Algebraic Topology I

Algebraic Topology I

Semester
WiSe 2021 / 22

Lecturer
Clara Löh

Type of course (Veranstaltungsart)
Vorlesung

German title
Algebraische Topologie I

Contents
Algebraic topology studies topological spaces via algebraic invariants -- by modelling certain aspects of topological spaces in the realm of algebra, e.g., by groups and group homomorphisms. Classical examples include homotopy groups and (co)homology theories.

Algebraic topology has various applications, both in theoretical and in applied mathematics, for instance, through fixed point theorems, (non-)embeddability results, topological data analysis, and many more. For example, Nash's proof of existence of certain equilibria in game theory is based on a topological argument. Topics covered in this course include:

What is algebraic topology?
The fundamental group and covering theory
The Eilenberg-Steenrod axioms
Singular homology
Cellular homology
Classical applications of (co)homology.

This course will be complemented with a course "Geometric Group Theory" in the summer 2022. The course in SS 2022 can also be attended independently of the present course on Algebraic Topology. Moreover, there probably will also be a continuation of the Algebraic Topology Series.

If all participants agree, this course can be held in German; solutions to the exercises can be handed in in German or English.

Literature
will be announced at the beginning of the course

Recommended previous knowledge
Analysis I,II,(IV), Algebra (groups, rings, modules; basic homological algebra is helpful, but not strictly required)

Time/Date
Tue/Fri 8--10

Location
M 101

Course homepage
http://www.mathematik.uni-r.de/loeh/teaching/topologie1_ws2122/
(Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)

Registration

Preliminary registration for the organisation of exercise classes: at the end of the previous
semester via EXA or LSF (see announcement by the department)
Registration for the exercise classes: via GRIPS in the first week of the lecture period
Registration for course work/examination/ECTS: FlexNow

Course work (Studienleistungen)

Successful participation in the exercise classes: 50% of the credits, presentation of a
solution in class

Examination (Prüfungsleistungen)

Oral exam: Duration: 25 minutes, Date: individual, re-exam: Date: individual

Modules
BV, MV, MGAGeo

ECTS
9