3-4 hours of lectures per week.
Lecturer
Johan Dupont
Content
We continue the study of deRham cohomology as developed in Topology 1. In particular in connection with manifolds having an action by a Lie group. We shall therefore first 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 connection and curvature in a principal fibre bundle i.e. a manifold with a particularly "nice" Lie group action (so-called "gauge theory"). This leads to the Chern-Weil theory of certain cohomological invariants, the "characteristic classes" for principal fibre 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 vector 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 connection and curvature in fibre 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
Notes, and for reference: F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, 1983.