Geometrische und getwistete algebraische Topologie

几何和扭曲代数拓扑

• 批准号: 5453524
• 负责人: Professor Dr. Ulrich Bunke
• 金额: --
• 依托单位: Fakultät für Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2005
• 资助国家: 德国
• 起止时间: 2004-12-31 至 2008-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5453524?language=en
• 关键词: Geometrische; und; getwistete; algebraische; Topologie

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).

该项目由两个相关部分组成。在光滑流形上，整系数的上同调可以精化为Deligne上同调（它的类也称为Cheeger-Simons微分特征标）。我们将把这样的精化称为光滑延拓。这种细化中的类携带关于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，向量丛的积分陈类可以自然地提升到Deligne上同调的类。第一部分的主题涉及系统的概括。给定一个乘法上同调理论和一个变换到具有真实的系数的上同调，人们可以考虑它的光滑扩张。如果上同调理论有一个很好的描述，在几何圈，然后根据我们的经验，它是可能的，以建设这样一个光滑的扩展。这适用于例如K理论或各种边界理论。该项目的第一部分的目标是使光滑扩张的结构，特别是复杂的配边理论精确。此外，它将调查的问题，是否自然变换和上同调操作有自然升降机的光滑扩张。一个相关的问题是定义光滑的特征类（例如Connor-Floyd类）的向量丛的联系（这是一个推广的经典情况下的定义Deligne上同调值陈类）。