Resumen de Deformacions i representacions de varietats hiperbòliques no orientables de baixa dimensió
Juan Luis Durán Batalla
In this work, we extend several known results on hyperbolic 3-manifolds and surfaces to the non-orientable case and compare them to its orientable counterpart. In particular, we focus on the deformation space of a non-orientable hyperbolic 3-manifold of finite volume, the metric completion of said deformations and the variety of representation of the Klein bottle and, more generally, of any closed non-orientable surface.
We are interested in studying the local structure of the deformation space. We approach the subject through two sides, from geometric ideal triangulations and the variety of representations. Our main result on the topic is that, for the one non-orientable cusped case, the deformation space of an ideal triangulation is homeomorphic to a half-open interval whereas deformations of representations are homeomorphic to an open interval of the real line. If we consider an orientable cusp (in a non-orientable manifold), the discrepancy is no longer observed and we obtain that its deformations are homeomorphic to an open set of $\mathbb{C}$.
The deformations of non-orientable cusps are related to different representations of a Klein bottle which we call type I and II. The completion of the end is either a solid Klein bottle or disc orbi-bundle for respective representations of type I and II. Furthermore, deformations of a geometric ideal triangulation can only yield representations of type I.
On the other hand, we study in depth the variety of representations of the Klein bottle and, more generally, we compute the number of connected components of the variety of representations of any closed non-orientable surfaces. For the surface of genus $k$, there are $2^{k+1}$ connected components, which are distinguished by the first and second Stiefel-Whitney class of the associated principal bundle.
