Smooth classifying spaces for differential $K$-theory

TL;DR

This paper develops smooth Banach manifold models for differential K-theory, providing a refined, geometric framework that captures the universal Chern character and allows classifying space-level computations without compactness assumptions.

Contribution

It introduces a new differential K-theory model based on smooth Banach manifolds, enabling computations directly on classifying spaces and refining the topological Chern character.

Findings

Models are norm completions of stable Grassmannian and unitary group.

Constructed groups are isomorphic to the unique differential extension with S^1-integration.

Work is done entirely on classifying spaces, avoiding compactness assumptions.

Abstract

We construct a version of differential $K$-theory based on smooth Banach manifold models for the homotopy types $B \mathrm U\times Z$ and $\mathrm U$ that appear in the topological $K$-theory spectrum. These manifolds carry natural differential forms that refine the topological universal Chern character, together with natural addition and inversion operations that induce the respective structure on $\hat K$. Our models are norm completions of the usual stable Grassmannian and the stable unitary group. Their regularity allows us to work completely on the level of classifying spaces, and therefore we do not need a compactness assumption on our manifolds that is present in many other descriptions. The constructed groups $\hat K(M)$ are isomorphic to the unique differential extension of $K$-theory that admits an $S^1$-integration.