Beschreibung
InhaltsangabeI 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 resolutio
