Triangulations and combinatorial differential geometry
Student No.:50
Time:【Schedule updated】Tue 15:10-17:00/Fri 10:10-12:00
Instructor:Hyam Rubinstein  [University of Melbourne]
Place:Conference Room 3, Floor 2, Jin Chun Yuan West Building
Starting Date:2013-3-26
Ending Date:2013-4-26




【Schedule updated】Lecture on April 5 will be changed to April 11 at 15:10-17:00. 




The aim is to construct explicit triangulations, which reflect geometric and topological properties of the underlying manifold. Particular emphasis is on dimension 3 but some of the methods work well in all dimensions.



       A method will be given to verify when a triangulation of a 3-manifold is minimal, i.e. has the smallest number of tetrahedra. Such triangulations are often related to the underlying geometric structure.


       Even triangulations satisfy every codimension 2 face has even order. Even triangulations have a holonomy representation and by passing to the holonomy cover, have an n-colouring of the codimension 2 faces.


       In dimension 3 there are many explicit constructions of even triangulations related to Z2 torsion in homology.


       In all dimensions, the n-colouring of an even triangulation leads to a simple condition implying the manifold has a polyhedral Cartan-Hadamard structure. In particular, the universal covering is Euclidean n-space.


       Essential triangulations have the property that there is a single finite vertex or only ideal vertices and each edge is homotopically non-trivial. If no two edges are homotopic, the triangulation is strongly essential. Geodesic triangulations of non-positively curved manifolds are strongly essential and, if a manifold has a metric of non-positive (constant) curvature then it has such an essential (strongly essential) triangulation.


       Haken 3-manifolds all have explicitly constructed strongly essential triangulations.


•       Angle structures introduced by Casson and Rivin are weak versions of Thurston’s ideal hyperbolic triangulations in dimension 3. Angle structures will be constructed, using Agol’s veering triangulations and Epstein-Penner ideal cell decompositions of cusped hyperbolic 3-manifolds. (There is a Z2 homology obstruction in the latter case).



       A canonical family of ideal triangulations with semi-angle structures will be used to show that the index structures introduced by Dimofte-Gaiotto-Gukov are topological invariants of cusped hyperbolic 3-manifolds.


•       A new construction of polyhedral almost Fuchsian surfaces will be given. In hyperbolic manifolds, smooth almost Fuchsian surfaces have all principal curvatures at most one and their fundamental groups inject into the fundamental group of the 3-manifold. Such surfaces play a crucial role in recent work of Kahn-Markovic and were previously studied by Uhlenbeck, Taubes, Schlenker etc.