Hyperbolic manifolds in high dimensions

Hyperbolic manifolds play a central role in three-dimensional topology, constituting the richest of the eight Thurston geometries, and arising as knot complements. Fibrations of hyperbolic 3-manifolds over the circle are also well understood, with tools such as the Thurston norm and Agol's virtual fibering theorem. On the other hand, the theory of hyperbolic manifolds in higher dimensions, as well as their role in the wider context of differential geometry, is not as clear, due to increased theoretical and computational complexities. The present thesis aims to shed some light on this vast field by means of three specific topics. Firstly, to study fibrations of odd-dimensional hyperbolic manifolds, it is natural to seek a generalization of the Thurston norm. A promising candidate for such an invariant is Friedl and Lück's twisted L^2-Euler characteristic, which also detects the Euler characteristic of a fiber. Our contribution is an algorithm that computes this invariant, given a CW complex and a class in its first cohomology group. Then, we consider the technique of constructing hyperbolic manifolds by gluing Coxeter polytopes, which is especially useful as it gives some control over the structure of the resulting manifolds. As a result, we construct a family of closed hyperbolic 5-manifolds with b_1 = 0 and volume < 250000. In a joint work with Edoardo Rizzi, we also find cusp-transitive hyperbolic 4-manifolds with every possible cusp section. Lastly, another line of research is based on a result of Kolpakov, Reid and Slavich, which gives codimension-1 geodesic embeddings of hyperbolic manifolds. By iterating a slight generalization of this result, and relating it to the Stiefel–Whitney characteristic classes, we prove the existence of closed orientable hyperbolic manifolds without spin^c structures in all dimensions ≥ 5; more generally, for all k ≥ 1, we find such manifolds with w_{4k-1} ≠ 0 in dimensions ≥ 4k+1. We also find closed non-cobordant hyperbolic manifolds in all dimensions ≥ 4 not of the form 4k+3.