Hyperbolic Manifolds and Coloured Polytopes

In this thesis, we explore the possibilities of Davis-Januszkiewicz (1991) techniques to build manifold covers of right-angled polytopes. Combining these techniques with Choi-Park (2017) work on the cohomology ring of real toric manifolds, and with computational algorithms in Sagemath, we classify up to isometry all possible cusp sections of finite-volume, orientable, hyperbolic 4-manifolds obtained as manifold covers of right-angled polytopes. We also obtain the first example of an orientable, cusped hyperbolic 4-manifold such that all cusp sections are rational homology spheres. This answers in particular to an open question from Golénia-Moroianu (2012) and provides a counter-example to Mazzeo-Phillips (1990).