• Physical Background to the K-Theory Classification of D-Branes: Introduction and References.- Physical Background to the K-Theory Classification of D-Branes: Introduction and References.- Bundles over a Space and Modules over an Algebra.- Generalities on Bundles and Categories.- Vector Bundles.- Relation Between Vector Bundles, Projective Modules, and Idempotents.- K-Theory of Vector Bundles, of Modules, and of Idempotents.- Principal Bundles and Sections of Fibre Bundles: Reduction of the Structure and the Gauge Group I.- Homotopy Classification of Bundles and Cohomology: Classifying Spaces.- Homotopy Classes of Maps and the Homotopy Groups.- The Milnor Construction: Homotopy Classification of Principal Bundles.- Fibrations and Bundles: Gauge Group II.- Cohomology Classes as Homotopy Classes: CW-Complexes.- Basic Characteristic Classes.- Characteristic Classes of Manifolds.- Spin Structures.- Versions of K-Theory and Bott Periodicity.- G-Spaces, G-Bundles, and G-Vector Bundles.- Equivariant K-Theory Functor KG : Periodicity, Thom Isomorphism, Localization, and Completion.- Bott Periodicity Maps and Clifford Algebras.- Gram–Schmidt Process, Iwasawa Decomposition, and Reduction of Structure in Principal Bundles.- Topological Algebras: G-Equivariance and KK-Theory.- Algebra Bundles: Twisted K-Theory.- Isomorphism Classification of Operator Algebra Bundles.- Brauer Group of Matrix Algebra Bundles and K-Groups.- Analytic Definition of Twisted K-Theory.- The Atiyah–Hirzebruch Spectral Sequence in K-Theory.- Twisted Equivariant K-Theory and the Verlinde Algebra.- Gerbes and the Three Dimensional Integral Cohomology Classes.- Bundle Gerbes.- Category Objects and Groupoid Gerbes.- Stacks and Gerbes.- Erratum.

From the reviews: "The book, according to the authors, originated in a course of lectures by D. Husemoller for physics students at the LMU in Munchen during summer term 2003. ... The book is suitable for those readers who want to learn this subject rapidly, even if they are newcomers to this field, and also for the sake of the interest in twisted K-theory this book will be stimulative even to the readers who have already studied Husemoller's book ... ." (Haruo Minami, Zentralblatt MATH, Vol. 1135 (13), 2008)