The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de Rham cocycles -- known as differential cohomology. We will discuss the analog in the case of a group action on the manifold: We will show the compatibility of the equivariant characteristic class in integral Borel cohomology with the equivariant characteristic form in the Cartan model. Motivated by this understanding of characteristic forms, we define equivariant differential cohomology as a refinement of equivariant integral cohomology by Cartan cocycles.

The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de Rham cocycles -- known as differential cohomology. We will discuss the analog in the case of a group action on the manifold: We will show the compatibility of the equivariant characteristic class in integral Borel cohomology with the equivariant characteristic form in the Cartan model. Motivated by this understanding of characteristic forms, we define equivariant differential cohomology as a refinement of equivariant integral cohomology by Cartan cocycles.

Im Mittelpunkt der Untersuchungen stehen Grundlagen für äquivariante motivische Homotopietheorie. Für eine neue Grothendieck-Topologie auf einer Kategorie von äquivarianten glatten k-Schemata werden unstabile und stabile motivische Homotopietheorie entwickelt. Im zweiten Teil der Arbeit wird als Anwendung der stabilen Theorie eine Adams-Spektralsequenz mit motivischer Borel-Kohomologie konstruiert.