### 2017　Special lectures on advanced topics in Mathematics I

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Instructor(s)
Endo Hisaaki
Class Format
Lecture
Media-enhanced courses
Day/Period(Room No.)
Intensive ()
Group
-
Course number
MTH.E533
Credits
2
2017
Offered quarter
2Q
Syllabus updated
2017/3/17
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

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.

### Student learning outcomes

・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

### Keywords

3-manifold, fundamental group, essential surface, JSJ decomposition, special cube complex, subgroup separability, sutured manifold

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

### Class flow

This is a standard lecture course. There will be some assignments.

### Course schedule/Required learning

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.

None required

### Reference books, course materials, etc.

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

### Assessment criteria and methods

Assignments (100%)

### Related courses

• MTH.E639 ： Special lectures on current topics in Mathematics I
• ZUA.E343 ： Special courses on advanced topics in Mathematics I

None required