• aspherical
• Coxeter systems
• manifolds

A connected n-manifold is called aspherical if its higher homotopy groups vanish, except possibly for the fundamental group.
Aspherical manifolds in dimensions 1, 2, and 3 are well-understood: they are characterized by their universal covers being homeomorphic to R^n. This naturally raises the question of whether the same property holds in higher dimensions. In 1983, Michael W. Davis proved that for every n>3, there exists a closed aspherical n-manifold whose universal cover is not homeomorphic to the Euclidean space.
In this thesis, I will present a proof of Davis’s theorem. The proof uses results from algebraic topology, the study of Coxeter systems, simplicial complexes and cell complexes.