Verfügbarkeit: Basic bundle theory and K-cohomology invariants

Basic bundle theory and K-cohomology invariants:

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Bibliographische Detailangaben
Beteiligte Personen:	Husemöller, Dale (VerfasserIn), Joachim, Michael (VerfasserIn), Jurco, Branislav (VerfasserIn), Schottenloher, Martin 1944- (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Berlin ; Heidelberg ; New York Springer 2008
Schriftenreihe:	Lecture notes in physics 726
Schlagwörter:	Faisceaux, Théorie des K-théorie Fiber bundles (Mathematics) K-theory Faserbündel K-Theorie
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016295577&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Umfang:	xv, 340 Seiten Illustrationen
ISBN:	9783540749554 9783540749561

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Datensatz im Suchindex

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adam_text	Contents Physical Background to the K-Theory Classification of D-Branes: Introduction and References ......................................... 1 Part I Bundles over a Space and Modules over an Algebra 1 Generalities on Bundles and Categories ........................... 9 1 Bundles Over a Space ....................................... 9 2 Examples of Bundles ....................................... 11 3 Two Operations on Bundles .................................. 13 4 Category Constructions Related to Bundles ..................... 14 5 Functors Between Categories ................................. 16 6 Morphisms of Functors or Natural Transformations .............. 18 7 Étale Maps and Coverings ................................... 20 References ..................................................... 22 2 Vector Bundles ................................................. 23 1 Bundles of Vector Spaces and Vector Bundles ................... 23 2 Isomorphisms of Vector Bundles and Induced Vector Bundles ..... 25 3 Image and Kernel of Vector Bundle Morphisms ................. 26 4 The Canonical Bundle Over the Grassmannian Varieties .......... 28 5 Finitely Generated Vector Bundles ............................ 29 6 Vector Bundles on a Compact Space ........................... 31 7 Collapsing and Clutching Vector Bundles on Subspaces .......... 31 8 Metrics on Vector Bundles ................................... 33 Reference ...................................................... 34 3 Relation Between Vector Bundles, Projective Modules, and Idempotents ................................................... 35 1 Local Coordinates of a Vector Bundle Given by Global Functions over a Normal Space ........................................ 36 2 The Full Embedding Property of the Cross Section Functor ....... 37 3 Finitely Generated Projective Modules ......................... 38 4 The Serre-Swan Theorem ................................... 40 5 Idempotent Classes Associated to Finitely Generated Projective Modules .................................................. 42 4 K-Theory of Vector Bundles, of Modules, and of Idempotents ....... 45 1 Generalities on Adding Negatives ............................. 45 2 if-Groups of Vector Bundles ................................. 47 3 /f-Groups of Finitely Generated Projective Modules .............. 48 4 if-Groups of Idempotents .................................... 50 5 ÃT-Theory of Topological Algebras ............................ 51 References ..................................................... 54 5 Principal Bundles and Sections of Fibre Bundles: Reduction of the Structure and the Gauge Group I ................................ 55 1 Bundles Defined by Transformation Groups .................... 55 2 Definition and Examples of Principal Bundles ................... 57 3 Fibre Bundles .............................................. 58 4 Local Coordinates for Fibre Bundles .......................... 58 5 Extension and Restriction of Structure Group ................... 60 6 Automorphisms of Principal Bundles and Gauge Groups .......... 62 Reference ...................................................... 62 Part II Homotopy Classification of Bundles and Cohomology: Classifying Spaces 6 Homotopy Classes of Maps and the Homotopy Groups ............. 65 1 The Space Мар(Х,У) ....................................... 65 2 Continuity of Substitution and Map(X χ Γ, ľ) ................... 66 3 Free and Based Homotopy Classes of Maps .................... 67 4 Homotopy Categories ....................................... 68 5 Homotopy Groups of a Pointed Space ......................... 69 6 Bundles on a Cylinder fix [0,1]............................... 72 7 The Milnor Construction: Homotopy Classification of Principal Bundles ....................................................... 75 1 Basic Data from a Numerable Principal Bundle ................. 75 2 Total Space of the Milnor Construction ........................ 76 3 Uniqueness up to Homotopy of the Classifying Map ............. 78 4 The Infinite Sphere as the Total Space of the Milnor Construction . . 80 References ..................................................... 81 8 Fibrations and Bundles: Gauge Group II ......................... 83 1 Factorization, Lifting, and Extension in Square Diagrams ......... 84 2 Fibrations and Cofibrations .................................. 85 3 Fibres and Cofibres: Loop Space and Suspension ................ 88 4 Relation Between Loop Space and Suspension Group Structures on Homotopy Classes of Maps [X,Y]* ......................... 90 Contents 5 Outline of the Fibre Mapping Sequence and Cofibre Mapping Sequence ................................................. 91 6 From Base to Fibre and From Fibre to Base ..................... 93 7 Homotopy Characterization of the Universal Bundle ............. 95 8 Application to the Classifying Space of the Gauge Group ......... 95 9 The Infinite Sphere as the Total Space of a Universal Bundle ...... 96 Reference ...................................................... 96 9 Cohomology Classes as Homotopy Classes: CW-Complexes ......... 97 1 Filtered Spaces and Cell Complexes ........................... 98 2 Whitehead s Characterization of Homotopy Equivalences ......... 99 3 Axiomatic Properties of Cohomology and Homology ............100 4 Construction and Calculation of Homology and Cohomology ...........................................103 5 Hurewicz Theorem .........................................105 6 Representability of Cohomology by Homotopy Classes ...........105 7 Products of Cohomology and Homology .......................106 8 Introduction to Morse Theory ................................107 References ......................................................109 10 Basic Characteristic Classes ..................................... Ill 1 Characteristic Classes of Line Bundles ......................... Ill 2 Projective Bundle Theorem and Splitting Principle ...............113 3 Chem Classes and Stiefel-Whitney Classes of Vector Bundles .....114 4 Elementary Properties of Characteristic Classes .................117 5 Chern Character and Related Multiplicative Characteristic Classes .118 6 Euler Class ................................................ 121 7 Thom Space, Thorn Class, and Thom Isomorphism ..............122 8 Stiefel- Whitney Classes in Terms of Steenrod Operations .........122 9 Pontrjagin classes ..........................................125 References .....................................................125 11 Characteristic Classes of Manifolds ..............................127 1 Orientation in Euclidean Space and on Manifolds ................127 2 Poincaré Duality on Manifolds ...............................129 3 Thom Class of the Tangent Bundle and Duality .................130 4 Euler Class and Euler Characteristic of a Manifold ...............131 5 Wu s Formula for the Stiefel-Whitney Classes of a Manifold ......132 6 Cobordism and Stiefel- Whitney Numbers ......................133 7 Introduction to Characteristic Classes and Riemann-Roch ........134 Reference ......................................................135 Uontents 12 Spin Structures ................................................137 1 The Groups Spin(n) and Spin (n) ..............................137 2 Orientation and the First Stiefel-Whitney Class .................139 3 Spin Structures and the Second Stiefel-Whitney Class ............140 4 Spin Structures and the Third Integral Stief el-Whitney Class ..... 141 5 Relation Between Characteristic Classes of Real and Complex Vector Bundles .................................142 6 Killing Homotopy Groups in a Fibration .......................142 Part III Versions of Х -Theory and Bott Periodicity 13 G-Spaces, G-Bundles, and G-Vector Bundles ......................149 1 Relations Between Spaces and G-Spaces: G-Homotopy ...........149 2 Generalities on G-Bundles ...................................152 3 Generalities on G-Vector Bundles .............................153 4 Special Examples of G-Vector Bundles ........................155 5 Extension and Homotopy Problems for G-Vector Bundles for G a Compact Group .....................................157 6 Relations Between Complex and Real G-Vector Bundles ..........158 7 KRc-Theory ...............................................159 References .....................................................161 14 Equivariant if-Theory Functor Kg · Periodicity, Thom Isomorphism, Localization, and Completion ......................163 1 Associated Projective Space Bundle to a G-Equivariant Bundle .... 163 2 Assertion of the Periodicity Theorem for a Line Bundle ..........164 3 Thom Isomorphism .........................................167 4 Localization Theorem of Atiyah and Segal .....................170 5 Equivariant ŕf-Theory Completion Theorem of Atiyah and Segal ... 172 References .....................................................173 15 Bott Periodicity Maps and Clifford Algebras ......................175 1 Vector Bundles and Their Principal Bundles and Metrics ..........175 2 Homotopy Representation of K-Theory ........................176 3 The Bott Maps in the Periodicity Series ........................179 4 KRG{X) and the Representation Ring RR(G) ....................180 5 Generalities on Clifford Algebras and Their Modules .............181 6 KRq4{) and Modules Over Clifford Algebras ..................184 7 Bott Periodicity and Morse Theory ............................185 8 The Graded Rings KU{) and KO{) ........................187 References .....................................................188 16 Gram-Schmidt Process, Iwasawa Decomposition, and Reduction of Structure ......................................................189 1 Classical Gram-Schmidt Process .............................189 2 Definition of Basic Linear Groups .............................190 3 Iwasawa Decomposition for GL and SL ........................191 4 Applications to Structure Group Reduction for Principal Bundles Related to Vector Bundles ...................................192 5 The Special Case of SL2 (K) and the Upper Half Plane ...........193 6 Relation Between SL2(R) and SL2(C) with the Lorentz Groups ... 194 A Appendix: A Novel Characterization of the Iwasawa Decomposition of a Simple Lie Group (by B. Krötz) .............195 References .....................................................201 17 Topological Algebras: G-Equivariance and .KK-Theory .............203 1 The Module of Cross Sections for a G-Equivariant Vector Bundle .. 204 2 G-Equivariant К -Theory and the ÍT-Theory of Cross Products ......205 3 Generalities on Topological Algebras: Stabilization ..............207 4 E11(X) and Ext(X) Pairing with ^-Theory to Z ...................209 5 Extensions: Universal Examples ..............................212 6 Basic Examples of Extensions for AT-Theory ....................215 7 Homotopy Invariant, Half Exact, and Stable Functors ............219 8 The Bivariant Functor kk,(A, B) .............................. 220 9 Bott Map and Bott Periodicity ................................221 A Appendix: The Green-Julg Theorem (by S. Echterhoff)...........223 References .....................................................226 Part IV Algebra Bundles: Twisted K-Theory 18 Isomorphism Classification of Operator Algebra Bundles ...........229 1 Vector Bundles and Algebra Bundles ..........................230 2 Principal Bundle Description and Classifying Spaces .............231 3 Homotopy Classification of Principal Bundles ..................233 4 Classification of Operator Algebra Bundles .....................235 References .....................................................239 19 Brauer Group of Matrix Algebra Bundles and tf-Groups ...........241 1 Properties of the Morphism an ...............................241 2 From Brauer Groups to Grothendieck Groups ...................243 3 Stability I: Vector Bundles ...................................244 4 Stability II: Characteristic Classes of Algebra Bundles and Projective Äľ-Group .....................................245 5 Rational Class Groups ......................................246 6 Sheaf Theory Interpretation ..................................247 Reference ......................................................249 xiv Contents 20 Analytic Definition of Twisted K-Theory ..........................251 1 Cross Sections and Fibre Homotopy Classes of Cross Sections .....251 2 Two Basic Analytic Results in Bundle Theory and K-Theory ......252 3 Twisted A -Theory in Terms of Fredholm Operators ..............253 21 The Atiyah-Hirzebruch Spectral Sequence in Aľ-Theory ............255 1 Exact Couples: Their Derivation and Spectral Sequences .........255 2 Homological Spectral Sequence for a Filtered Object .............256 3 /r-Theory Exact Couples for a Filtered Space ...................258 4 Atiyah-Hirzebruch Spectral Sequence for /ÍT-Theory .............260 5 Formulas for Differentials ...................................262 6 Calculations for Products of Real Projective Spaces ..............263 7 Twisted /í -Theory Spectral Sequence ..........................264 Reference ......................................................264 22 Twisted Equivariant if-Theory and the Verlinde Algebra ...........265 1 The Verlinde Algebra as the Quotient of the Representation Ring ... 266 2 The Verlinde Algebra for SU(2) and s[(2) ......................268 3 The G-Bundles on G with the Adjoint G-Action .................271 4 A Version of the Freed-Hopkins-Teleman Theorem .............273 References .....................................................274 Part V Gerbes and the Three Dimensional Integral Cohomology Classes 23 Bundle Gerbes .................................................277 1 Notation for Gluing of Bundles ...............................277 2 Definition of Bundle Gerbes .................................280 3 The Gerbe Characteristic Class ...............................281 4 Stability Properties of Bundle Gerbes ..........................283 5 Extensions of Principal Bundles Over a Central Extension ........284 6 Modules Over Bundle Gerbes and Twisted X-Theory .............284 Reference ......................................................286 24 Category Objects and Groupoid Gerbes ..........................287 1 Simplicial Objects in a Category ..............................287 2 Categories in a Category .....................................290 3 The Nerve of the Classifying Space Functor and Definition of Algebraic ЛГ -Theory ...................................... 293 4 Groupoids in a Category .....................................295 5 The Groupoid Associated to a Covering ........................297 6 Gerbes on Groupoids .......................................298 7 The Groupoid Gerbe Characteristic Class ......................300 Contents 25 Stacks and Gerbes ............................................. ^ 1 Presheaves and Sheaves with Values in Category ................304 2 Generalities on Adjoint Functors .............................. 3^ 3 Categories Over Spaces (Fibred Categories) ....................309 4 Prestacks Over a Space ...................................... 3 l 5 Descent Data .............................................. 3l5 6 The Stack Associated to a Prestack ............................ 316 7 Gerbes as Stacks of Groupoids ............................... 318 8 Cohomological Classification of Principal G-Sheaves ............318 9 Cohomological Classification of Bands Associated with a Gerbe . . . 320 323 Bibliography ....................................................... 327 Index of Notations .................................................. Notation for Examples of Categories .................................. JJJ T , .............335 Index ................................................
any_adam_object	1
author	Husemöller, Dale Joachim, Michael Jurco, Branislav Schottenloher, Martin 1944-
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series	Lecture notes in physics
series2	Lecture notes in physics
spellingShingle	Husemöller, Dale Joachim, Michael Jurco, Branislav Schottenloher, Martin 1944- Basic bundle theory and K-cohomology invariants Lecture notes in physics Faisceaux, Théorie des ram K-théorie ram Fiber bundles (Mathematics) K-theory Faserbündel (DE-588)4135582-9 gnd K-Theorie (DE-588)4033335-8 gnd
subject_GND	(DE-588)4135582-9 (DE-588)4033335-8
title	Basic bundle theory and K-cohomology invariants
title_auth	Basic bundle theory and K-cohomology invariants
title_exact_search	Basic bundle theory and K-cohomology invariants
title_full	Basic bundle theory and K-cohomology invariants D. Husemöller ; M. Joachim ; B. Jurco ; M. Schottenloher
title_fullStr	Basic bundle theory and K-cohomology invariants D. Husemöller ; M. Joachim ; B. Jurco ; M. Schottenloher
title_full_unstemmed	Basic bundle theory and K-cohomology invariants D. Husemöller ; M. Joachim ; B. Jurco ; M. Schottenloher
title_short	Basic bundle theory and K-cohomology invariants
title_sort	basic bundle theory and k cohomology invariants
topic	Faisceaux, Théorie des ram K-théorie ram Fiber bundles (Mathematics) K-theory Faserbündel (DE-588)4135582-9 gnd K-Theorie (DE-588)4033335-8 gnd
topic_facet	Faisceaux, Théorie des K-théorie Fiber bundles (Mathematics) K-theory Faserbündel K-Theorie
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016295577&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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