Rational Homotopy Theory

Yves Félix, Stephen Halperin, Jean-Claude Thomas (auth.)9780387950686, 0387950680

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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Table of contents :
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
The cochain algebra C * ( X ; $$Bbbk $$ )….Pages 65-67
(R, d)-modules and semifree resolutions….Pages 68-76
Semifree cochain models of a fibration….Pages 77-87
Semifree chain models of a G —fibration….Pages 88-101
p—local and rational spaces….Pages 102-114
Front Matter….Pages N1-N1
Commutative cochain algebras for spaces and simplicial sets….Pages 115-130
Smooth Differential Forms….Pages 131-137
Sullivan models….Pages 138-164
Adjunction spaces, homotopy groups and Whitehead products….Pages 165-180
Relative Sullivan algebras….Pages 181-194
Fibrations, homotopy groups and Lie group actions….Pages 195-222
The loop space homology algebra….Pages 223-236
Spatial realization….Pages 237-259
Front Matter….Pages N3-N3
Spectral sequences….Pages 260-267
The bar and cobar constructions….Pages 268-272
Front Matter….Pages N3-N3
Projective resolutions of graded modules….Pages 273-282
Front Matter….Pages N5-N5
Graded (differential) Lie algebras and Hopf algebras….Pages 283-298
The Quillen functors C * and ℒ….Pages 299-312
The commutative cochain algebra, C * (L,d L )….Pages 313-321
Lie models for topological spaces and CW complexes….Pages 322-336
Chain Lie algebras and topological groups….Pages 337-342
The dg Hopf algebra C * (ΩX)….Pages 343-350
Front Matter….Pages N7-N7
Lusternik-Schnirelmann category….Pages 351-369
Rational LS category and rational cone-length….Pages 370-380
LS category of Sullivan algebras….Pages 381-405
Rational LS category of products and fibrations….Pages 406-414
The homotopy Lie algebra and the holonomy representation….Pages 415-433
Front Matter….Pages N9-N9
Elliptic spaces….Pages 434-451
Growth of Rational Homotopy Groups….Pages 452-463
The Hochschild-Serre spectral sequence….Pages 464-473
Grade and depth for fibres and loop spaces….Pages 474-491
Lie algebras of finite depth….Pages 492-500
Cell Attachments….Pages 501-510
Poincaré Duality….Pages 511-515
Seventeen Open Problems….Pages 516-520
Back Matter….Pages 521-539