Universität Regensburg

WiSe 23/24: Algebraic Topology I

Lectures: Tuesday 08-10 and Thursday 10-12 in M101
Exercises: Friday 12-14 in M101

GRIPS Seite

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:

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

Exercises

Lecture notes

From WiSe 20/21:

References