Topological approaches to Euler characteristics in odd-dimensional orbifolds

Abstract

This master thesis deals with orbifolds, a generalization of manifolds. On one hand, for compact manifolds of odd dimension one has a pretty interesting formula: the Euler characteristic of the manifold is half the characteristic of its boundary. On the other hand, Ichiro Satake stated and proved in 1957 that the Euler characteristic of an odd-dimensional compact Riemannian orbifold without boundary is 0. From this last result it can be proven a generalisation of the formula for odd-dimensional compact smooth orbifolds. Nevertheless, the proof given by Satake uses the Chern-Gauss-Bonnet formula, so the objective of this Master Thesis is to give a purely topological proof of the formula described. For this, the idea is to dissect an orbifold into smaller parts where the study of this formula becomes easier. In the following we define the main characteristics and properties of orbifolds, as well as some of their topological and geometrical features.