Contents

1 Introduction
1.1 Differential cohomology and physics

2 Axioms for differential cohomology
2.1 Variations of the axiomatic approach
2.2 Homotopy formula
2.3 Eflat* as generalized cohomology theory ER/Z

3 Uniqueness of differential extensions
3.1 Uniqueness of differential K-theory

4 Models for differential K-theory
4.1 Vector bundles with connection
4.2 Classifying maps
4.3 Geometric families of elliptic operators
4.4 Differential characters
4.5 Multiplicative K-theory
4.6 Currential K-theory
4.7 Differential K-theory via bordism
4.8 Geometric cycle models for R/Z-K-theory via the flat part of differential K-theory
4.9 Real K-theory
4.10 Differential K-homology and bivariant differential K-theory

5 Orientation and integration
5.1 Differential K-orientations
5.2 Integration
5.3 S1-integration

6 Index theory and natural transformations
6.1 Differential Chern character
6.2 Differential Riemann-Roch theorem
6.3 Differential Atiyah-Singer index theorem

7 Twisted differential K-theory
7.1 Twists for ordinary K-theory
7.2 Twisted differential K-theory
7.3 T-duality
7.4 T-duality and differential K-theory

8 Applications of differential K-theory
8.1 Holonomy
8.3 Reduced eta-invariants
8.4 e-invariant
8.5 Secondary invariants for string bordism