Differential K-theory. A survey

TL;DR

This survey reviews the development of differential K-theory, a refined cohomology theory combining topology and differential geometry, highlighting axiomatic characterizations, explicit constructions, and key properties including index theorems.

Contribution

It provides a comprehensive overview of recent advances in differential K-theory, including axiomatic frameworks, explicit models, and fundamental theorems, consolidating knowledge in this evolving field.

Findings

Axiomatic characterization of differential K-theory

Explicit constructions using vector bundles and homotopy theory

Formulation of Riemann-Roch and Atiyah-Singer index theorems in differential K-theory

Abstract

Generalized differential cohomology theories, in particular differential K-theory (often called "smooth K-theory"), are becoming an important tool in differential geometry and in mathematical physics. In this survey, we describe the developments of the recent decades in this area. In particular, we discuss axiomatic characterizations of differential K-theory (and that these uniquely characterize differential K-theory). We describe several explicit constructions, based on vector bundles, on families of differential operators, or using homotopy theory and classifying spaces. We explain the most important properties, in particular about the multiplicative structure and push-forward maps and will state versions of the Riemann-Roch theorem and of Atiyah-Singer family index theorem for differential K-theory.