The subject of the course is the fundamental groups of 3-manifolds. In the resolution of the virtual Haken conjecture by Agol and Wise the fundamental group of every hyperbolic 3-manifold is shown to enjoy subgroup separability. After introducing canonical decompositions of 3-manifolds we discuss key results and ideas for the recent progress on their fundamental groups. We also overview a proof of the virtual fibering conjecture.
The fundamental group is a fundamental notion not only in topology but widely in mathematics. Taking a survey of the topology of 3-manifolds, with an emphasis on their fundamental groups, students are expected to gain more insight into the significance of the study of the fundamental group and its applications.
・Understand basic properties of surface groups
・Understand canonical decompositions of 3-manifolds along spheres and tori
・Understand the relation between special cube complexes and right-angled Artin groups
・Understand the statements of the virtual Haken conjecture and the virtual fibering conjecture
3-manifold, fundamental group, essential surface, JSJ decomposition, special cube complex, subgroup separability, sutured manifold
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
This is a standard lecture course. There will be some assignments.
|Course schedule||Required learning|
|Class 1||The following topics will be covered in this order: -- surface groups, subgroup separability -- prime decomposition, essential surfaces -- JSJ decomposition, geometrization -- special cube complexes, right-angled Artin groups -- overview of a proof of the virtual Haken conjecture -- Thurston norm, sutured manifolds -- overview of a proof of the virtual fibering conjecture||Details will be provided during each class.|
M. Aschenbrenner, S. Friedl and H. Wilton, 3-Manifold Groups, EMS Series of Lectures in Mathematics, 2015
B. Martelli, An Introduction to Geometric Topology, CreateSpace Independent Publishing Platform, 2016