Preface page ................................................. xiii
Acknowledgments .............................................. xix
Introduction to unstable homotopy theory ....................... 1
1 Homotopy groups with coefficients .......................... 11
1.1 В asic 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-primarу 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 al gebras ............................. 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 Kunneth
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 Kiinneth 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
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