Ebook: Rational Homotopy Theory

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.

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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.

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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.
Content:
Front Matter....Pages i-xxxii
Front Matter....Pages xxxiii-xxxiii
Topological spaces....Pages 1-3
CW complexes, homotopy groups and cofibrations....Pages 4-22
Fibrations and topological monoids....Pages 23-39
Graded (differential) algebra....Pages 40-50
Singular chains, homology and Eilenberg- MacLane spaces....Pages 51-64
(R, d)-modules and semifree resolutions....Pages 65-67
Semifree cochain models of a fibration....Pages 68-76
Semifree chain models of a G—fibration....Pages 77-87
p—local and rational spaces....Pages 88-101
Front Matter....Pages 102-114
Commutative cochain algebras for spaces and simplicial sets....Pages N1-N1
Smooth Differential Forms....Pages 115-130
Sullivan models....Pages 131-137
Adjunction spaces, homotopy groups and Whitehead products....Pages 138-164
Relative Sullivan algebras....Pages 165-180
Fibrations, homotopy groups and Lie group actions....Pages 181-194
The loop space homology algebra....Pages 195-222
Spatial realization....Pages 223-236
Front Matter....Pages 237-259
Spectral sequences....Pages N3-N3
The bar and cobar constructions....Pages 260-267
Front Matter....Pages 268-272
Projective resolutions of graded modules....Pages N3-N3
Front Matter....Pages 273-282
Graded (differential) Lie algebras and Hopf algebras....Pages N5-N5
The Quillen functors C* and ?....Pages 283-298
Lie models for topological spaces and CW complexes....Pages 299-312
Chain Lie algebras and topological groups....Pages 313-321
Front Matter....Pages 322-336
Lusternik-Schnirelmann category....Pages 337-342
Rational LS category and rational cone-length....Pages 343-350
LS category of Sullivan algebras....Pages N7-N7
Rational LS category of products and fibrations....Pages 351-369
The homotopy Lie algebra and the holonomy representation....Pages 370-380
Front Matter....Pages 381-405
Elliptic spaces....Pages 406-414
Growth of Rational Homotopy Groups....Pages 415-433
The Hochschild-Serre spectral sequence....Pages N9-N9
Grade and depth for fibres and loop spaces....Pages 434-451
Lie algebras of finite depth....Pages 452-463
Cell Attachments....Pages 464-473
Poincar? Duality....Pages 474-491
Seventeen Open Problems....Pages 492-500
Back Matter....Pages 501-510
....Pages 511-515