Baas / Jahren / Friedlander Algebraic Topology

The Abel Symposium 2007
1. Auflage 2009
ISBN: 978-3-642-01200-6
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 409 Seiten

The 2007 Abel Symposium took place at the University of Oslo in August 2007.  The goal of the symposium was to bring together mathematicians whose research efforts have led to recent advances in algebraic geometry, algebraic K-theory, algebraic topology, and mathematical physics.  A common theme of this symposium was the development of new perspectives and new constructions with a categorical flavor.  As the lectures at the symposium and the papers of this volume demonstrate, these perspectives and constructions have enabled a broadening of vistas, a synergy between once-differentiated subjects, and solutions to mathematical problems both old and new.

Autoren/Hrsg.

Baas, Nils

Jahren, Björn

Friedlander, Eric

Inhaltsverzeichnis

1;Preface to the Series;5
2;Preface;6
3;Contents;9
4;The Classifying Space of a Topological 2-Group;13
4.1;1 Introduction;13
4.2;2 Overview;15
4.3;3 Topological 2-Groups;20
4.4;4 Nonabelian Cohomology;24
4.5;5 Proofs;28
4.5.1;5.1 Proof of Theorem 1;28
4.5.2;5.2 Remarks on Theorem 1;31
4.5.3;5.3 Proof of Lemma 1;32
4.5.4;5.4 Proof of Lemma 2;35
4.5.5;5.5 Proof of Lemma 3;38
4.5.6;5.6 Proof of Theorem 2;41
4.6;References;42
5;String Topology in Dimensions Two and Three;44
5.1;1 Introduction;44
5.2;2 Algebraic Characterization of Simple Closed Curves on Surfaces;44
5.3;3 Three Manifolds;46
5.3.1;3.1 The Statement About Groups for Closed Surfaces and Surfaces with Boundary;46
5.3.2;3.2 Two Properties of the Kernel;47
5.3.3;3.3 Irreducible Three Manifolds;47
5.4;References;48
6;Floer Homotopy Theory, Realizing Chain Complexes by Module Spectra, and Manifolds with Corners;49
6.1;1 Introduction;49
6.2;2 Floer Homotopy Theory;53
6.2.1;2.1 Preliminaries from Morse Theory;53
6.2.2;2.2 Smooth Floer Theories;57
6.3;3 Realizing Chain Complexes by E-module Spectra;58
6.4;4 Manifolds with Corners, E*-orientations of Flow Categories, and Floer E*-homology;63
6.5;References;69
7;Relative Chern Characters for Nilpotent Ideals;70
7.1;1 Introduction;70
7.1.1;1.1 Notation;71
7.2;2 Cyclic Homology of Cocommutative Hopf Algebras;71
7.2.1;2.1 Bar Resolution and Bar Complex of an Augmented Algebra;72
7.2.2;2.2 The Cyclic Module of a Cocommutative Coalgebra;72
7.2.3;2.3 The Case of Hopf Algebras;73
7.2.4;2.4 Cyclic Complexes of Cocommutative Hopf Algebras;73
7.2.5;2.5 Adic Filtrations and Completion;75
7.3;3 Comparison with the Cyclic Module of the Algebra H;77
7.3.1;3.1 A Natural Section of the Projection HN(M'(H))HH(M'(H));77
7.3.2;3.2 The Lift HH(B(H)) -3muc HN(H);79
7.3.3;3.3 Passage to Completion;80
7.4;4 The Case of Universal Enveloping Algebras of Lie Algebras;81
7.4.1;4.1 Chevalley–Eilenberg Complex;81
7.4.2;4.2 The Loday–Quillen Map;83
7.5;5 Nilpotent Lie Algebras and Nilpotent Groups;84
7.6;6 The Relative Chern Character of a Nilpotent Ideal;86
7.6.1;6.1 The Absolute Chern Character;86
7.6.2;6.2 Volodin Models for the Relative Chern Characterof Nilpotent Ideals;87
7.6.3;6.3 The Relative Chern Character for Rational Nilpotent Ideals;88
7.6.4;6.4 The Rational Homotopy Theory Character for Nilpotent Ideals;89
7.6.5;6.5 Main Theorem;89
7.6.6;6.6 Naturality;90
7.7;References;91
8;Algebraic Differential Characters of FlatConnections with Nilpotent Residues;92
8.1;1 Introduction;92
8.2;2 Filtrations;94
8.3;3 -Splittings;97
8.4;References;103
9;Norm Varieties and the Chain Lemma (After Markus Rost);104
9.1;0.1 Rost Varieties;106
9.2;1 Forms on Vector Bundles;107
9.3;2 The Chain Lemma When n=2;109
9.3.1;2.1 The p-Forms;111
9.3.2;2.2 Norm Principle for n=2;112
9.4;3 The Symbol Chain;113
9.5;4 Model Pn-1 for Moves of Type Cn;116
9.6;5 Model for p Moves;119
9.7;6 Nice G-Actions;122
9.8;7 G-Fixed Point Equivalences;124
9.9;8 A n-Variety;129
9.10;9 The Norm Principle;130
9.11;10 Expressing Norms;133
9.12;A Appendix: The DN Theorem;135
9.13;References;139
10;On the Whitehead Spectrum of the Circle;140
10.1;1 The Groups TRqn(A;p);144
10.2;2 The Fundamental Theorem;147
10.3;3 Topological Cyclic Homology;150
10.4;4 The Skeleton Spectral Sequence;153
10.5;5 The Groups TRqn(S;2);156
10.6;6 The Groups TRqn(Z;2);164
10.7;7 The Groups TRqn(S,I;2);174
10.8;8 The Groups WhqTop(S1) for q 3;184
10.9;References;192
11;Cocycle Categories;194
11.1;1 Introduction;194
11.2;2 Cocycles;197
11.3;3 Torsors;202
11.3.1;3.1 Torsors for Sheaves of Groups;202
11.3.2;3.2 Diagrams and Torsors;206
11.3.3;3.3 Stack Completion;208
11.3.4;3.4 Homotopy Colimits;212
11.4;4 Abelian Sheaf Cohomology;214
11.5;5 Group Extensions and 2-Groupoids;216
11.6;6 Classification of Gerbes;219
11.7;7 The Parabolic Groupoid;220
11.8;References;226
12;A Survey of Elliptic Cohomology;228
12.1;1 Elliptic Cohomology;228
12.1.1;1.1 Cohomology Theories;228
12.1.2;1.2 Formal Groups from Cohomology Theories;230
12.1.3;1.3 Elliptic Cohomology;237
12.2;2 Derived Algebraic Geometry;241
12.2.1;2.1 E-Rings;244
12.2.2;2.2 Derived Schemes;247
12.3;3 Derived Group Schemes and Orientations;250
12.3.1;3.1 Orientations of the Multiplicative Group;254
12.3.2;3.2 Orientations of the Additive Group;257
12.3.3;3.3 The Geometry of Preorientations;259
12.3.4;3.4 Equivariant A-Cohomology for Abelian Groups;260
12.3.5;3.5 The Nonabelian Case;263
12.4;4 Oriented Elliptic Curves;265
12.4.1;4.1 Construction of the Moduli Stack;266
12.4.2;4.2 The Proof of Theorem 4.1: The Local Case;269
12.4.3;4.3 Elliptic Cohomology near ;272
12.5;5 Applications;274
12.5.1;5.1 2-Equivariant Elliptic Cohomology;274
12.5.2;5.2 Loop Group Representations;276
12.5.3;5.3 The String Orientation;277
12.5.4;5.4 Higher Equivariance;281
12.5.5;5.5 Elliptic Cohomology and Geometry;283
12.6;References;285
13;On Voevodsky's Algebraic K-Theory Spectrum;287
13.1;1 Preliminaries;287
13.1.1;1.1 Recollections on Motivic Homotopy Theory;289
13.1.2;1.2 A Construction of BGL;290
13.1.3;1.3 The Periodicity Element;294
13.1.4;1.4 Uniqueness of BGL;296
13.1.5;1.5 Preliminary Computations I;298
13.1.6;1.6 Vanishing of Certain Groups I;301
13.2;2 Smash-Product, Pull-backs, Topological Realization;302
13.2.1;2.1 The Smash Product;303
13.2.2;2.2 A Monoidal Structure on BGL;304
13.2.3;2.3 Preliminary Computations II;306
13.2.4;2.4 Vanishing of Certain Groups II;307
13.2.5;2.5 Vanishing of Certain Groups III;308
13.2.6;2.6 BGL as an Oriented Commutative bold0mu mumu PPequationPPPPbold0mu mumu 11equation1111-Ring Spectrum;308
13.2.7;2.7 BGL*,* as an Oriented Ring Cohomology Theory;309
13.3;A Motivic Homotopy Theory;310
13.3.1;A.1 Categories of Motivic Spaces;311
13.3.2;A.2 Model Categories;312
13.3.3;A.3 Model Structures for Motivic Spaces;314
13.3.4;A.4 Topological Realization;321
13.3.5;A.5 Spectra;323
13.3.6;A.6 Symmetric Spectra;328
13.3.7;A.7 Stable Topological Realization;332
13.4;B Some Results on K-Theory;335
13.4.1;B.1 Cellular Schemes;335
13.5;References;338
14;Chern Character, Loop Spaces and Derived Algebraic Geometry;339
14.1;1 Motivations and Objectives;339
14.1.1;1.1 From Elliptic Cohomology to Categorical Sheaves;340
14.1.2;1.2 Towards a Theory of Categorical Sheaves in Algebraic Geometry;342
14.1.3;1.3 The Chern Character and the Loop Space;344
14.1.4;1.4 Plan of the Paper;345
14.2;2 Categorification of Homological Algebra and dg-Categories;345
14.3;3 Loop Spaces in Derived Algebraic Geometry;351
14.4;4 Construction of the Chern Character;355
14.5;5 Final Comments;357
14.6;References;361
15;Voevodsky's Lectures on Motivic Cohomology 2000/2001;363
15.1;1 Introduction;363
15.2;2 Motivic Cohomology and Motivic Homotopy Category;364
15.2.1;2.1 Last Year;364
15.2.2;2.2 Motivic Homotopy Category;367
15.2.3;2.3 Derived Categories Vs. Homotopy Categories;369
15.2.4;2.4 Application to Presheaves with Transfers;373
15.2.5;2.5 End of the Proof of Theorem 2;374
15.2.6;2.6 Appendix: Localization;376
15.3;3 A1-Equivalences of Simplicial Sheaves on G-Schemes;380
15.3.1;3.1 Sheaves on a Site of G-Schemes;380
15.3.2;3.2 The Brown–Gersten Closed Model Structure on Simplicial Sheaves on G-Schemes;382
15.3.3;3.3 -Closed Classes;386
15.3.4;3.4 The Class of A1-Equivalences Is -Closed;386
15.3.5;3.5 The Class of A1-Equivalences as a -Closure;391
15.3.6;3.6 One More Characterization of Equivalences;398
15.4;4 Solid Sheaves;401
15.4.1;4.1 Open Morphisms and Solid Morphisms of Sheaves;401
15.4.2;4.2 A Criterion for Preservation of Local Equivalences;408
15.5;5 Two Functors;410
15.5.1;5.1 The Functor XX/G;410
15.5.2;5.2 The Functor XXW;412
15.6;References;417