 Singular chains，homology and Eilenberg-MacLane spaces51

（a）Basic definitions，（normalized）singular chains52

（b）Topological products，tensor products and the dgc，C（X;k）53

（c）Pairs，excision，homotopy and the Hurewicz homomorphism56

（d）Weak homotopy equivalences58

（e）Cellular homology and the Hurewicz theorem59

（f）Eilenberg-MacLane spaces62

5 The cochain algebra C（X;k）65

6 （R，d）-modules and semifree resolutions68

（a）Semifree models68

（b）Quasi-isomorphism theorems72

7 Semifree cochain models of a fibration77

8 Semifree chain models of a G-fibration88

（a）The chain algebra of a topological monoid88

（b）Semifree chain models89

（c）The quasi-isomorphism theorem92

（d）The Whitehead-Serre theorem94

9 P-local and rational spaces102

（a）P-local spaces102

（b）Localization107

（c）Rational homotopy type110

Ⅱ Sullivan Models115

10 Commutative cochain algebras for spaces and simplicial sets115

（a）Simplicial sets and simplicial cochain algebras116

（b）The construction of A（K）118

（c）The simplicial commutative cochain algebra APL，and APL（X）121

（d）The simplicial cochain algebra CPL，and the main theorem124

（e）Integration and the de Rham theorem128

11 Smooth Differential Forms131

（a）Smooth manifolds131

（b）Smooth differential forms132

（c）Smooth singular simplices133

（b）（d）The weak equivalence ADR（M）？APL（M;R）134

12 Sullivan models138

（a）Sullivan algebras and models:constructions and examples140

（b）Homotopy in Sullivan algebras148

（c）Quasi-isomorphisms，Sullivan representatives，uniqueness of mini-mal models and formal spaces152

（d）Computational examples156

（e）Differential forms and geometric examples160

13 Adjunction spaces，homotopy groups and Whitehead products165

（a）Morphisms and quasi-isomorphisms166

（b）Adjunction spaces168

（c）Homotopy groups171

（d）Cell attachments173

（e）Whitehead product and the quadratic part of the differential175

14 Relative Sullivan algebras181

（a）The semifree property，existence of models and homotopy182

（b）Minimal Sullivan models186

15 Fibrations，homotopy groups and Lie group actions195

（a）Models of fibrations195

（b）Loops on spheres，Eilenberg-MacLane spaces and spherical fibrations200

（c）Pullbacks and maps of fibrations203

（d）Homotopy groups208

（e）The long exact homotopy sequence213

（f）Principal bundles，homogeneous spaces and Lie group actions216

16 The loop space homology algebra223

（a）The loop space homology algebra224

（b）The minimal Sullivan model of the path space fibration226

（c）The rational product decomposition of ΩX228

（d）The primitive subspace of H（ΩX;k）230

（e）Whitehead products，commutators and the algebra structure of H（ΩX;k）232

17 Spatial realization237

（a）The Milnor realization of a simplicial set238

（b）Products and fibre bundles243

（c）The Sullivan realization of a commutative cochain algebra247

（d）The spatial realization of a Sullivan algebra249

（e）Morphisms and continuous maps255

（f）Integration，chain complexes and products256

Ⅲ Graded Differential Algebra（continued）260

18 Spectral sequences260

（a）Bigraded modules and spectral sequences260

（b）Filtered differential modules261

（c）Convergence263

（d）Tensor products and extra structure265

19 The bar and cobar constructions268

20 Projective resolutions of graded modules273

（a）Projective resolutions273

（b）Graded Ext and Tor275

（c）Projective dimension278

（d）Semifree resolutions278

Ⅳ Lie Models283

21 Graded（differential）Lie algebras and Hopf algebras283

（a）Universal enveloping algebras285

（b）Graded Hopf algebras288

（c）Free graded Lie algebras289

（d）The homotopy Lie algebra of a topological space292

（e）The homotopy Lie algebra of a minimal Sullivan algebra294

（f）Differential graded Lie algebras and differential graded Hopf algebras296

22 The Quillen functors C and L299

（a）Graded coalgebras299

（b）The construction of C（L）and of C（L;M）301

（c）The properties of C（L;UL）302

（d）The quasi-isomorphism C（L）？ BUL305

（e）The construction L（C，d）306

（f）Free Lie models309

23 The commutative cochain algebra，C（L，dL）313

（a）The constructions C（L，dL），and L（A，d）313

（b）The homotopy Lie algebra and the Milnor-Moore spectral sequence317

（c）Cohomology with coefficients319

24 Lie models for topological spaces and CW complexes322

（a）Free Lie models of topological spaces324

（b）Homotopy and homology in a Lie model325

c）Suspensions and wedges of spheres326

（d）Lie models for adjunction spaces328

（e）CW complexes and chain Lie algebras331

（f）Examples331

（g）Lie model for a homotopy fibre334

25 Chain Lie algebras and topological groups337

（a）The topological group，|ΓL|337

（b）The principal fibre bundle，338

（c）|ΓL| as a model for the topological monoid，ΩX340

（d）Morphisms of chain Lie algebras and the holonomy action341

26 The dg Hopf algebra C（ΩX）343

（a）Dga homotopy344

（b）The dg Hopf algebra C（ΩX）and the statement of the theorem346

（c）The chain algebra quasi-isomorphism θ：（ULV，d）347

（d）The proof of Theorem 26.5349

Ⅴ Rational Lusternik Schnirelmann Category351

27 Lusternik-Schnirelmann category351

（a）LS category of spaces and maps352

（b）Ganea's fibre-cofibre construction355

（c）Ganea spaces and LS category357

（d）Cone-length and LS category:Ganea's theorem359

（e）Cone-length and LS category:Cornea's theorem361

（f）Cup-length，c（X;k）and Toomer's invariant，e（X;k）366

28 Rational LS category and rational cone-length370

（a）Rational LS category371

（b）Rational cone-length372

（c）The mapping theorem375

（d）Gottlieb groups377

29 LS category of Sullivan algebras381

（a）The rational cone-length of spaces and the product length of models382

（b）The LS category of a Sullivan algebra384

（c）The mapping theorem for Sullivan algebras389

（d）Gottlieb elements392

（e）Hess'theorem393

（f）The model of（？V，d）→（？V/？＞mV，d）396

（g）The Milnor-Moore spectral sequence and Ginsburg's theorem399

（h）The invariants mcat and e for（？V，d）-modules401

30 Rational LS category of products and fibrations406

（a）Rational LS category of products406

（b）Rational LS category of fibrations408

（c）The mapping theorem for a fibre inclusion411

31 The homotopy Lie algebra and the holonomy representation415

（a）The holonomy representation for a Sullivan model418

（b）Local nilpotence and local conilpotence420

（c）Jessup's theorem424

（d）Proof of Jessup's theorem425

（e）Examples430

（f）Iterated Lie brackets432

Ⅵ The Rational Dichotomy:Elliptic and Hyperbolic Spaces and Other Applications434

32 Elliptic spaces434

（a）Pure Sullivan algebras435

（b）Characterization of elliptic Sullivan algebras438

（c）Exponents and formal dimension441

（d）Euler-Poincaré characteristic444

（e）Rationally elliptic topological spaces447

（f）Decomposability of the loop spaces of rationally elliptic spaces449

33 Growth of Rational Homotopy Groups452

（a）Exponential growth of rational homotopy groups453

（b）Spaces whose rational homology is finite dimensional455

（c）Loop space homology458

34 The Hochschild-Serre spectral sequence464

（a）Hom，Ext，tensor and Tor for UL-modules465

（b）The Hochschild-Serre spectral sequence467

（c）Coefficients in UL469

35 Grade and depth for fibres and loop spaces474

（a）Complexes of finite length475

（b）ΩY-spaces and C（ΩY）-modules476

（c）The Milnor resolution of k478

（d）The grade theorem for a homotopy fibre481

（e）The depth of H（ΩX）486

（f）The depth of UL486

（g）The depth theorem for Sullivan algebras487

36 Lie algebras of finite depth492

（a）Depth and grade493

（b）Solvable Lie algebras and the radical495

（c）Noetherian enveloping algebras496

（d）Locally nilpotent elements497

（e）Examples497

37 Cell Attachments501

（a）The homology of the homotopy fibre，X ×Y PY502

（b）Whitehead products and G-fibrations502

（c）Inert element503

（d）The homotopy Lie algebra of a spherical 2-cone505

（e）Presentations of graded Lie algebras507

（f）The L？fwall-Roos example509

38 Poincaré Duality511

（b）Properties of Poincaré duality511

（b）Elliptic spaces512

（c）LS category513

（d）Inert elements513

39 Seventeen Open Problems516

References521

Index531