Title of project:
Euler Characteristic and the Classification of Surfaces

Code:
UP2122B

Supervisor: 
Dr Ulrich Pennig

Project description:
In the first lecture of the Algebraic Topology course we encountered the Euler characteristic of polytopes as a topological invariant. Polytopes are special cases of combinatorially defined topological spaces called simplicial complexes and the definition of the Euler characteristic can be adapted to work for these as well. If we triangulate a surface, it gives rise to a simplicial complex and therefore always has an associated Euler characteristic, which is independent of the choice of triangulation as we will see. Another important property of surfaces is their orientability, which tells us whether we can pick a full set of coordinate patches, such that the handedness of the coordinate system does not change going from one patch to the other. The torus from the lecture is orientable, while the Klein bottle is not.

The goal of this project is to understand the Euler characteristic of simplicial complexes in full generality and of surfaces in particular. We will then look at the orientability of surfaces. It turns out that the pair of Euler characteristic and orientability of a surface without boundary provides a complete homeomorphism invariant. This means that two surfaces without boundary are homeomorphic if and only if their Euler characteristics agree and they are both either orientable or not orientable. After having understood the definitions we will focus on proving this theorem.

Prerequisite Modules:
Algebraic Topology (MA3008)
Groups (MA0213) or Rings and Fields (MA3003)

Number of students who could be supervised for this project:
1