• Baues Oliver

• Baues Oliver

In the lecture course we will introduce the participants to  the basic ideas and methods of differential topology.  The idea of differential topology is to understand the interaction of smooth manifolds and their maps with the topology of manifolds using differentiable methods. For example, suitable invariants will be constructed which allow do distinguish manifolds up to diffeomorphism or to calculate topological information from differentiable data, 

Examples in this direction are the mapping degree of smooths maps or  the Poincaré-Hopf theorem which shows how the index of a vector field maybe used to calculate the Euler characteristic of a manifold.

The methods which we will encouter are important for the consideration of  many problems in topology, geometry and global analysis. 

Knowledge of basic methods used in differential topology. Familiarity with embedding and transversality theorems, cohomology of manifolds, fiber bundles and characteristic classes. .A good understanding of how to apply these methods to problems in geometry, topology and analysis.

counts towards Algebra/Geometry/Topology and Analysis

Literature:

J. Milnor, “Topology from a differential viewpoint”, Virginia University Press, 1965. 
T. Bröcker, K. Jänich , “Einführung in die Differentialtopologie”, Springer Verlag 1973. 
M. Hirsch, “Differential topology”, Springer Verlag 1976.
G. Bredon, “Topology and Geometry”, Springer Verlag 1993.
R. Bott, L.W. Tu, “Differential forms in algebraic topology”, Springer Verlag 1982 
F.W. Warner, “Foundations of differentiable manifolds and Lie groups”, Springer 1983

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