Basic bundle theory and K-cohomology invariants:
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg ; New York
Springer
2008
|
| Series: | Lecture notes in physics
726 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xv, 340 Seiten Illustrationen |
| ISBN: | 9783540749554 9783540749561 |
Staff View
MARC
| LEADER | 00000nam a22000008cb4500 | ||
|---|---|---|---|
| 001 | BV023092735 | ||
| 003 | DE-604 | ||
| 005 | 20211214 | ||
| 007 | t | ||
| 008 | 080122s2008 gw a||| |||| 00||| eng d | ||
| 020 | |a 9783540749554 |c Print |9 978-3-540-74955-4 | ||
| 020 | |a 9783540749561 |c Online |9 978-3-540-74956-1 | ||
| 035 | |a (OCoLC)173239484 | ||
| 035 | |a (DE-599)BVBBV023092735 | ||
| 040 | |a DE-604 |b ger |e rda | ||
| 041 | 0 | |a eng | |
| 044 | |a gw |c DE | ||
| 049 | |a DE-384 |a DE-29T |a DE-355 |a DE-19 |a DE-11 |a DE-20 |a DE-83 | ||
| 050 | 0 | |a QA612.6 | |
| 082 | 0 | |a 514.224 |2 22 | |
| 084 | |a UD 8220 |0 (DE-625)145543: |2 rvk | ||
| 084 | |a 19K99 |2 msc | ||
| 084 | |a 55R10 |2 msc | ||
| 084 | |a 46M20 |2 msc | ||
| 245 | 1 | 0 | |a Basic bundle theory and K-cohomology invariants |c D. Husemöller ; M. Joachim ; B. Jurco ; M. Schottenloher |
| 264 | 1 | |a Berlin ; Heidelberg ; New York |b Springer |c 2008 | |
| 300 | |a xv, 340 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in physics |v 726 | |
| 650 | 7 | |a Faisceaux, Théorie des |2 ram | |
| 650 | 7 | |a K-théorie |2 ram | |
| 650 | 4 | |a Fiber bundles (Mathematics) | |
| 650 | 4 | |a K-theory | |
| 650 | 0 | 7 | |a Faserbündel |0 (DE-588)4135582-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a K-Theorie |0 (DE-588)4033335-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Faserbündel |0 (DE-588)4135582-9 |D s |
| 689 | 0 | 1 | |a K-Theorie |0 (DE-588)4033335-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Husemöller, Dale |0 (DE-588)117713058 |4 aut | |
| 700 | 1 | |a Joachim, Michael |4 aut | |
| 700 | 1 | |a Jurco, Branislav |4 aut | |
| 700 | 1 | |a Schottenloher, Martin |d 1944- |0 (DE-588)138917760 |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-540-74956-1 |
| 830 | 0 | |a Lecture notes in physics |v 726 |w (DE-604)BV000003166 |9 726 | |
| 856 | 4 | 2 | |m Digitalisierung UB Augsburg |q application/pdf |u 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 |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016295577 | ||
Record in the Search Index
| _version_ | 1804137346500657152 |
|---|---|
| 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
KR*G{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
................................................
|
| adam_txt |
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
KR*G{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 |
| any_adam_object_boolean | 1 |
| author | Husemöller, Dale Joachim, Michael Jurco, Branislav Schottenloher, Martin 1944- |
| author_GND | (DE-588)117713058 (DE-588)138917760 |
| author_facet | Husemöller, Dale Joachim, Michael Jurco, Branislav Schottenloher, Martin 1944- |
| author_role | aut aut aut aut |
| author_sort | Husemöller, Dale |
| author_variant | d h dh m j mj b j bj m s ms |
| building | Verbundindex |
| bvnumber | BV023092735 |
| callnumber-first | Q - Science |
| callnumber-label | QA612 |
| callnumber-raw | QA612.6 |
| callnumber-search | QA612.6 |
| callnumber-sort | QA 3612.6 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | UD 8220 |
| ctrlnum | (OCoLC)173239484 (DE-599)BVBBV023092735 |
| dewey-full | 514.224 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 514 - Topology |
| dewey-raw | 514.224 |
| dewey-search | 514.224 |
| dewey-sort | 3514.224 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| discipline_str_mv | Physik Mathematik |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>02085nam a22005418cb4500</leader><controlfield tag="001">BV023092735</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20211214 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">080122s2008 gw a||| |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9783540749554</subfield><subfield code="c">Print</subfield><subfield code="9">978-3-540-74955-4</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9783540749561</subfield><subfield code="c">Online</subfield><subfield code="9">978-3-540-74956-1</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)173239484</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV023092735</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rda</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="044" ind1=" " ind2=" "><subfield code="a">gw</subfield><subfield code="c">DE</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-384</subfield><subfield code="a">DE-29T</subfield><subfield code="a">DE-355</subfield><subfield code="a">DE-19</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-20</subfield><subfield code="a">DE-83</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">QA612.6</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">514.224</subfield><subfield code="2">22</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">UD 8220</subfield><subfield code="0">(DE-625)145543:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">19K99</subfield><subfield code="2">msc</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">55R10</subfield><subfield code="2">msc</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">46M20</subfield><subfield code="2">msc</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Basic bundle theory and K-cohomology invariants</subfield><subfield code="c">D. Husemöller ; M. Joachim ; B. Jurco ; M. Schottenloher</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Berlin ; Heidelberg ; New York</subfield><subfield code="b">Springer</subfield><subfield code="c">2008</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">xv, 340 Seiten</subfield><subfield code="b">Illustrationen</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Lecture notes in physics</subfield><subfield code="v">726</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Faisceaux, Théorie des</subfield><subfield code="2">ram</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">K-théorie</subfield><subfield code="2">ram</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Fiber bundles (Mathematics)</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">K-theory</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Faserbündel</subfield><subfield code="0">(DE-588)4135582-9</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">K-Theorie</subfield><subfield code="0">(DE-588)4033335-8</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Faserbündel</subfield><subfield code="0">(DE-588)4135582-9</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">K-Theorie</subfield><subfield code="0">(DE-588)4033335-8</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Husemöller, Dale</subfield><subfield code="0">(DE-588)117713058</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Joachim, Michael</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Jurco, Branislav</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Schottenloher, Martin</subfield><subfield code="d">1944-</subfield><subfield code="0">(DE-588)138917760</subfield><subfield code="4">aut</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Erscheint auch als</subfield><subfield code="n">Online-Ausgabe</subfield><subfield code="z">978-3-540-74956-1</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Lecture notes in physics</subfield><subfield code="v">726</subfield><subfield code="w">(DE-604)BV000003166</subfield><subfield code="9">726</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Augsburg</subfield><subfield code="q">application/pdf</subfield><subfield code="u">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</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-016295577</subfield></datafield></record></collection> |
| id | DE-604.BV023092735 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:41:41Z |
| indexdate | 2024-07-09T21:10:50Z |
| institution | BVB |
| isbn | 9783540749554 9783540749561 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016295577 |
| oclc_num | 173239484 |
| open_access_boolean | |
| owner | DE-384 DE-29T DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-20 DE-83 |
| owner_facet | DE-384 DE-29T DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-20 DE-83 |
| physical | xv, 340 Seiten Illustrationen |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in physics |
| series2 | Lecture notes in physics |
| spelling | Basic bundle theory and K-cohomology invariants D. Husemöller ; M. Joachim ; B. Jurco ; M. Schottenloher Berlin ; Heidelberg ; New York Springer 2008 xv, 340 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Lecture notes in physics 726 Faisceaux, Théorie des ram K-théorie ram Fiber bundles (Mathematics) K-theory Faserbündel (DE-588)4135582-9 gnd rswk-swf K-Theorie (DE-588)4033335-8 gnd rswk-swf Faserbündel (DE-588)4135582-9 s K-Theorie (DE-588)4033335-8 s DE-604 Husemöller, Dale (DE-588)117713058 aut Joachim, Michael aut Jurco, Branislav aut Schottenloher, Martin 1944- (DE-588)138917760 aut Erscheint auch als Online-Ausgabe 978-3-540-74956-1 Lecture notes in physics 726 (DE-604)BV000003166 726 Digitalisierung UB Augsburg application/pdf 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 Inhaltsverzeichnis |
| 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_exact_search_txtP | 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 |
| volume_link | (DE-604)BV000003166 |
| work_keys_str_mv | AT husemollerdale basicbundletheoryandkcohomologyinvariants AT joachimmichael basicbundletheoryandkcohomologyinvariants AT jurcobranislav basicbundletheoryandkcohomologyinvariants AT schottenlohermartin basicbundletheoryandkcohomologyinvariants |