Smooth extensions of generalized cohomology theories

广义上同调理论对光滑流形范畴的限制允许一个光滑的细化。第一个例子是由Cheeger-Simons在带积分系数的普通上同的情况下给出的。Cheeger-Simons定义的一个动机是，在存在连接的情况下，向量束的积分chen类可以自然地提升到积分上同调的光滑扩展中的类。它在算术几何中也有重要的应用。近年来，其他广义上同调理论的光滑扩展成为数学和数学物理的研究热点。光滑k理论在弦理论的场描述中起着重要的作用。其他上同调理论的光滑扩展类以场理论的拉格朗日量出现，例如wzw模型。关键论文论文[HS05]提供了广义上同调理论光滑扩展的一般方法。目前对广义上同调理论的光滑扩展理论的理解是一个拼凑的例子和部分结果。一个理论缺少的要素是：1。一个公理式的设置。光滑扩展及其附加结构（如乘积）的唯一性Umkehr地图理论自然变换的理论。

The restriction of a generalized cohomology theory to the category of smooth manifolds admits a smooth refinement. The first example has been introduced by Cheeger-Simons in the case of ordinary cohomology with integral coefficients. One motivation of the definition of Cheeger-Simons was that in the presence of connections the integral Chern classes of a vector bundle can naturally be lifted to classes in the smooth extension of integral cohomology. It also has important applications in arithmetic geometry. Recently, smooth extensions of other generalized cohomology theories came into the focus of research in mathematics and mathematical physics. Smooth K-theory plays a role in the description of fields in string theory. Classes in smooth extensions of other cohomology theories appeared as Lagrangians of field theories, e.g. the WZW-model. The key paper paper [HS05] provides a general approach to smooth extensions of generalized cohomology theories.The current understanding of the theory of smooth extensions of generalized cohomology theories is a patchwork of examples and partial results. Missing elements of a theory are:1. an axiomatic set-up2. uniqueness of smooth extensions and their additional structures like products3. the theory of Umkehr maps4. a theory of natural transformations.