微分$K$ -理论的光滑分类空间
Smooth classifying spaces for differential $K$-theory
摘要：
我们构造了一个版本的微分$K$ -理论的基础上光滑的巴拿赫流形模型的同伦类型$B \mathrm U\times Z$和$\mathrm U$出现在拓扑$K$ -理论谱。这些流形带有自然的微分形式，这些微分形式细化了拓扑普适的陈特征，以及自然的加法和求逆运算，这些运算在$\hat K$上诱导出相应的结构。我们的模型是通常的稳定格拉斯曼群和稳定酉群的范数完备化。它们的正则性使我们能够完全在分类空间的水平上工作，因此我们不需要在流形上进行紧性假设，而这在许多其他描述中是存在的。所构造的群$\hat K(M)$同构于$K$ -理论的唯一微分扩张，该扩张允许$S^1$ -积分。
英文摘要：
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.
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