Algebraic Topology (eBook)

Preface to the Series 5
Preface 6
Contents 9
The Classifying Space of a Topological 2-Group 13
1 Introduction 13
2 Overview 15
3 Topological 2-Groups 20
4 Nonabelian Cohomology 24
5 Proofs 28
5.1 Proof of Theorem 1 28
5.2 Remarks on Theorem 1 31
5.3 Proof of Lemma 1 32
5.4 Proof of Lemma 2 35
5.5 Proof of Lemma 3 38
5.6 Proof of Theorem 2 41
References 42
String Topology in Dimensions Two and Three 44
1 Introduction 44
2 Algebraic Characterization of Simple Closed Curves on Surfaces 44
3 Three Manifolds 46
3.1 The Statement About Groups for Closed Surfaces and Surfaces with Boundary 46
3.2 Two Properties of the Kernel 47
3.3 Irreducible Three Manifolds 47
References 48
Floer Homotopy Theory, Realizing Chain Complexes by Module Spectra, and Manifolds with Corners 49
1 Introduction 49
2 Floer Homotopy Theory 53
2.1 Preliminaries from Morse Theory 53
2.2 Smooth Floer Theories 57
3 Realizing Chain Complexes by E-module Spectra 58
4 Manifolds with Corners, E*-orientations of Flow Categories, and Floer E*-homology 63
References 69
Relative Chern Characters for Nilpotent Ideals 70
1 Introduction 70
1.1 Notation 71
2 Cyclic Homology of Cocommutative Hopf Algebras 71
2.1 Bar Resolution and Bar Complex of an Augmented Algebra 72
2.2 The Cyclic Module of a Cocommutative Coalgebra 72
2.3 The Case of Hopf Algebras 73
2.4 Cyclic Complexes of Cocommutative Hopf Algebras 73
2.5 Adic Filtrations and Completion 75
3 Comparison with the Cyclic Module of the Algebra H 77
3.1 A Natural Section of the Projection HN(M'(H))HH(M'(H)) 77
3.2 The Lift HH(B(H)) -3muc HN(H) 79
3.3 Passage to Completion 80
4 The Case of Universal Enveloping Algebras of Lie Algebras 81
4.1 Chevalley–Eilenberg Complex 81
4.2 The Loday–Quillen Map 83
5 Nilpotent Lie Algebras and Nilpotent Groups 84
6 The Relative Chern Character of a Nilpotent Ideal 86
6.1 The Absolute Chern Character 86
6.2 Volodin Models for the Relative Chern Characterof Nilpotent Ideals 87
6.3 The Relative Chern Character for Rational Nilpotent Ideals 88
6.4 The Rational Homotopy Theory Character for Nilpotent Ideals 89
6.5 Main Theorem 89
6.6 Naturality 90
References 91
Algebraic Differential Characters of FlatConnections with Nilpotent Residues 92
1 Introduction 92
2 Filtrations 94
3 -Splittings 97
References 103
Norm Varieties and the Chain Lemma (After Markus Rost) 104
0.1 Rost Varieties 106
1 Forms on Vector Bundles 107
2 The Chain Lemma When n=2 109
2.1 The p-Forms 111
2.2 Norm Principle for n=2 112
3 The Symbol Chain 113
4 Model Pn-1 for Moves of Type Cn 116
5 Model for p Moves 119
6 Nice G-Actions 122
7 G-Fixed Point Equivalences 124
8 A n-Variety 129
9 The Norm Principle 130
10 Expressing Norms 133
A Appendix: The DN Theorem 135
References 139
On the Whitehead Spectrum of the Circle 140
1 The Groups TRqn(A p)
2 The Fundamental Theorem 147
3 Topological Cyclic Homology 150
4 The Skeleton Spectral Sequence 153
5 The Groups TRqn(S 2)
6 The Groups TRqn(Z 2)
7 The Groups TRqn(S,I 2)
8 The Groups WhqTop(S1) for q 3 184
References 192
Cocycle Categories 194
1 Introduction 194
2 Cocycles 197
3 Torsors 202
3.1 Torsors for Sheaves of Groups 202
3.2 Diagrams and Torsors 206
3.3 Stack Completion 208
3.4 Homotopy Colimits 212
4 Abelian Sheaf Cohomology 214
5 Group Extensions and 2-Groupoids 216
6 Classification of Gerbes 219
7 The Parabolic Groupoid 220
References 226
A Survey of Elliptic Cohomology 228
1 Elliptic Cohomology 228
1.1 Cohomology Theories 228
1.2 Formal Groups from Cohomology Theories 230
1.3 Elliptic Cohomology 237
2 Derived Algebraic Geometry 241
2.1 E-Rings 244
2.2 Derived Schemes 247
3 Derived Group Schemes and Orientations 250
3.1 Orientations of the Multiplicative Group 254
3.2 Orientations of the Additive Group 257
3.3 The Geometry of Preorientations 259
3.4 Equivariant A-Cohomology for Abelian Groups 260
3.5 The Nonabelian Case 263
4 Oriented Elliptic Curves 265
4.1 Construction of the Moduli Stack 266
4.2 The Proof of Theorem 4.1: The Local Case 269
4.3 Elliptic Cohomology near 272
5 Applications 274
5.1 2-Equivariant Elliptic Cohomology 274
5.2 Loop Group Representations 276
5.3 The String Orientation 277
5.4 Higher Equivariance 281
5.5 Elliptic Cohomology and Geometry 283
References 285
On Voevodsky's Algebraic K-Theory Spectrum 287
1 Preliminaries 287
1.1 Recollections on Motivic Homotopy Theory 289
1.2 A Construction of BGL 290
1.3 The Periodicity Element 294
1.4 Uniqueness of BGL 296
1.5 Preliminary Computations I 298
1.6 Vanishing of Certain Groups I 301
2 Smash-Product, Pull-backs, Topological Realization 302
2.1 The Smash Product 303
2.2 A Monoidal Structure on BGL 304
2.3 Preliminary Computations II 306
2.4 Vanishing of Certain Groups II 307
2.5 Vanishing of Certain Groups III 308
2.6 BGL as an Oriented Commutative bold0mu mumu PPequationPPPPbold0mu mumu 11equation1111-Ring Spectrum 308
2.7 BGL*,* as an Oriented Ring Cohomology Theory 309
A Motivic Homotopy Theory 310
A.1 Categories of Motivic Spaces 311
A.2 Model Categories 312
A.3 Model Structures for Motivic Spaces 314
A.4 Topological Realization 321
A.5 Spectra 323
A.6 Symmetric Spectra 328
A.7 Stable Topological Realization 332
B Some Results on K-Theory 335
B.1 Cellular Schemes 335
References 338
Chern Character, Loop Spaces and Derived Algebraic Geometry 339
1 Motivations and Objectives 339
1.1 From Elliptic Cohomology to Categorical Sheaves 340
1.2 Towards a Theory of Categorical Sheaves in Algebraic Geometry 342
1.3 The Chern Character and the Loop Space 344
1.4 Plan of the Paper 345
2 Categorification of Homological Algebra and dg-Categories 345
3 Loop Spaces in Derived Algebraic Geometry 351
4 Construction of the Chern Character 355
5 Final Comments 357
References 361
Voevodsky's Lectures on Motivic Cohomology 2000/2001 363
1 Introduction 363
2 Motivic Cohomology and Motivic Homotopy Category 364
2.1 Last Year 364
2.2 Motivic Homotopy Category 367
2.3 Derived Categories Vs. Homotopy Categories 369
2.4 Application to Presheaves with Transfers 373
2.5 End of the Proof of Theorem 2 374
2.6 Appendix: Localization 376
3 A1-Equivalences of Simplicial Sheaves on G-Schemes 380
3.1 Sheaves on a Site of G-Schemes 380
3.2 The Brown–Gersten Closed Model Structure on Simplicial Sheaves on G-Schemes 382
3.3 -Closed Classes 386
3.4 The Class of A1-Equivalences Is -Closed 386
3.5 The Class of A1-Equivalences as a -Closure 391
3.6 One More Characterization of Equivalences 398
4 Solid Sheaves 401
4.1 Open Morphisms and Solid Morphisms of Sheaves 401
4.2 A Criterion for Preservation of Local Equivalences 408
5 Two Functors 410
5.1 The Functor XX/G 410
5.2 The Functor XXW 412
References 417