Rational Homotopy Theory
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1 Yves Felix Stephen Halperin Jean-Claude Thomas Rational Homotopy Theory Springer
2 Contents Introduction Table of Examples vii xxvii I Homotopy Theory, Resolutions for Fibrations, and P- local Spaces 0 Topological spaces 1 1 CW complexes, homotopy groups and cofibrations 4 (a) CW complexes 4 (b) Homotopy groups 10 (c) Weak homotopy type 12 (d) Cofibrations and NDR pairs 15 (e) Adjunction spaces 18 (f) Cones, suspensions, joins and smashes 20 2 Fibrations and topological monoids 23 (a) Fibrations 24 (b) Topological monoids and G-fibrations 28 (c) The homotopy fibre and the holonomy action 30 (d) Fibre bundles and principal bundles 32 (e) Associated bundles, classifying spaces, the Borel construction and the holonomy fibration 36 3 Graded (differential) algebra 40 (a) Graded modules and complexes 40 (b)-graded algebras 43 (c) Differential graded algebras 46 (d) Graded coalgebras 47 (e) When h is a field...' Singular chains, homology and Eilenberg-MacLane spaces 51 (a) Basic definitions, (normalized) singular chains.... '. 52 (b) Topological products, tensor products and the dgc, C,(X; Jc) (c) Pairs, excision, homotopy and the Hurewicz homomorphism (d) Weak homotopy equivalences 58 (e) Cellular homology and the Hurewicz theorem 59 (f) Eilenberg-MacLane spaces 62 5 The cochain algebra C*(X;k) 65
3 xx CONTENTS 6 (i?, d) modules and semifree resolutions 68 N (a) Semifree models 68 (b) Quasi-isomorphism theorems 72 7 Semifree cochain models of a fibration 77 8 Semifree chain models of a G fibration 88 (a) The chain algebra of a topological monoid 88 (b) Semifree chain models 89 (c) The quasi-isomorphism theorem 92 (d) The Whitehead-Serre theorem 94 9 V local and rational spaces, 102 (a) P-local spaces 102 (b) Localization 107 (c) Rational homotopy type 110 II Sullivan Models 10 Commutative cochain algebras for spaces and simplicial sets 115 (a) Simplicial sets and simplicial cochain algebras 116 (b) The construction of A(K) 118 (c) The simplicial commutative cochain algebra APL, and APL(X) (d) The simplicial cochain algebra Gpi, and the main theorem (e) Integration and the de Rham theorem Smooth Differential Forms 131 (a) Smooth manifolds 131 (b) Smooth differential forms 132 (c) Smooth singular simplices 133 (b.)-(d) The weak equivalence A DR {M) ~ A PL (M;R) Sullivan models 138 (a) Sullivan algebras and models: constructions and examples 140 (b) Homotopy in Sullivan algebras. : 148 (c) Quasi-isomorphisms, Sullivan representatives, uniqueness of minimal models and formal spaces....' 152 (d) Computational examples. 156 (e) Differential forms and geometric examples Adjunction spaces, homotopy groups and Whitehead products 165 (a) Morphisms and quasi-isomorphisms (b) Adjunction spaces 168 (c) Homotopy groups 171 (d) Cell attachments : 173
4 Rational Homotopy Theory xxi (e) Whitehead product and the quadratic part of the differential Relative Sullivan algebras 181 (a) The semifree property, existence of models and homotopy 182 (b) Minima! Sullivan models Fibrations, homotopy groups and Lie group actions 195 (a) Models of fibrations 195 (b) Loops on spheres, Eilenberg-MacLane spaces and spherical fibrations200 (c) Pullbacks and maps of fibrations 203 (d) Homotopy groups 208 (e) The long exact homotopy sequence 213 (f) Principal bundles, homogeneous spaces and Lie group actions The loop space homology algebra 223 (a) The loop space homology algebra 224 (b) The minimal Sullivan model of the path space fibration 226 (c) The rational product decomposition of VlX 228 (d) The primitive subspace of H*(QX;]k) 230 (e) Whitehead products, commutators and the algebra structure of z) Spatial realization 237 (a) The Milnor realization of a simplicial set 238 (b) Products and fibre bundles 243 (c) The Sullivan realization of a commutative cochain algebra 247 (d) The spatial realization of a Sullivan algebra 249 (e) Morphisms and continuous maps 255 (f) Integration, chain complexes and products 256 III Graded Differential Algebra (continued) 18 Spectral sequences 260 (a) Bigraded modules and spectral sequences ; 260 (b) Filtered differential modules 261 (c) Convergence 263 (d) Tensor products and extra structure The bar and cobar constructions Projective resolutions of graded modules 273 (a) Projective resolutions." (b) Graded Ext and Tor (c) Projective dimension 278 (d) Semifree resolutions 278
5 xxii CONTENTS IV Lie Models 21 Graded (differential) Lie algebras and Hopf algebras 283 (a) Universal enveloping algebras 285 (b) Graded Hopf algebras 288 (c) Free graded Lie algebras 289 (d) The homotopy Lie algebra of a topological space 292 (e) The homotopy Lie algebra of a minimal Sullivan algebra 294 (f) Differential graded Lie algebras and differential graded Hopf algebras The Quillen functors C* and C 299 (a) Graded coalgebras 299 (b) The construction of C*(L) and of C»(L; M) 301 (c) The properties of C, (L; UL) 302 (d) The quasi-isomorphism C,(L) = > BUL 305 (e) The construction C(C,d) 306 (f) Free Lie models The commutative cochain algebra, C*(L,dL) 313 (a) The constructions C*(L,di), and (A,d) 313 (b) The homotopy Lie algebra and the Milnor-Moore spectral sequence 317 (c) Cohomology with coefficients Lie models for topological spaces and CW complexes 322 (a) Free Lie models of topological spaces 324 (b) Homotopy and homology in a Lie model 325 c) Suspensions and wedges of spheres 326 (d) Lie models for adjunction spaces 328 (e) CW complexes and chain Lie algebras 331 (f) Examples 331 (g) Lie model for a homotopy fibre Chain Lie algebras and topological groups 337 (a) The topological group, FL 337 (b) The principal fibre bundle, 338 (c) FL as a model for the topological monoid, ftx 340 (d) Morphisms of chain Lie algebras and the holonomy action The dg Hopf algebra C,(fiX) 343 (a) Dga homotopy... V- 344 (b) The dg Hopf algebra C*(nX) and the statement of the theorem. 346 (c) The chain algebra quasi-isomorphism 8 : (Uhy,d) 347 (d) The proof of Theorem
6 Rational Homotopy Theory xxiii V Rational Lusternik Schnirelmann Category 27 Lusternik-Schnirelmann category 351 (a) LS category of spaces and maps 352 (b) Ganea's fibre-cofibre construction 355 (c) Ganea spaces and LS category 357 (d) Cone-length and LS category: Ganea's theorem 359 (e) Cone-length and LS category: Cornea's theorem 361 (f) Cup-length, c(x;k) and Toomer's invariant, e(x;k) Rational LS category and rational cone-length 370 (a) Rational LS category 371 (b) Rational cone-length 372 (c) The mapping theorem 375 (d) Gottlieb groups LS category of Sullivan algebras 381 (a) The rational cone-length of spaces and the product length of models 382 (b) The LS category of a Sullivan algebra 384 (c) The mapping theorem for Sullivan algebras 389 (d) Gottlieb elements 392 (e) Hess' theorem 393 (f) The model of {AV,d) -> {KV/K >m V,d) 396 (g) The Milnor-Moore spectral sequence and Ginsburg's theorem (h) The invariants meat and e for (AV, (i)-modules Rational LS category of products and fibrations 406 (a) Rational LS category of products 406 (b) Rational LS category of fibrations 408 (c) The mapping theorem for a fibre inclusion The homotopy Lie algebra and the holonomy representation 415 (a) The holonomy representation for a Sullivan model 418 (b) Local nilpotence and local conilpotence 420 (c) Jessup's theorem, 424 (d) Proof of Jessup's theorem 425 (e) Examples 430 (f) Iterated Lie brackets 432 VI The Rational Dichotomy: Elliptic and Hyperbolic Spaces and Other Applications 32 Elliptic spaces 434
7 xxiv CONTENTS (a) Pure Sullivan algebras (b) Characterization of elliptic Sullivan algebras 438 (c) Exponents and formal dimension 441 (d) Euler-Poincare characteristic 444 (e) Rationally elliptic topological spaces 447 (f) Decomposability of the loop spaces of rationally elliptic spaces Growth of Rational Homotopy Groups 452 (a) Exponential growth of rational homotopy groups 453 (b) Spaces whose rational homology is finite dimensional 455 (c) Loop space homology The Hochschild-Serre spectral sequence 464 (a) Horn, Ext, tensor and Tor for t/l-modules 465 (b) The Hochschild-Serre spectral sequence 467 (c) Coefficients in UL Grade and depth for fibres and loop spaces 474 (a) Complexes of finite length 475 (b) fty-spaces and C*(nY)-modules 476 (c) The Milnor resolution of A 478 (d) The grade theorem for a homotopy fibre 481 (e) The depth of ff.(nx) '.* 486 (f) The depth of UL 486 (g) The depth theorem for Sullivan algebras Lie algebras of finite depth 492 (a) Depth and grade 493 (b) Solvable Lie algebras and the radical 495 (c) Noetherian enveloping algebras 496 (d) Locally nilpotent elements 497 (ef Examples Cell Attachments 501 (a) The homology of the homotopy fibre, X x Y PY. 502 (b) Whitehead products and G-fibrations 502 (c) Inert element 503 (d) The homotopy Lie algebra of a spherical 2-cone 505 (e) Presentations of graded Lie algebras 507 (f) The Lofwall-Roos example Poincare Duality 511 (b) Properties of Poincare duality 511 (b) Elliptic spaces 512 (c) LS category 513 (d) Inert elements 513
8 Rational Homotopy Theory xxv 39 Seventeen Open Problems 516 References 521 Index 531
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