This is a long awaited book on rational homotopy theory which contains all the main theorems with complete proofs, and more elementary proofs for many results that were proved ten or fifteen years ago. The authors added a frist section on classical algebraic topology to make the book accessible to students with only little background in algebraic topology.
as well as by the list of open problems in the final section of this monograph. The computational power of rational homotopy theory is due to the discovery by Quillen [135] and by Sullivan [144] of an explicit algebraic formulation. In each case the rational homotopy type of a topological space is the same as the isomorphism class of its algebraic model and the rational homotopy type of a continuous map is the same as the algebraic homotopy class of the correspond ing morphism between models. These models make the rational homology and homotopy of a space transparent. They also (in principle, always, and in prac tice, sometimes) enable the calculation of other homotopy invariants such as the cup product in cohomology, the Whitehead product in homotopy and rational Lusternik-Schnirelmann category. In its initial phase research in rational homotopy theory focused on the identi of these models. These included fication of rational homotopy invariants in terms the homotopy Lie algebra (the translation of the Whitehead product to the homo topy groups of the loop space OX under the isomorphism 11'+1 (X) ~ 1I. (OX», LS category and cone length. Since then, however, work has concentrated on the properties of these in variants, and has uncovered some truly remarkable, and previously unsuspected phenomena. For example - If X is an n-dimensional simply connected finite CW complex, then either its rational homotopy groups vanish in degrees 2': 2n, or else they grow exponentially.
Inhaltsverzeichnis
I Homotopy Theory, Resolutions for Fibrations, and P- local Spaces. - 0 Topological spaces. - 1 CW complexes, homotopy groups and cofibrations. - 2 Fibrations and topological monoids. - 3 Graded (differential) algebra. - 4 Singular chains, homology and Eilenberg-MacLane spaces. - 5 The cochain algebra C*(X; $$\Bbbk $$. - 6 (R, d) modules and semifree resolutions. - 7 Semifree cochain models of a fibration. - 8 Semifree chain models of a G fibration. - 9 P local and rational spaces. - II Sullivan Models. - 10 Commutative cochain algebras for spaces and simplicial sets. - 11 Smooth Differential Forms. - 12 Sullivan models. - 13 Adjunction spaces, homotopy groups and Whitehead products. - 14 Relative Sullivan algebras. - 15 Fibrations, homotopy groups and Lie group actions. - 16 The loop space homology algebra. - 17 Spatial realization. - III Graded Differential Algebra (continued). - 18 Spectral sequences. - 19 The bar and cobar constructions. - 20 Projective resolutions of graded modules. - IV Lie Models. - 21 Graded (differential) Lie algebras and Hopf algebras. - 22 The Quillen functors C* and C. - 23 The commutative cochain algebra, C*(L, dL). - 24 Lie models for topological spaces and CW complexes. - 25 Chain Lie algebras and topological groups. - 26 The dg Hopf algebra C*(? X. - V Rational Lusternik Schnirelmann Category. - 27 Lusternik-Schnirelmann category. - 28 Rational LS category and rational cone-length. - 29 LS category of Sullivan algebras. - 30 Rational LS category of products and flbrations. - 31 The homotopy Lie algebra and the holonomy representation. - VI The Rational Dichotomy: Elliptic and Hyperbolic Spaces and Other Applications. - 32 Elliptic spaces. - 33 Growth of Rational Homotopy Groups. - 34 The Hochschild-Serre spectral sequence. - 35 Grade and depth for fibres and loop spaces. - 36Lie algebras of finite depth. - 37 Cell Attachments. - 38 Poincaré Duality. - 39 Seventeen Open Problems. - References.
