Differential Topology

Course Objective

At the end of this course:
-the student understands the introduced concepts such as coverings,
degree of a
function, intersection number, 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.

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:
-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

This is covering theory plus the content of Chapter 1.5, 1.7, 2, 3 of
the accompanying book.

Additional Information 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

Additional Information Target Audience

Bachelor Mathematics, year 3

Recommended background knowledge

Topology