Rational Homotopy Theory

ISBN-13: 9780387950686 / Angielski / Twarda / 2000 / 539 str.

Yves Felix; Y. Felix; Stephen Halperin

Opis

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.

Recenzje

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

Spis treści

I 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 resolutions of graded modules.- (a) Projective resolutions.- (b) Graded Ext and Tor.- (c) Projective dimension.- (d) Semifree resolutions.- IV Lie Models.- 21 Graded (differential) Lie algebras and Hopf algebras.- (a) Universal enveloping algebras.- (b) Graded Hopf algebras.- (c) Free graded Lie algebras.- (d) The homotopy Lie algebra of a topological space.- (e) The homotopy Lie algebra of a minimal Sullivan algebra.- (f) Differential graded Lie algebras and differential graded Hopf algebras.- 22 The Quillen functors C* and C.- (a) Graded coalgebras.- (b) The construction of C*(L) and of C*(L;M).- (c) The properties of C*(L;UL).- (d) The quasi-isomorphism C* (L) ?? BUL.- (e) The construction L(C, d).- (f) Free Lie models.- 23 The commutative cochain algebra, C*(L,dL).- (a) The constructions C*(L,DL), and L(A,d).- (b) The homotopy Lie algebra and the Milnor-Moore spectral sequence.- (c) Cohomology with coefficients.- 24 Lie models for topological spaces and CW complexes.- (a) Free Lie models of topological spaces.- (b) Homotopy and homology in a Lie model.- (c) Suspensions and wedges of spheres.- (d) Lie models for adjunction spaces.- (e) CW complexes and chain Lie algebras.- (f) Examples.- (g) Lie model for a homotopy fibre.- 25 Chain Lie algebras and topological groups.- (a) The topological group !?L!.- (b) The principal fibre bundle,.- (c) \?L\ as a model for the topological monoid, ?X.- (d) Morphisms of chain Lie algebras and the holonomy action.- 26 The dg Hopf algebra C*(?X.- (a) Dga homotopy.- (b) The dg Hopf algebra C*(?X) and the statement of the theorem.- (c) The chain algebra quasi-isomorphism ? : (ULv ,d).- (d) The proof of Theorem 26.5.- V Rational Lusternik Schnirelmann Category.- 27 Lusternik-Schnirelmann category.- (a) LS category of spaces and maps.- (b) Ganea’s fibre-cofibre construction.- (c) Ganea spaces and LS category.- (d) Cone-length and LS category: Ganea’s theorem.- (e) Cone-length and LS category: Cornea’s theorem.- (f) Cup-length, c(X; $$\Bbbk $$) and Toomer’s invariant, e(X; $$\Bbbk $$).- 28 Rational LS category and rational cone-length.- (a) Rational LS category.- (b) Rational cone-length.- (c) The mapping theorem.- (d) Gottlieb groups.- 29 LS category of Sullivan algebras.- (a) The rational cone-length of spaces and the product length of models.- (b) The LS category of a Sullivan algebra.- (c) The mapping theorem for Sullivan algebras.- (d) Gottlieb elements.- (e) Hess’ theorem.- (f) The model of (?V,d) ? (?V/?>mV,d).- (g) The Milnor-Moore spectral sequence and Ginsburg’s theorem.- (h) The invariants meat and e for (?V, d)-modules.- 30 Rational LS category of products and flbrations.- (a) Rational LS category of products.- (b) Rational LS category of fibrations.- (c) The mapping theorem for a fibre inclusion.- 31 The homotopy Lie algebra and the holonomy representation.- (a) The holonomy representation for a Sullivan model.- (b) Local nilpotence and local conilpotence.- (c) Jessup’s theorem.- (d) Proof of Jessup’s theorem.- (e) Examples.- (f) Iterated Lie brackets.- VI The Rational Dichotomy: Elliptic and Hyperbolic Spaces and Other Applications.- 32 Elliptic spaces.- (a) Pure Sullivan algebras.- (b) Characterization of elliptic Sullivan algebras.- (c) Exponents and formal dimension.- (d) Euler-Poincaré characteristic.- (e) Rationally elliptic topological spaces.- (f) Decomposability of the loop spaces of rationally elliptic spaces.- 33 Growth of Rational Homotopy Groups.- (a) Exponential growth of rational homotopy groups.- (b) Spaces whose rational homology is finite dimensional.- (c) Loop space homology.- 34 The Hochschild-Serre spectral sequence.- (a) Horn, Ext, tensor and Tor for UL-modules.- (b) The Hochschild-Serre spectral sequence.- (c) Coefficients in UL.- 35 Grade and depth for fibres and loop spaces.- (a) Complexes of finite length.- (b) ?Y-spaces and C*(?Y)-modules.- (c) The Milnor resolution of $$\Bbbk $$.- (d) The grade theorem for a homotopy fibre.- (e) The depth of H*(?X).- (f) The depth of UL.- (g) The depth theorem for Sullivan algebras.- 36 Lie algebras of finite depth.- (a) Depth and grade.- (b) Solvable Lie algebras and the radical.- (c) Noetherian enveloping algebras.- (d) Locally nilpotent elements.- (e) Examples.- 37 Cell Attachments.- (a) The homology of the homotopy fibre, X ×YPY.- (b) Whitehead products and G-fibrations.- (c) Inert element.- (d) The homotopy Lie algebra of a spherical 2-cone.- (e) Presentations of graded Lie algebras.- (f) The Löfwall-Roos example.- 38 Poincaré Duality.- (b) Properties of Poincaré duality.- (b) Elliptic spaces.- (c) LS category.- (d) Inert elements.- Rational Homotopy Theory.- 39 Seventeen Open Problems.- References.