Geometrische und getwistete algebraische Topologie

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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).