This text organizes a range of results in chromatic homotopy theory, running a single thread through theorems in bordism and a detailed understanding of the moduli of formal groups. It emphasizes the naturally occurring algebro-geometric models that presage the topological results, taking the reader through a pedagogical development of the field. In addition to forming the backbone of the stable homotopy category, these ideas have found application in other fields: the daughter subject 'elliptic cohomology' abuts mathematical physics, manifold geometry, topological analysis, and the representation theory of loop groups. The common language employed when discussing these subjects showcases their unity and guides the reader breezily from one domain to the next, ultimately culminating in the construction of Witten's genus for String manifolds. This text is an expansion of a set of lecture notes for a topics course delivered at Harvard University during the spring term of 2016.
Author(s): Eric Peterson
Series: Cambridge Studies in Advanced Mathematics 177
Publisher: Cambridge University Press
Year: 2019
Language: English
Pages: 421
Contents......Page 8
Preface......Page 9
Foreword......Page 13
Introduction......Page 16
Conventions......Page 22
1 Unoriented Bordism......Page 24
1.1 Thom Spectra and the Thom Isomorphism......Page 25
1.2 Cohomology Rings and Affine Schemes......Page 31
1.3 The Steenrod Algebra......Page 37
1.4 Hopf Algebra Cohomology......Page 46
1.5 The Unoriented Bordism Ring......Page 54
2 Complex Bordism......Page 64
2.1 Calculus on Formal Varieties......Page 66
2.2 Divisors on Formal Curves......Page 75
2.3 Line Bundles Associated to Thom Spectra......Page 80
2.4 Power Operations for Complex Bordism......Page 88
2.5 Explicitly Stabilizing Cyclic MU–Power Operations......Page 98
2.6 The Complex Bordism Ring......Page 106
3 Finite Spectra......Page 112
3.1 Descent and the Context of a Spectrum......Page 114
3.2 The Structure of Mfg I: The Affine Cover......Page 126
3.3 The Structure of Mfg II: Large Scales......Page 136
3.4 The Structure of Mfg III: Small Scales......Page 147
3.5 Nilpotence and Periodicity in Finite Spectra......Page 155
3.6 Chromatic Fracture and Convergence......Page 169
4 Unstable Cooperations......Page 180
4.1 Unstable Contexts and the Steenrod Algebra......Page 182
4.2 Algebraic Mixed Unstable Cooperations......Page 193
4.3 Unstable Cooperations for Complex Bordism......Page 203
4.4 Dieudonné Modules......Page 209
4.5 Ordinary Cooperations for Landweber Flat Theories......Page 218
4.6 Cooperations among Geometric Points on Mfg......Page 226
5 The σ-Orientation......Page 234
5.1 Coalgebraic Formal Schemes......Page 235
5.2 Special Divisors and the Special Splitting Principle......Page 241
5.3 Chromatic Analysis of BU[6, ∞)......Page 253
5.4 Analysis of BU[6, ∞) at Infinite Height......Page 262
5.5 Modular Forms and MU[6, ∞)-Manifolds......Page 271
5.6 Chromatic Spin and String Orientations......Page 285
Appendix A Power Operations......Page 298
A.1 Rational Chromatic Phenomena (Nathaniel Stapleton)......Page 300
A.2 Orientations and Power Operations......Page 322
A.3 The Spectrum of Modular Forms......Page 337
A.4 Orientations by E∞ Maps......Page 348
B.1 Historical Retrospective (Michael Hopkins)......Page 364
B.2 The Road Ahead......Page 367
References......Page 394
Index......Page 416
