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Low-dimensional topology
Spring semester 2022 in Fribourg

Lecture: Tuesday 10h-12h and Thursday 8h-9h.
Exercises: Thursday 9h-10h.
Location: room 2.52 in PER08 on Tuesdays and room 1.309 in PER07 on Thursdays.

Content

The aim of this course is to give an introduction to low-dimensional topology. The objects we study are one- two- and three-dimensional topological manifolds, their symmetries and the ways in which they embed in each other.

Particular topics we cover are the topological classification of compact surfaces, knot theory and classical knot invariants such as colorability, the signature or the Alexander polynomial, the mapping class group, and the Lickorish-Wallace theorem which states that a compact 3-manifold can be obtained by surgery on a link in the three-dimensional sphere.

Prerequisites

We assume knowledge of basic topological notions, including the fundamental group. Further knowledge of concepts from algebraic topology (such as homology groups) is advantageous, but will not be assumed.

Program

For our weekly topics and the exercise series, see the course's Moodle page.

Literature

The following references are the main sources for the course:

Burde, Zieschang, Heusener "Knots"
Farb, Margalit "A primer on mapping class groups"
Katok, Climenhaga "Lectures on surfaces"
Lickorish "An introduction to knot theory"

Further references for specific topics are given as the lecture progresses.