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.
By:
Yves Felix,
Stephen Halperin,
J.-C. Thomas,
J.-C. Thomas
Imprint: Springer-Verlag New York Inc.
Country of Publication: United States
Edition: Softcover reprint of the original 1st ed. 2001
Volume: 205
Dimensions:
Height: 235mm,
Width: 155mm,
Spine: 30mm
Weight: 890g
ISBN: 9781461265160
ISBN 10: 1461265169
Series: Graduate Texts in Mathematics
Pages: 539
Publication Date: 13 October 2012
Audience:
Professional and scholarly
,
Undergraduate
Format: Paperback
Publisher's Status: Active
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.
Reviews for Rational Homotopy Theory
From the reviews: MATHEMATICAL REVIEWS In 535 pages, the authors give a complete and thorough development of rational homotopy theory as well as a review (of virtually) all relevant notions of from basic homotopy theory and homological algebra. This is a truly remarkable achievement, for the subject comes in many guises. Y. Felix, S. Halperin, and J.-C. Thomas Rational Homotopy Theory A complete and thorough development of rational homotopy theory as well as a review of (virtually) all relevant notions from basic homotopy theory and homological algebra. This is truly a magnificent achievement ... a true appreciation for the goals and techniques of rational homotopy theory, as well as an effective toolkit for explicit computation of examples throughout algebraic topology. -AMERICAN MATHEMATICAL SOCIETY
