3-4 hours of lectures per week.
Lecturer
Johan Dupont
Content
We continue the study of deRham cohomology as developed in Topology 1, now with emphasis on manifolds having an action by a Lie group. We shall therefore rst study Lie groups and the associated homogeneous manifolds and calculate the cohomology of these in favourable cases. Then we shall proceed with the theory of connections and curvature in a principal bre bundle i.e. a manifold with a particularly \nice" Lie group action (socalled "gauge theory"). This leads to the Chern-Weil theory of certain cohomological invariants, the "characteristic classes" for principal bre bundles. In the case of the classical Lie groups this gives representatives in the deRham cohomology of the Euler and Chern classes for real respectively complex vec-tor bundles. Since characteristic classes are topological invariants the theory can be viewed as a generalization of the Gauss-Bonnet theorem in classical differential geometry. Furthermore the theory of connections and curvature in bre bundles is the basic ingredient in the constructions of the famous invariants for 3- and 4-manifolds by Donaldson, Seiberg and Witten.
Prerequisites
Topology 1
Literature
Lecture Notes
Evaluation
Students who do not intend to take a degree in Mathematics or Statistics from the University of Aarhus, but wish to earn credits for a 2.dels course from the Department of Mathematics, should indicate at the beginning of the course that they wish to be examined.
The form of examination for these students will be active participation together with oral or written contributions.
Credits
10 ECTS
Semester
Spring 2003