2017 Special lectures on advanced topics in Mathematics I

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Academic unit or major
Graduate major in Mathematics
Endo Hisaaki   
Class Format
Media-enhanced courses
Day/Period(Room No.)
Intensive ()  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
Language used
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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


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

Prerequisites (i.e., required knowledge, skills, courses, etc.)

None required

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