Rational Homotopy Theory

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 $$\Bbbk $$ 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;$$\Bbbk $$).- (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;$$\Bbbk $$.- 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;?).- 12 Sullivan models.- (a) Sullivan algebras and models: constructions and examples.- (b) Homotopy in Sullivan algebras.- (c) Quasi-isomorphisms, Sullivan representatives, uniqueness of minimal 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 sphericalflbrations.- (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 ?X.- (d) The primitive subspace of H*(?X;$$\Bbbk $$).- (e) Whitehead products, commutators and the algebra structure of H*(?X;$$\Bbbk $$).- 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 C.- (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 !?L!.- (b) The principal fibre bundle,.- (c) \?L\ as a model for the topological monoid, ?X.- (d) Morphisms of chain Lie algebras and the holonomy action.- 26 The dg Hopf algebra C*(?X.- (a) Dga homotopy.- (b) The dg Hopf algebra C*(?X) and the statement of the theorem.- (c) The chain algebra quasi-isomorphism ?: (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; $$\Bbbk $$) and Toomer's invariant, e(X; $$\Bbbk $$).- 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 meat and e for (?V, d)-modules.- 30 Rational LS category of products and flbrations.- (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) Horn, 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) ?Y-spaces and C*(?Y)-modules.- (c) The Milnor resolution of $$\Bbbk $$.- (d) The grade theorem for a homotopy fibre.- (e) The depth of H*(?X).- (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 ×YPY.- (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 Löfwall-Roos example.- 38 Poincaré Duality.- (b) Properties of Poincaré duality.- (b) Elliptic spaces.- (c) LS category.- (d) Inert elements.- Rational Homotopy Theory.- 39 Seventeen Open Problems.- References.