# Elementary Topology (dt. Elementare Topologie)

 Level, degree of commitment Advanced module, depends on importing study program 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. Origin 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

The module 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 B.Sc. Mathematics, this module can be attended in the study area Compulsory Elective Modules in Mathematics.

The module can also be used in other study programs (export module).

• 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).