|Ideal triangulations of 3-manifolds and the deformation variety|
|Time：||Wed. 13:00-14:50; Fri. 8:00-9:50 (Last lecture on Apr. 23: 8:00-9:50)|
|Instructor：||Henry Segerman [University of Melbourne]|
|Place：||Conference Room 3, Floor 2, Jin Chun Yuan West Building|
This course will complement the course given by Professor Luo. We will start with the combinatorial structure of an ideal triangulation of a 3-manifold, and see how the complement of the figure 8 knot admits an ideal triangulation with two tetrahedra. Next we will define Thurston's gluing equations on an ideal triangulation, and the resulting deformation variety. The deformation variety is one way (currently, the most computationally effective way) of describing hyperbolic structures on the manifold. We will spend the remainder of the time on properties of the deformation variety, and its relation to combinatorial and topological properties of the triangulation and manifold. In particular, a solution to the gluing equations implies that all edges of the triangulation are essential, and ideal points of the deformation variety (often) correspond to incompressible surfaces in the manifold.
Some knowledge of algebraic topology and hyperbolic geometry would be useful.
Reference for the course:
William P. Thurston, The Geometry and Topology of Three-Manifolds (chapters 3 and 4), available from http://library.msri.org/books/gt3m/
Henry Segerman and Stephan Tillmann, Pseudo-Developing Maps for Ideal Triangulations I: Essential Edges and Generalised Hyperbolic Gluing Equations, http://arxiv.org/abs/1107.1030
Stephan Tillmann, Degenerations of ideal hyperbolic triangulations, http://arxiv.org/abs/math/0508295