Algebraic Topology I, WS 2026/27

News

Registration for the exercise classes: in the first days of the winter term, via GRIPS
If you plan to write a bachelor thesis under my supervision in SS 2027 (in Topology/Geometry), you should participate in a seminar in the Global Analysis and Geometry group before SS 2027.

Algebraic Topology

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.

Moebius

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.

There will be suitable follow-up courses in summer 2027.

Lecture notes (updated after each lecture):

pdf;
ocg-free pdf;
correction log.
Intro slides: pdf

Time/Location

Monday, 10:15--12:00, M 104,
Thursday, 8:25--10:00, M 104.
If you absolutely cannot attend in person: A zoom stream will be available (access data on GRIPS). No recordings will be provided and I do not consent to recordings being made of these lectures.

Exercise classes

Time/Location:
Mo 8--10, M 104
Tue 8--10, M 009

Please register before October 14, 10:00, via GRIPS for the exercise classes!
The exercise classes start in the second week.
Organisational matters

Exercise sheets

Solutions can be submitted in English or German and in teams of up to two people. Please do not forget to add your name to all your submissions and adhere to the standards outlined here!

Sheet 1 of October 15, 2026. Submission by October 22, 2026.

Literature

This course will not follow a single book. Therefore, you should individually compose your own favourite selection of books.
A list of suitable books can be found in the lecture notes.
Please note that throughout Algebraic Topology different conventions are in use and that there are substantially different routes to developing the canon of fundamental results in Algebraic Topology.

Prerequisites

All participants should have a firm background in Analysis I/II (in particular, basic point set topology, e.g., as in Analysis II in WS 2011/12), in Linear Algebra I/II, and basic knowledge in group theory (as covered in the lectures on Algebra).
Knowledge about manifolds as in Analysis IV is not necessary, but helpful.
Knowledge about basic homological algebra (as in the last two weeks of Kommutative Algebra) is not necessary, but helpful.

Exams

Oral exams (25 minutes). Dates: tba

Last change: October 15, 2026