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光滑加细。第一个例子是Cheeger-Simons在常上同调的整系数情形下引入的。Cheeger-Simons定义的一个动机是，在存在联系的情况下，向量丛的积分陈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张成为数学和数学物理研究的热点。光滑K-理论在弦论的场描述中起着重要的作用。其他上同调理论的光滑扩张类出现在场论的拉格朗日中，例如WZW模型。[HS 05]是研究广义上同调理论光滑扩张的一个重要文献，目前对广义上同调理论光滑扩张理论的理解是一些例子和部分结果的拼凑。一个理论的缺失要素是：1。不言自明的安排2.光滑扩张的唯一性及其类似乘积的附加结构3. Umkehr映射理论4.自然转化理论

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.