D. Husemöller, M. Joachim, B. Jurčo, M. Schottenloher (auth.)9783540749554, 3540749551
Based on several recent courses given to mathematical physics students, this volume is an introduction to bundle theory with the aim to provide newcomers to the field with solid foundations in topological K-theory. A fundamental theme, emphasized in the book, centers around the gluing of local bundle data related to bundles into a global object.
One renewed motivation for studying this subject, which has developed for almost 50 years in many directions, comes from quantum field theory, especially string theory, where topological invariants play an important role.
Table of contents :
Front Matter….Pages I-XV
Front Matter….Pages 1-1
Physical Background to the K-Theory Classification of D-Branes: Introduction and References….Pages 1-6
Front Matter….Pages 7-8
Generalities on Bundles and Categories….Pages 9-22
Vector Bundles….Pages 23-34
Relation Between Vector Bundles, Projective Modules, and Idempotents….Pages 35-44
K-Theory of Vector Bundles, of Modules, and of Idempotents….Pages 45-54
Principal Bundles and Sections of Fibre Bundles: Reduction of the Structure and the Gauge Group I….Pages 55-62
Front Matter….Pages 63-64
Homotopy Classes of Maps and the Homotopy Groups….Pages 65-74
The Milnor Construction: Homotopy Classification of Principal Bundles….Pages 75-81
Fibrations and Bundles: Gauge Group II….Pages 83-96
Cohomology Classes as Homotopy Classes: CW-Complexes….Pages 97-109
Basic Characteristic Classes….Pages 111-125
Characteristic Classes of Manifolds….Pages 127-135
Spin Structures….Pages 137-145
Front Matter….Pages 146-147
G-Spaces, G-Bundles, and G-Vector Bundles….Pages 149-161
Equivariant K -Theory Functor K G : Periodicity, Thom Isomorphism, Localization, and Completion….Pages 163-173
Bott Periodicity Maps and Clifford Algebras….Pages 175-188
Gram–Schmidt Process, Iwasawa Decomposition, and Reduction of Structure in Principal Bundles….Pages 189-201
Topological Algebras: G -Equivariance and KK -Theory….Pages 203-226
Front Matter….Pages 227-228
Isomorphism Classification of Operator Algebra Bundles….Pages 229-239
Brauer Group of Matrix Algebra Bundles and K -Groups….Pages 241-249
Front Matter….Pages 227-228
Analytic Definition of Twisted K -Theory….Pages 251-253
The Atiyah–Hirzebruch Spectral Sequence in K -Theory….Pages 255-264
Twisted Equivariant K -Theory and the Verlinde Algebra….Pages 265-274
Front Matter….Pages 275-276
Bundle Gerbes….Pages 277-286
Category Objects and Groupoid Gerbes….Pages 287-302
Stacks and Gerbes….Pages 303-322
Erratum….Pages E1-E11
Back Matter….Pages 323-340
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