Algebraic Topology I Autumn 2020

Paul Biran
Patricia Dietzsch


The course will be delivered entirely online via live zoom sessions without any frontal presence. The lectures will take place

Link to zoom for the lectures:
Meeting ID: 929 4609 1050
Password: atop1HS20

Please MUTE YOUR MICROPHONES during the lectures. In case you have a question, use the zoom option to "raise your hand".

Video recordings of the sessions will be provided.

There is a forum which can be used for discussion about the course: Link to the forum.


This is an introductory course in algebraic topology. Topics covered include:

Along the way we will introduce the basics of homological algebra and category theory.


You should know the basics of point-set topology.

Useful to have is a basic knowledge of the fundamental group and covering spaces (at the level usually covered in the course "topology"). Students not familiar with this topic can look this up, for example in Chapter 3, Section 1-6 and Section 8 in G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. (Members of ETH can legally download the ebook trough the ETH network.)

Some knowledge of differential geometry and differential topology is useful but not necessary.
Some (elementary) group theory and algebra will also be needed.

Lectures Content

Week Content
Week 1 Homotopy - definitions, basic constructions and properties
Week 2 Singular homology - definition, basic properties and first calculations
Week 3 Hurewicz theorem and a quick introduction to homological algebra - chain complexes and exact sequences
Week 4 Homology long exact sequence, relative homology and reduced homology
Week 5 Axiomatic approach to homology, singular homology with coefficients
Week 6 The homology of the sphere and related calculations, degrees of maps and applications (Brouwer fixed-point theorem)
Week 7 Calculation of degrees, local degrees
Week 8 Fundamental theorem of algebra, CW-complexes - definitions and examples
Week 9 Cellular homology
Week 10 Cellular maps, Euler characteristic and applications
Week 11 Homotopy axiom, cross product
Week 12 Excision axiom
Week 13 The Mayer-Vietoris long exact sequence, applications of singular homology
Week 14 Jordan separation theorem, invariance of domain, invariance of dimension


The new exercises will be posted here.

If you have any questions concerning the exercises, please don't hesitate to contact Patricia Dietzsch. She will also offer regular zoom office hours, during which you can ask questions. Office hours will take place in the weeks after a new exercise sheet is posted, on Thursdays at 16:15-17:00. If possible, please ask questions in advance, via email or in the forum. You may also ask questions spontaneously during the zoom session.

Exercise sheets Solutions Office hours
Exercise sheet 1 Solutions 1 Video and pdf
Exercise sheet 2 Solutions 2 Notes
Exercise sheet 3 Solutions 3 Notes
Exercise sheet 4 Solutions 4 Notes
Exercise sheet 5 Solutions 5 Notes
Exercise sheet 6 Solutions 6 Notes

Old Exams

Exam Solutions
Exam HS15 Solutions HS15
Exam FS16 Solutions FS16
Exam HS18 Solutions HS18