Complex Cobordism and Stable Homotopy Groups of Spheres (eBook)

Front Cover 1
COMPLEX COBORDISM AND STABLE HOMOTOPY GROUPS OF SPHERES 4
Copyright Page 5
Contents 8
Preface 16
Notations and Commonly Used Abbreviations 20
Chapter 1. An Introduction to the Homotopy Groups of Spheres 22
1. Classical Theorems Old and New 24
2. Methods of Computing p(Sn) 28
3. The Adams–Novikov E2-Term, Formal Group Laws, and the Greek Letter Construction 37
4. More Formal Group Law Theory, BP-Theory, Morava's Point of View, and the Chromatic Spectral Sequence 44
5. Unstable Homotopy Groups and the EHP Spectral Sequence 50
Chapter 2. Setting Up the Adams Spectral Sequence 68
1. The Classical Adams Spectral Sequence 69
2. The Adams Spectral Sequence Based on a Generalized Homology Theory 78
3. The Smash Product Pairing and the Generalized Connecting Homomorphism 83
Chapter 3. The Classical Adams Spectral Sequence 90
1. The Steenrod Algebra and Some Easy Calculations 90
2. The May Spectral Sequence 100
3. The Lambda Algebra 111
4. Some General Properties of Ext 121
5. Survey and Further Reading 131
Chapter 4. BP-Theory and the Adams–Novikov Spectral Sequence 140
1. Quillen’s Theorem and the Structure of BP*(BP) 140
2. A Survey of BP-Theory 150
3. Some Calculations in BP*(BP) 158
4. Beginning Calculations with the Adams–Novikov Spectral Sequence 171
Chapter 5. The Chromatic Spectral Sequence 191
1. The Algebraic Construction 193
2. Ext1(BP*/In) and Hopf Invariant One 205
3. Ext(M1) and the J-Homomorphism 213
4. Ext2 and the Thom Reduction 222
5. Periodic Families in Ext2 229
6. Elements in Ext3 and Beyond 236
Chapter 6. Morava Stabilizer Algebras 240
1. The Change-of-Rings Isomorphism 241
2. The Structure of S(n) 247
3. The Cohomology of S(n) 252
4. The Odd Primary Kervaire Invariant Elements 265
5. The Spectra T(n) 274
Chapter 7. Computing Stable Homotopy Groups with the Adams – Novikov Spectral Sequence 278
1. The Method of Infinite Descent 279
2. The Complex C1,1 283
3. The Homotopy Groups of a Complex with p Cells 289
4. The ‘‘Algorithm’’ and Computations at p = 3 302
5. Computations for p = 5 310
Appendix 1. Hopf Algebras and Hopf Algebroids 327
1. Basic Definitions 330
2. Homological Algebra 340
3. Some Spectral Sequences 346
4. Massey Products 356
5. Algebraic Steenrod Operations 367
Appendix 2. Formal Group Laws 375
1. Universal Formal Group Laws and Strict Isomorphisms 376
2. Classification and Endomorphism Rings 390
Appendix 3. Tables of Homotopy Groups of Spheres 401
References 416
Index 428