Differential Topology

Course Objective
At the end of this course:
-the student understands the introduced concepts such as degree of a
function, intersection number, coverings, and is able to use these to
prove some fundamental results;
-the student understands the meaning of the theorems and knows how to
derive them;
-the student can relate different properties to each other, for example
relating the zeros of transversal vector fields to the genus of the
surface;
-the student knows examples of investigating the same mathematical
content from different mathematical perspectives.

Course Content
Differential topology studies differentiable manifolds and
differentiable functions from a topological viewpoint. In contrast to
differential geometry properties arising from metrics are not studied,
and in contrast to the general topology course, the spaces are
restricted to nicely behaving manifolds. Studied properties of the
manifolds are rather global than local. The classification of coverings
can be seen as an alternative formulation of the fundamental theorem of
Galois theory, allowing to derive an algebraic theorem from a
topological perspective.

The following topics will be covered during the course:
-smooth manifolds;
-classification of coverings;
-Sard’s theorem;
-Brouwer’s fixed point theorem;
-transversality;
-intersection theory and degree theory;
-Euler characteristic;
-index of vector fields;
-Hopf theorem

Teaching Methods
Lectures and tutorials (2+2 hours per week)

Method of Assessment
For this course there is a midterm
examination (50%) and a final exam (50%). There will also be a resit
examination.

Literature
Guilluimin and Pollack: Differential Topology

Target Audience
Bachelor Mathematics, year 3

Additional Information
This course is offered for the first time. Master students are welcome
to participate as well.

Recommended background knowledge
Topology
Differential geometry