2019 Advanced topics in Geometry C1

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Academic unit or major
Graduate major in Mathematics
Endo Hisaaki 
Class Format
Media-enhanced courses
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Syllabus updated
Lecture notes updated
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Course description and aims

The aim of this lecture is to familiarize the students with the basic language of
and some fundamental theorems for mapping class groups of surfaces.
This course will be succeeded by [MTH.B408 : Advanced topics in Geometry D].

Student learning outcomes

Students are expected to
・understand the definitions of mapping class groups, Dehn twists, and braid groups.


Mapping class groups, simple closed curves, Dehn twists, braid groups, Birman's theorem.

Competencies that will be developed

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

Class flow

Standard lecture course

Course schedule/Required learning

  Course schedule Required learning
Class 1 Manifolds and isotopies Details will be provided in class.
Class 2 Connected sums and handle decompositions Details will be provided in class.
Class 3 Construction and classification of surfaces Details will be provided in class.
Class 4 Mapping class groups of surfaces Details will be provided in class.
Class 5 The Dehn-Nielsen-Baer theorem Details will be provided in class.
Class 6 Curves on surfaces and Dehn twists Details will be provided in class .
Class 7 Braid groups and Birman's theorem Details will be provided in class.
Class 8 Evaluation of progress Details will be provided in class.


None required

Reference books, course materials, etc.

B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton University Press.

Assessment criteria and methods

Exams and reports. Details will be provided in class.

Related courses

  • MTH.B301 : Geometry I
  • MTH.B302 : Geometry II
  • ZUA.B301 : Geometry I
  • MTH.B341 : Topology
  • MTH.B408 : Advanced topics in Geometry D1

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

Students are expected to have passed [Geometry I], [Geometry II] and [Topology].

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