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Algebraic Methods in Unstable Homotopy Theory

By: Joseph Neisendorfer

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Hardcover | 18 February 2010

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576 Pages

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The most modern and thorough treatment of unstable homotopy theory available. The focus is on those methods from algebraic topology which are needed in the presentation of results, proven by Cohen, Moore, and the author, on the exponents of homotopy groups. The author introduces various aspects of unstable homotopy theory, including: homotopy groups with coefficients; localization and completion; the Hopf invariants of Hilton, James, and Toda; Samelson products; homotopy Bockstein spectral sequences; graded Lie algebras; differential homological algebra; and the exponent theorems concerning the homotopy groups of spheres and Moore spaces. This book is suitable for a course in unstable homotopy theory, following a first course in homotopy theory. It is also a valuable reference for both experts and graduate students wishing to enter the field.

Industry Reviews

'It is very well written and organized, as would be expected of the author, who is renowned for an excellent expository style ... the book could, and should, be used as a text for students aiming to do research in unstable homotopy theory. By the end of a thorough reading, a student would be well grounded in a suite of contemporary methods and would be ready to tackle research problems. the book's comprehensive nature also means that it is a wonderful reference for expects in the area.' Bulletin of the London Mathematical Society
'... provides a new generation of topologists with a readable and workmanlike collection of important techniques and results ...' Mathematical Reviews

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Non-Fiction Mathematics Topology Algebraic Topology

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Non-Fiction Mathematics Topology Algebraic Topology

Preface	p. xiii
Acknowledgments	p. xix
Introduction to unstable homotopy theory	p. 1
Homotopy groups with coefficients	p. 11
Basic definitions	p. 12
Long exact sequences of pairs and fibrations	p. 15
Universal coefficient exact sequences	p. 16
Functor properties	p. 18
The Bockstein long exact sequence	p. 20
Nonfinitely generated coefficient groups	p. 23
The mod k Hurewicz homomorphism	p. 25
The mod k Hurewicz isomorphism theorem	p. 27
The mod k Hurewicz isomorphism theorem for pairs	p. 32
The third homotopy group with odd coefficients is abelian	p. 34
A general theory of localization	p. 35
Dror Farjoun-Bousfield localization	p. 37
Localization of abelian groups	p. 46
Classical localization of spaces: inverting primes	p. 47
Limits and derived functors	p. 53
Hom and Ext	p. 55
p-completion of abelian groups	p. 58
p-completion of simply connected spaces	p. 63
Completion implies the mod k Hurewicz isomorphism	p. 68
Fracture lemmas	p. 70
Killing Eilenberg-MacLane spaces: Miller's theorem	p. 74
Zabrodsky mixing: the Hilton-Roitberg examples	p. 83
Loop structures on p-completions of spheres	p. 88
Serre's C-theory and finite generation	p. 91
Fibre extensions of squares and the Peterson-Stein formula	p. 94
Homotopy theoretic fibres	p. 95
Fibre extensions of squares	p. 96
The Peterson-Stein formula	p. 99
Totally fibred cubes	p. 101
Spaces of the homotopy type of a CW complex	p. 104
Hilton-Hopf invariants and the EHP sequence	p. 107
The Bott-Samelson theorem	p. 108
The James construction	p. 111
The Hilton-Milnor theorem	p. 113
The James fibrations and the EHP sequence	p. 118
James's 2-primary exponent theorem	p. 121
The 3-connected cover of S³ and its loop space	p. 124
The first odd primary homotopy class	p. 126
Elements of order 4	p. 128
Computations with the EHP sequence	p. 132
James-Hopf invariants and Toda-Hopf invariants	p. 135
Divided power algebras	p. 136
James-Hopf invariants	p. 141
p-th Hilton-Hopf invariants	p. 145
Loops on filtrations of the James construction	p. 148
Toda-Hopf invariants	p. 151
Toda's odd primary exponent theorem	p. 155
Samelson products	p. 158
The fibre of the pinch map and self maps of Moore spaces	p. 160
Existence of the smash decomposition	p. 166
Samelson and Whitehead products	p. 167
Uniqueness of the smash decomposition	p. 171
Lie identities in groups	p. 177
External Samelson products	p. 179
Internal Samelson products	p. 186
Group models for loop spaces	p. 190
Relative Samelson products	p. 198
Universal models for relative Samelson products	p. 202
Samelson products over the loops on an H-space	p. 210
Bockstein spectral sequences	p. 221
Exact couples	p. 222
Mod p homotopy Bockstein spectral sequences	p. 225
Reduction maps and extensions	p. 229
Convergence	p. 230
Samelson products in the Bockstein spectral sequence	p. 232
Mod p homology Bockstein spectral sequences	p. 235
Mod p cohomology Bockstein spectral sequences	p. 238
Torsion in H-spaces	p. 241
Lie algebras and universal enveloping algebras	p. 251
Universal enveloping algebras of graded Lie algebras	p. 252
The graded Poincare-Birkhoff-Witt theorem	p. 257
Consequences of the graded Poincare-Birkhoff-Witt theorem	p. 264
Nakayama's lemma	p. 267
Free graded Lie algebras	p. 270
The change of rings isomorphism	p. 274
Subalgebras of free graded Lie algebras	p. 278
Applications of graded Lie algebras	p. 283
Serre's product decomposition	p. 284
Loops of odd primary even dimensional Moore spaces	p. 286
The Hilton-Milnor theorem	p. 290
Elements of mod p Hopf invariant one	p. 294
Cycles in differential graded Lie algebras	p. 299
Higher order torsion in odd primary Moore spaces	p. 303
The homology of acyclic free differential graded Lie algebras	p. 306
Differential homological algebra	p. 313
Augmented algebras and supplemented coalgebras	p. 315
Universal algebras and coalgebras	p. 323
Bar constructions and cobar constructions	p. 326
Twisted tensor products	p. 329
Universal twisting morphisms	p. 332
Acyclic twisted tensor products	p. 335
Modules over augmented algebras	p. 337
Tensor products and derived functors	p. 340
Comodules over supplemented coalgebras	p. 345
Injective classes	p. 349
Cotensor products and derived functors	p. 356
Injective resolutions, total complexes, and differential Cotor	p. 363
Cartan's constructions	p. 369
Homological invariance of differential Cotor	p. 374
Alexander-Whitney and Eilenberg-Zilber maps	p. 378
Eilenberg-Moore models	p. 383
The Eilenberg-Moore spectral sequence	p. 387
The Eilenberg-Zilber theorem and the Künneth formula	p. 390
Coalgebra structures on differential Cotor	p. 393
Homotopy pullbacks and differential Cotor of several variables	p. 395
Eilenberg-Moore models of several variables	p. 400
Algebra structures and loop multiplication	p. 403
Commutative multiplications and coalgebra structures	p. 407
Fibrations which are totally nonhomologous to zero	p. 409
Suspension in the Eilenberg-Moore models	p. 413
The Bott-Samelson theorem and double loops of spheres	p. 416
Special unitary groups and their loop spaces	p. 425
Special orthogonal groups	p. 430
Odd primary exponent theorems	p. 437
Homotopies, NDR pairs, and H-spaces	p. 438
Spheres, double suspensions, and power maps	p. 444
The fibre of the pinch map	p. 447
The homology exponent of the loop space	p. 453
The Bockstein spectral sequence of the loop space	p. 456
The decomposition of the homology of the loop space	p. 460
The weak product decomposition of the loop space	p. 465
The odd primary exponent theorem for spheres	p. 473
H-space exponents	p. 478
Homotopy exponents of odd primary Moore spaces	p. 480
Nonexistence of H-space exponents	p. 485
Differential homological algebra of classifying spaces	p. 489
Projective classes	p. 490
Differential graded Hopf algebras	p. 494
Differential Tor	p. 495
Classifying spaces	p. 502
The Serre filtration	p. 504
Eilenberg-Moore models for Borel constructions	p. 505
Differential Tor of several variables	p. 508
Eilenberg-Moore models for several variables	p. 512
Coproducts in differential Tor	p. 515
Kunneth theorem	p. 517
Products in differential Tor	p. 518
Coproducts and the geometric diagonal	p. 520
Suspension and transgression	p. 525
Eilenberg-Moore spectral sequence	p. 528
Euler class of a vector bundle	p. 530
Grassmann models for classifying spaces	p. 534
Homology and cohomology of classifying spaces	p. 537
Axioms for Stiefel-Whitney and Chern classes	p. 539
Applications of Stiefel-Whitney classes	p. 542
Bibliography	p. 545
Index	p. 550
Table of Contents provided by Ingram. All Rights Reserved.

Algebraic Methods in Unstable Homotopy Theory

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