Geometrische und getwistete algebraische Topologie

该项目由两个相关部分组成。在光滑流形上，带积分系数的上同调可以细化为Deligne上同调（其类也称为Cheeger-Simons微分特征）。我们将这种细化称为平滑扩展。此细化中的类包含有关表示微分形式的附加信息。Cheeger-Simons的一个有趣的观察是，在存在连接的情况下，向量束的整Chern类可以自然地提升为Deligne上同的类。第一部分的主题是一个系统的概括。给定一个乘法上同调理论和一个到实系数上同调的变换，我们可以考虑它的光滑扩展。如果上同调理论在几何循环方面有一个很好的描述，那么根据我们的经验，就有可能构造这样一个光滑的扩展。例如，这适用于k理论或各种bordism理论。项目第一部分的目标是使光滑扩展的构造，特别是复杂协同理论的构造精确。进一步研究了自然变换和上同运算对光滑扩展是否有自然提升的问题。一个相关的问题是定义具有连接的向量束的光滑特征类（如connor - floyd类）（这是delign -上同调值Chern类定义的经典情况的推广）。

The project consists of two related parts. On smooth manifolds cohomology with integral coefficients can be refined to Deligne cohomology (its classes are also called Cheeger-Simons differential characters). We will call such a refinement a smooth extension. A class in this refinement carries additional information about a representing differential form. It is an interesting observation of Cheeger-Simons that in the presence of connections the integral Chern classes of a vector bundle can naturally be lifted to classes in Deligne cohomology. The topic of the first part concerns a systematic generalization. Given a multiplicative cohomology theory together with a transformation to cohomology with real coefficients one can consider its smooth extensions. If the cohomology theory has a nice description in terms of geometric cycles then by our experience it is possible to construct such a smooth extension. This applies e.g. to K-theory or various bordism theories. The goal of the first part of the project is to make the constructions of smooth extensions, in particular of the complex cobordism theory precise. Furthermore it shall investigate the question whether natural transformations and cohomology operations have natural lifts to the smooth extensions. A related question is to define smooth characteristic classes (e.g. Connor-Floyd-classes) for vector bundles with connections (this is a generalization of the classical case of the definition of Deligne-cohomology valued Chern classes).