Table of Contents
Preface xiii
Acknowledgments xix
Introduction to unstable homotopy theory 1
1 Homotopy groups with coefficients 11
1.1 Basic definitions 12
1.2 Long exact sequences of pairs and fibrations 15
1.3 Universal coefficient exact sequences 16
1.4 Functor properties 18
1.5 The Bockstein long exact sequence 20
1.6 Nonfinitely generated coefficient groups 23
1.7 The mod k Hurewicz homomorphism 25
1.8 The mod k Hurewicz isomorphism theorem 27
1.9 The mod k Hurewicz isomorphism theorem for pairs 32
1.10 The third homotopy group with odd coefficients is abelian 34
2 A general theory of localization 35
2.1 Dror Farjoun-Bousfield localization 37
2.2 Localization of abelian groups 46
2.3 Classical localization of spaces: inverting primes 47
2.4 Limits and derived functors 53
2.5 Hom and Ext 55
2.6 p-completion of abelian groups 58
2.7 p-completion of simply connected spaces 63
2.8 Completion implies the mod k Hurewicz isomorphism 68
2.9 Fracture lemmas 70
2.10 Killing Eilenberg-MacLane spaces: Miller's theorem 74
2.11 Zabrodsky mixing: the Hilton-Roitberg examples 83
2.12 Loop structures on p-completions of spheres 88
2.13 Serre's C-theory and finite generation 91
3 Fibre extensions of squares and the Peterson-Stein formula 94
3.1 Homotopy theoretic fibres 95
3.2 Fibre extensions of squares 96
3.3 The Peterson-Stein formula 99
3.4 Totally fibred cubes 101
3.5 Spaces of the homotopy type of a CW complex 104
4 Hilton-Hopf invariants and the EHP sequence 107
4.1 The Bott-Samelson theorem 108
4.2 The James construction 111
4.3 The Hilton-Milnor theorem 113
4.4 The James fibrations and the EHP sequence 118
4.5 James's 2-primary exponent theorem 121
4.6 The 3-connected cover of S3 and its loop space 124
4.7 The first odd primary homotopy class 126
4.8 Elements of order 4 128
4.9 Computations with the EHP sequence 132
5 James-Hopf invariants and Toda-Hopf invariants 135
5.1 Divided power algebras 136
5.2 James-Hopf invariants 141
5.3 p-th Hilton-Hopf invariants 145
5.4 Loops on filtrations of the James construction 148
5.5 Toda-Hopf invariants 151
5.6 Toda's odd primary exponent theorem 155
6 Samelson products 158
6.1 The fibre of the pinch map and self maps of Moore spaces 160
6.2 Existence of the smash decomposition 166
6.3 Samelson and Whitehead products 167
6.4 Uniqueness of the smash decomposition 171
6.5 Lie identities in groups 177
6.6 External Samelson products 179
6.7 Internal Samelson products 186
6.8 Group models for loop spaces 190
6.9 Relative Samelson products 198
6.10 Universal models for relative Samelson products 202
6.11 Samelson products over the loops on an H-space 210
7 Bockstein spectral sequences 221
7.1 Exact couples 222
7.2 Mod p homotopy Bockstein spectral sequences 225
7.3 Reduction maps and extensions 229
7.4 Convergence 230
7.5 Samelson products in the Bockstein spectral sequence 232
7.6 Mod p homology Bockstein spectral sequences 235
7.7 Mod p cohomology Bockstein spectral sequences 238
7.8 Torsion in H-spaces 241
8 Lie algebras and universal enveloping algebras 251
8.1 Universal enveloping algebras of graded Lie algebras 252
8.2 The graded Poincare-Birkhoff-Witt theorem 257
8.3 Consequences of the graded Poincare-Birkhoff-Witt theorem 264
8.4 Nakayama's lemma 267
8.5 Free graded Lie algebras 270
8.6 The change of rings isomorphism 274
8.7 Subalgebras of free graded Lie algebras 278
9 Applications of graded Lie algebras 283
9.1 Serre's product decomposition 284
9.2 Loops of odd primary even dimensional Moore spaces 286
9.3 The Hilton-Milnor theorem 290
9.4 Elements of mod p Hopf invariant one 294
9.5 Cycles in differential graded Lie algebras 299
9.6 Higher order torsion in odd primary Moore spaces 303
9.7 The homology of acyclic free differential graded Lie algebras 306
10 Differential homological algebra 313
10.1 Augmented algebras and supplemented coalgebras 315
10.2 Universal algebras and coalgebras 323
10.3 Bar constructions and cobar constructions 326
10.4 Twisted tensor products 329
10.5 Universal twisting morphisms 332
10.6 Acyclic twisted tensor products 335
10.7 Modules over augmented algebras 337
10.8 Tensor products and derived functors 340
10.9 Comodules over supplemented coalgebras 345
10.10 Injective classes 349
10.11 Cotensor products and derived functors 356
10.12 Injective resolutions, total complexes, and differential Cotor 363
10.13 Cartan's constructions 369
10.14 Homological invariance of differential Cotor 374
10.15 Alexander-Whitney and Eilenberg-Zilber maps 378
10.16 Eilenberg-Moore models 383
10.17 The Eilenberg-Moore spectral sequence 387
10.18 The Eilenberg-Zilber theorem and the Künneth formula 390
10.19 Coalgebra structures on differential Cotor 393
10.20 Homotopy pullbacks and differential Cotor of several variables 395
10.21 Eilenberg-Moore models of several variables 400
10.22 Algebra structures and loop multiplication 403
10.23 Commutative multiplications and coalgebra structures 407
10.24 Fibrations which are totally nonhomologous to zero 409
10.25 Suspension in the Eilenberg-Moore models 413
10.26 The Bott-Samelson theorem and double loops of spheres 416
10.27 Special unitary groups and their loop spaces 425
10.28 Special orthogonal groups 430
11 Odd primary exponent theorems 437
11.1 Homotopies, NDR pairs, and H-spaces 438
11.2 Spheres, double suspensions, and power maps 444
11.3 The fibre of the pinch map 447
11.4 The homology exponent of the loop space 453
11.5 The Bockstein spectral sequence of the loop space 456
11.6 The decomposition of the homology of the loop space 460
11.7 The weak product decomposition of the loop space 465
11.8 The odd primary exponent theorem for spheres 473
11.9 H-space exponents 478
11.10 Homotopy exponents of odd primary Moore spaces 480
11.11 Nonexistence of H-space exponents 485
12 Differential homological algebra of classifying spaces 489
12.1 Projective classes 490
12.2 Differential graded Hopf algebras 494
12.3 Differential Tor 495
12.4 Classifying spaces 502
12.5 The Serre filtration 504
12.6 Eilenberg-Moore models for Borel constructions 505
12.7 Differential Tor of several variables 508
12.8 Eilenberg-Moore models for several variables 512
12.9 Coproducts in differential Tor 515
12.10 Kunneth theorem 517
12.11 Products in differential Tor 518
12.12 Coproducts and the geometric diagonal 520
12.13 Suspension and transgression 525
12.14 Eilenberg-Moore spectral sequence 528
12.15 Euler class of a vector bundle 530
12.16 Grassmann models for classifying spaces 534
12.17 Homology and cohomology of classifying spaces 537
12.18 Axioms for Stiefel-Whitney and Chern classes 539
12.19 Applications of Stiefel-Whitney classes 542
Bibliography 545
Index 550
