# Elementary Topology (dt. Elementare Topologie)

 Level, degree of commitment Advanced module, compulsory elective module Forms of teaching and learning,workload Lecture (3 SWS), recitation class (1 SWS), 180 hours (60 h attendance, 120 h private study) Credit points,formal requirements 6 CP Course requirement(s): Successful completion of at least 50 percent of the points from the weekly exercises. Examination type: Written or oral examination (individual examination) Language,Grading German,The grading is done with 0 to 15 points according to the examination regulations for the degree program B.Sc. Mathematics. Duration,frequency One semester, irregular Person in charge of the module's outline Prof. Dr. Ilka Agricola

## Contents

• Topological spaces and manifolds
• Elementary properties of topological spaces: compactness, orientability, boundary. Many examples: Möbius band, Klein's bottle, projective space etc.
• Classification of surfaces, genus of a surface, triangulations, Boy's surface
• Euler Characteristic and Euler's polyhedron theorem
• Fundamental group, mapping degree and coverings

## Qualification Goals

Students will

• understand basic principles of topological structures and recognize that such structures are found in many parts of mathematics,
• have practiced axiomatic procedures and trained their ability to abstract,
• have practiced mathematical ways of working (developing mathematical intuition and its formal justification, reasoning),
• have improved their oral communication skills in exercises by practicing free speech in front of an audience and in discussion.

## Prerequisites

None. The competences taught in the following modules are recommended: either Foundations of Mathematics and Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis.

## Applicability

Module imported from B.Sc. Mathematics.

It can be attended at FB12 in study program(s)

• B.Sc. Mathematics
• M.Sc. Computer Science
• M.Sc. Mathematics
• LAaG Mathematics
• BA Minor Mathematics

When studying BA Minor Mathematics, this module can be attended in the study area Compulsory Elective Modules.

The module is assigned to variant A of the BSc minor program in Mathematics.

## Recommended Reading

• Boltjanskij, V.G. und Efremovic, V.A.: Anschauliche kombinatorische Topologie. VEB Deutscher Verlag der Wissenschaften (1986).
• Hatcher, A.: Algebraic topology. Cambridge University Press (2002).
• Hu, S.-T.: Homotopy Theory. Academic Press (1959).
• Ossa, E.: Topologie. Vieweg-Verlag (1992).
• Pontrjagin, L.S.: Grundzüge der kombinatorischen Topologie. VEB Deutscher Verlag der Wissenschaften (1956).
• Stöcker, R. und Zieschang, H.: Algebraische Topologie. Eine Einführung. Teubner-Verlag (1988).

## Please note:

This page describes a module according to the latest valid module guide in Winter semester 2023/24. Most rules valid for a module are not covered by the examination regulations and can therefore be updated on a semesterly basis. The following versions are available in the online module guide:

• Winter 2016/17 (no corresponding element)
• Summer 2018 (no corresponding element)
• Winter 2018/19 (no corresponding element)
• Winter 2019/20 (no corresponding element)
• Winter 2020/21 (no corresponding element)
• Summer 2021 (no corresponding element)
• Winter 2021/22 (no corresponding element)
• Winter 2022/23
• Winter 2023/24

The module guide contains all modules, independent of the current event offer. Please compare the current course catalogue in Marvin.

The information in this online module guide was created automatically. Legally binding is only the information in the examination regulations (Prüfungsordnung). If you notice any discrepancies or errors, we would be grateful for any advice.