Algebraic Methods in Unstable Homotopy Theory (New Mathematical Monographs)

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
What is in this book and what is not......Page 15
Prerequisites......Page 17
Ways to use this book......Page 18
Acknowledgments......Page 21
Introduction to unstable homotopy theory......Page 23
1 Homotopy groups with coefficients......Page 33
1.1 Basic definitions......Page 34
1.2 Long exact sequences of pairs and fibrations......Page 37
1.3 Universal coefficient exact sequences......Page 38
1.4 Functor properties......Page 40
1.5 The Bockstein long exact sequence......Page 42
1.6 Nonfinitely generated coefficient groups......Page 45
1.7 The mod k Hurewicz homomorphism......Page 47
1.8 The mod k Hurewicz isomorphism theorem......Page 49
1.9 The mod k Hurewicz isomorphism theorem for pairs......Page 54
1.10 The third homotopy group with odd coefficients is abelian......Page 56
2 A general theory of localization......Page 57
2.1 Dror Farjoun-Bousfield localization......Page 59
2.2 Localization of abelian groups......Page 68
2.3 Classical localization of spaces: inverting primes......Page 69
2.4 Limits and derived functors......Page 75
2.5 Hom and Ext......Page 77
2.6 p-completion of abelian groups......Page 80
2.7 p-completion of simply connected spaces......Page 85
2.8 Completion implies the mod k Hurewicz isomorphism......Page 90
2.9 Fracture lemmas......Page 92
2.10 Killing Eilenberg-MacLane spaces: Miller\'s theorem......Page 96
2.11 Zabrodsky mixing: the Hilton-Roitberg examples......Page 105
2.12 Loop structures on p-completions of spheres......Page 110
2.13 Serre\'s C-theory and finite generation......Page 113
3 Fibre extensions of squares and the Peterson-Stein formula......Page 116
3.1 Homotopy theoretic fibres......Page 117
3.2 Fibre extensions of squares......Page 118
3.3 The Peterson-Stein formula......Page 121
3.4 Totally fibred cubes......Page 123
3.5 Spaces of the homotopy type of a CW complex......Page 126
4 Hilton-Hopf invariants and the EHP sequence......Page 129
4.1 The Bott-Samelson theorem......Page 130
4.2 The James construction......Page 133
4.3 The Hilton-Milnor theorem......Page 135
4.4 The James fibrations and the EHP sequence......Page 140
4.5 James\'s 2-primary exponent theorem......Page 143
4.6 The 3-connected cover of S3 and its loop space......Page 146
4.7 The first odd primary homotopy class......Page 148
4.8 Elements of order 4......Page 150
4.9 Computations with the EHP sequence......Page 154
5 James-Hopf invariants and Toda-Hopf invariants......Page 157
5.1 Divided power algebras......Page 158
5.2 James-Hopf invariants......Page 163
5.3 p-th Hilton-Hopf invariants......Page 167
5.4 Loops on filtrations of the James construction......Page 170
5.5 Toda-Hopf invariants......Page 173
5.6 Toda\'s odd primary exponent theorem......Page 177
6 Samelson products......Page 180
6.1 The fibre of the pinch map and self maps of Moore spaces......Page 182
6.2 Existence of the smash decomposition......Page 188
6.3 Samelson and Whitehead products......Page 189
6.4 Uniqueness of the smash decomposition......Page 193
6.5 Lie identities in groups......Page 199
6.6 External Samelson products......Page 201
6.7 Internal Samelson products......Page 208
6.8 Group models for loop spaces......Page 212
6.9 Relative Samelson products......Page 220
6.10 Universal models for relative Samelson products......Page 224
6.11 Samelson products over the loops on an H-space......Page 232
Twisted anti-commutativity......Page 234
Relative Samelson products......Page 235
Bocksteins and Φ-based Samelson products......Page 236
Three-variable models......Page 237
The Jacobi identity......Page 238
7 Bockstein spectral sequences......Page 243
7.1 Exact couples......Page 244
7.2 Mod p homotopy Bockstein spectral sequences......Page 247
Meaning of a nonzero Bockstein differential......Page 251
7.4 Convergence......Page 252
7.5 Samelson products in the Bockstein spectral sequence......Page 254
7.6 Mod p homology Bockstein spectral sequences......Page 257
7.7 Mod p cohomology Bockstein spectral sequences......Page 260
7.8 Torsion in H-spaces......Page 263
8 Lie algebras and universal enveloping algebras......Page 273
8.1 Universal enveloping algebras of graded Lie algebras......Page 274
8.2 The graded Poincare-Birkhoff-Witt theorem......Page 279
8.3 Consequences of the graded Poincare-Birkhoff-Witt theorem......Page 286
8.4 Nakayama\'s lemma......Page 289
8.5 Free graded Lie algebras......Page 292
8.6 The change of rings isomorphism......Page 296
8.7 Subalgebras of free graded Lie algebras......Page 300
9 Applications of graded Lie algebras......Page 305
9.1 Serre\'s product decomposition......Page 306
9.2 Loops of odd primary even dimensional Moore spaces......Page 308
Geometric realization of UK......Page 309
Geometric realization via multiplying maps......Page 311
Filtering by powers of the augmentation ideal......Page 312
What commutators do in homology......Page 313
A filtration compatible with the powers of the augmentation ideal......Page 314
Conclusion of the proof of the Hilton–Milnor theorem......Page 315
9.4 Elements of mod p Hopf invariant one......Page 316
9.5 Cycles in differential graded Lie algebras......Page 321
9.6 Higher order torsion in odd primary Moore spaces......Page 325
Algebras generated by acyclic modules......Page 328
Useful operations on even dimensional classes......Page 329
10 Differential homological algebra......Page 335
10.1 Augmented algebras and supplemented coalgebras......Page 337
10.2 Universal algebras and coalgebras......Page 345
10.3 Bar constructions and cobar constructions......Page 348
Alternate notation......Page 349
Alternate notation......Page 350
10.4 Twisted tensor products......Page 351
10.5 Universal twisting morphisms......Page 354
10.6 Acyclic twisted tensor products......Page 357
10.7 Modules over augmented algebras......Page 359
10.8 Tensor products and derived functors......Page 362
10.9 Comodules over supplemented coalgebras......Page 367
10.10 Injective classes......Page 371
Reflection via adjoint functors 10.10.6......Page 373
The proper injective class for nondifferential comodules 10.10.7......Page 374
The proper injective class for differential comodules 10.10.8......Page 375
10.11 Cotensor products and derived functors......Page 378
10.12 Injective resolutions, total complexes, and differential Cotor......Page 385
Differential comodule structures in the total complex......Page 386
Chain maps and chain homotopies in the total complex......Page 387
The total complex of a relative injective complex......Page 388
10.13 Cartan\'s constructions......Page 391
10.14 Homological invariance of differential Cotor......Page 396
10.15 Alexander-Whitney and Eilenberg-Zilber maps......Page 400
10.16 Eilenberg-Moore models......Page 405
10.17 The Eilenberg-Moore spectral sequence......Page 409
10.18 The Eilenberg-Zilber theorem and the Kunneth formula......Page 412
10.19 Coalgebra structures on differential Cotor......Page 415
10.20 Homotopy pullbacks and differential Cotor of several variables......Page 417
10.21 Eilenberg-Moore models of several variables......Page 422
10.22 Algebra structures and loop multiplication......Page 425
10.23 Commutative multiplications and coalgebra structures......Page 429
10.24 Fibrations which are totally nonhomologous to zero......Page 431
10.25 Suspension in the Eilenberg-Moore models......Page 435
10.26 The Bott-Samelson theorem and double loops of spheres......Page 438
10.27 Special unitary groups and their loop spaces......Page 447
10.28 Special orthogonal groups......Page 452
11 Odd primary exponent theorems......Page 459
11.1 Homotopies, NDR pairs, and H-spaces......Page 460
11.2 Spheres, double suspensions, and power maps......Page 466
11.3 The fibre of the pinch map......Page 469
11.4 The homology exponent of the loop space......Page 475
11.5 The Bockstein spectral sequence of the loop space......Page 478
11.6 The decomposition of the homology of the loop space......Page 482
11.7 The weak product decomposition of the loop space......Page 487
11.8 The odd primary exponent theorem for spheres......Page 495
11.9 H-space exponents......Page 500
11.10 Homotopy exponents of odd primary Moore spaces......Page 502
11.11 Nonexistence of H-space exponents......Page 507
12 Differential homological algebra of classifying spaces......Page 511
12.1 Projective classes......Page 512
12.2 Differential graded Hopf algebras......Page 516
12.3 Differential Tor......Page 517
12.4 Classifying spaces......Page 524
12.5 The Serre filtration......Page 526
12.6 Eilenberg-Moore models for Borel constructions......Page 527
12.7 Differential Tor of several variables......Page 530
12.8 Eilenberg-Moore models for several variables......Page 534
12.9 Coproducts in differential Tor......Page 537
12.10 Kunneth theorem......Page 539
12.11 Products in differential Tor......Page 540
12.12 Coproducts and the geometric diagonal......Page 542
12.13 Suspension and transgression......Page 547
12.14 Eilenberg-Moore spectral sequence......Page 550
12.15 Euler class of a vector bundle......Page 552
12.16 Grassmann models for classifying spaces......Page 556
12.17 Homology and cohomology of classifying spaces......Page 559
12.18 Axioms for Stiefel-Whitney and Chern classes......Page 561
Axioms for Stiefel–Whitney classes 12.18.2.......Page 562
Whitney product formula 12.18.2.......Page 563
12.19 Applications of Stiefel-Whitney classes......Page 564
Bibliography......Page 567
Index......Page 572