Differential cohomology
Ulrich Bunke
TL;DR
The notes provide a comprehensive treatment of differential cohomology, starting from secondary characteristic forms and smooth Deligne cohomology, and advancing to a stable homotopy framework for differential extensions of generalized cohomology theories. They develop both the concrete differential-geometric toolkit (connections, characteristic forms, transgression, integral refinements) and the abstract homotopy-theoretic framework (differential extensions, Umkehr maps), with explicit examples in K-theory and bordism. Key contributions include a detailed account of integral refinements, Chern–Simons invariants, and the Deligne cohomology machinery that unifies topology, geometry, and arithmetic through differential refinements of characteristic classes. The material lays a foundation for applying differential cohomology to geometric quantization, gauge theory, and twisted generalized cohomology theories, providing both computational techniques and conceptual frameworks.
Abstract
These course note first provide an introduction to secondary characteristic classes and differential cohomology. They continue with a presentation of a stable homotopy theoretic approach to the theory of differential extensions of generalized cohomology theories including products and Umkehr maps.