Universität Regensburg   IMPRESSUM   DATENSCHUTZ
Fakultät für Mathematik Universität Regensburg

Hinweis Bitte informieren Sie sich auf den jeweiligen GRIPS-Seiten über den digitalen Ablauf der Lehrveranstaltungen.

English Note For our digital courses all relevant information can be found on the appropriate GRIPS sites.

Algebraic Topology I
Semester
WiSe 2020 / 21

Lecturer
Marc Hoyois

Type of course (Veranstaltungsart)
Vorlesung

Contents

Algebraic topology studies topological spaces by means of algebraic invariants (groups, vector spaces, etc.), which allow us to reduce questions in topology to questions in algebra. Algebraic topology has many applications, both in theoretical and in applied mathematics. Nowadays, a basic knowledge of algebraic topology is essential in most other fields of pure mathematics, including analysis, algebraic geometry, and number theory. In applied mathematics, topological data analysis is a relatively new field that relies heavily on tools from algebraic topology.

In this first course on algebraic topology, we will study in depth two important invariants of a topological space: its fundamental group and its (co)homology groups. We will also see how to use these algebraic invariants to answer some interesting topological questions.

Topics covered in this course include:

  • Covering spaces and the fundamental group
  • Simplicial sets and singular (co)homology
  • CW complexes and cellular (co)homology
  • Miscellaneous applications (the fundamental theorem of algebra, Brouwer's fixed point theorem and invariance of domain, the hedgehog theorem, etc.)

This course is complemented by the seminar "de Rham cohomology". This seminar explores another approach to the cohomology of smooth manifolds via differential forms and proves Poincaré duality, which is a fundamental homological feature of smooth manifolds not shared by more general topological spaces.



Literature
A. Hatcher, Algebraic Topology, 2001
W. Lück, Algebraische Topologie: Homologie une Mannigfaltigkeiten, 2005

Recommended previous knowledge
Algebra (groups, rings, modules), topological spaces

Time/Date
Mi 10-12, Fr 08-10

Location
online

Course homepage
http://www.mathematik.ur.de/hoyois/WS21/algtop1/index.html
(Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)

Registration
  • Registration for the exercise classes: via GRIPS in the first week of lecture period
  • Registration for course work/examination/ECTS: FlexNow
Course work (Studienleistungen)
  • Successful participation in the exercise classes: 50% of points in the exercises, presentation
    of a solution in class
Examination (Prüfungsleistungen)
  • Oral exam: Duration: 25 minutes, Date: first week after the lecture period, re-exam: Date: by
    appointment
Additional comments
There will be a weekly exercise session, Fr 10-12

Modules
BV, MV, MArGeo, MGAGeo

ECTS
9