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

0 Topological spaces

1 CW complexes，homotopy groups and cofibrations

（a）CW complexes

（b）Homotopy groups

（c）Weak homotopy type

（d）Cofibrations and NDR pairs

（e）Adjunction spaces

（f）Cones，suspensions，joins and smashes

2 Fibrations and topological monoids

（a）Fibrations

（b）Topological monoids and G-fibrations

（c）The homotopy fibre and the holonomy action

（d）Fibre bundles and principal bundles

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

3 Graded（differential）algebra

（a）Graded modules and complexes

（b）Graded algebras

（c）Differential graded algebras

（d）Graded coalgebras

（e）When k is a field

4 Singular chains，homology and Eilenberg-MacLane spaces

（a）Basic definitions，（normalized）singular chains

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

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

（d）Weak homotopy equivalences

（e）Cellular homology and the Hurewicz theorem

（f）Eilenberg-MacLane spaces

5 The cochain algebra C（X;k）

6 （R，d）-modules and semifree resolutions

（a）Semifree models

（b）Quasi-isomorphism theorems

7 Semifree cochain models of a fibration

8 Semifree chain models of a G-fibration

（a）The chain algebra of a topological monoid

（b）Semifree chain models

（c）The quasi-isomorphism theorem

（d）The Whitehead-Serre theorem

9 P-local and rational spaces

（a）P-local spaces

（b）Localization

（c）Rational homotopy type

Ⅱ Sullivan Models

10 Commutative cochain algebras for spaces and simplicial sets

（a）Simplicial sets and simplicial cochain algebras

（b）The construction of A（K）

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

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

（e）Integration and the de Rham theorem

11 Smooth Differential Forms

（a）Smooth manifolds

（b）Smooth differential forms

（c）Smooth singular simplices

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

12 Sullivan models

（a）Sullivan algebras and models:constructions and examples

（b）Homotopy in Sullivan algebras

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

（d）Computational examples

（e）Differential forms and geometric examples

13 Adjunction spaces，homotopy groups and Whitehead products

（a）Morphisms and quasi-isomorphisms

（b）Adjunction spaces

（c）Homotopy groups

（d）Cell attachments

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

14 Relative Sullivan algebras

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

（b）Minimal Sullivan models

15 Fibrations，homotopy groups and Lie group actions

（a）Models of fibrations

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

（c）Pullbacks and maps of fibrations

（d）Homotopy groups

（e）The long exact homotopy sequence

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

16 The loop space homology algebra

（a）The loop space homology algebra

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

（c）The rational product decomposition of ΩX

（d）The primitive subspace of H（ΩX;k）

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

17 Spatial realization

（a）The Milnor realization of a simplicial set

（b）Products and fibre bundles

（c）The Sullivan realization of a commutative cochain algebra

（d）The spatial realization of a Sullivan algebra

（e）Morphisms and continuous maps

（f）Integration，chain complexes and products

Ⅲ Graded Differential Algebra（continued）

18 Spectral sequences

（a）Bigraded modules and spectral sequences

（b）Filtered differential modules

（c）Convergence

（d）Tensor products and extra structure

19 The bar and cobar constructions

20 Projective resolutions of graded modules

（a）Projective resolutions

（b）Graded Ext and Tor

（c）Projective dimension

（d）Semifree resolutions

Ⅳ Lie Models

21 Graded（differential）Lie algebras and Hopf algebras

（a）Universal enveloping algebras

（b）Graded Hopf algebras

（c）Free graded Lie algebras

（d）The homotopy Lie algebra of a topological space

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

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

22 The Quillen functors C and L

（a）Graded coalgebras

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

（c）The properties of C（L;UL）

（d）The quasi-isomorphism C（L）？ BUL

（e）The construction L（C，d）

（f）Free Lie models

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

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

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

（c）Cohomology with coefficients

24 Lie models for topological spaces and CW complexes

（a）Free Lie models of topological spaces

（b）Homotopy and homology in a Lie model

c）Suspensions and wedges of spheres

（d）Lie models for adjunction spaces

（e）CW complexes and chain Lie algebras

（f）Examples

（g）Lie model for a homotopy fibre

25 Chain Lie algebras and topological groups

（a）The topological group，|ΓL|

（b）The principal fibre bundle，

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

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

26 The dg Hopf algebra C（ΩX）

（a）Dga homotopy

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

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

（d）The proof of Theorem 26.5

Ⅴ Rational Lusternik Schnirelmann Category

27 Lusternik-Schnirelmann category

（a）LS category of spaces and maps

（b）Ganea's fibre-cofibre construction

（c）Ganea spaces and LS category

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

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

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

28 Rational LS category and rational cone-length

（a）Rational LS category

（b）Rational cone-length

（c）The mapping theorem

（d）Gottlieb groups

29 LS category of Sullivan algebras

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

（b）The LS category of a Sullivan algebra

（c）The mapping theorem for Sullivan algebras

（d）Gottlieb elements

（e）Hess'theorem

（f）The model of（？V，d）→（？V/？＞mV，d）

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

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

30 Rational LS category of products and fibrations

（a）Rational LS category of products

（b）Rational LS category of fibrations

（c）The mapping theorem for a fibre inclusion

31 The homotopy Lie algebra and the holonomy representation

（a）The holonomy representation for a Sullivan model

（b）Local nilpotence and local conilpotence

（c）Jessup's theorem

（d）Proof of Jessup's theorem

（e）Examples

（f）Iterated Lie brackets

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

32 Elliptic spaces

（a）Pure Sullivan algebras

（b）Characterization of elliptic Sullivan algebras

（c）Exponents and formal dimension

（d）Euler-Poincaré characteristic

（e）Rationally elliptic topological spaces

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

33 Growth of Rational Homotopy Groups

（a）Exponential growth of rational homotopy groups

（b）Spaces whose rational homology is finite dimensional

（c）Loop space homology

34 The Hochschild-Serre spectral sequence

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

（b）The Hochschild-Serre spectral sequence

（c）Coefficients in UL

35 Grade and depth for fibres and loop spaces

（a）Complexes of finite length

（b）ΩY-spaces and C（ΩY）-modules

（c）The Milnor resolution of k

（d）The grade theorem for a homotopy fibre

（e）The depth of H（ΩX）

（f）The depth of UL

（g）The depth theorem for Sullivan algebras

36 Lie algebras of finite depth

（a）Depth and grade

（b）Solvable Lie algebras and the radical

（c）Noetherian enveloping algebras

（d）Locally nilpotent elements

（e）Examples

37 Cell Attachments

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

（b）Whitehead products and G-fibrations

（c）Inert element

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

（e）Presentations of graded Lie algebras

（f）The L？fwall-Roos example

38 Poincaré Duality

（b）Properties of Poincaré duality

（b）Elliptic spaces

（c）LS category

（d）Inert elements

39 Seventeen Open Problems

References

Index