Abstract
A polyhedral manifold is a manifold with a metric induced by a constant curvature triangulation. Polyhedral manifolds naturally inherit a Riemannian structure, which is well-defined outside of a subset of codimension at least 2 called the singular locus. The fundamental group of the complement to this singular locus has a natural representation called the holonomy map, whose image we term the holonomy group. The main aim of this thesis is to investigate how restrictions on the holonomy group of a polyhedral 3-manifold relate to properties of its singular locus.In Chapter 2, we give most of the essential definitions and elementary results used throughout the thesis. These include the precise definitions of a polyhedral manifold, the singular locus, and holonomy.
In Chapter 3, we consider how restrictions on the holonomy group of a polyhedral 3-manifold affect the local and global properties of its singular locus. We study Euclidean polyhedral 3-manifolds that are nonnegatively curved and integral, two conditions motivated by Thurston’s work in [Thu98]. In Theorem 1, we classify the 32 isometry types of codimension 3 singularities in such manifolds. We also show, in Theorem 2, that the number of these singularities is bounded.
Lastly, in Chapter 4, we consider the reverse problem: how restrictions on the topology of the singular locus result in constraints on the holonomy group. We study spherical polyhedral manifolds homeomorphic to the 3-sphere, and we require that the singular locus form a Seifert link—this is a slight generalisation of a torus link. Motivated by Panov’s work in [Pan09], we investigate when such polyhedral 3-spheres can be shown to have unitary holonomy. In the case of the Hopf link, the investigation is comprehensive, allowing us in Theorem 3 to show that a polyhedral 3-sphere singular along the Hopf link has a very simple geometric structure in almost all cases. For a more general Seifert link, we impose only a very mild condition on the length of a singular component to show in Theorem 4 that the holonomy is unitary. This allows us to produce useful geometric formulae that apply to almost all polyhedral 3-spheres singular along Seifert links, generalising work of Kolpakov and Mednykh in [KM09], among others.
| Date of Award | 1 Mar 2023 |
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| Original language | English |
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| Supervisor | Dmitri Panov (Supervisor) & affiliated academic (Supervisor) |