Report Title: Fundamental Groups and Covering Spaces
Speaker: Andrey Lazarev, Lancaster University
Location: The Fifth Seminar Room, Mathematics Building, Jilin University
Abstract: This course covers the definition and basic properties of homotopy groups of topological spaces, a fundamental concept in algebraic topology. The special role of the first homotopy group, known as the fundamental group, will be explained. One of the most effective methods for computing the fundamental group relies on constructing the universal covering space. The theory of covering spaces will be developed up to and including their classification. Various concrete computational examples will be provided.
10:00–11:00, June 25, 2025 – Reminder on homotopy groups, examples. Commutativity of higher homotopy groups
09:00–10:00, June 26, 2025 – Covering spaces and their relation to the fundamental group of the base space
09:00–10:00, June 27, 2025 – Computing the fundamental group via universal covering spaces. Examples: wedges of circles, tori, Klein bottle, real projective spaces
09:00–10:00, June 28, 2025 – Classification of covering spaces
Speaker Biography: Andrey Lazarev is a Professor at Lancaster University, UK. His research focuses on algebraic topology and homotopy theory. He served as Editor-in-Chief of the Bulletin of the London Mathematical Society and has published numerous high-level papers in journals including Advances in Mathematics, Proceedings of the London Mathematical Society, and Journal of Noncommutative Geometry.
