Lecturer

Paul Biran

Coordinator

Patricia Dietzsch

Lectures

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

• Wednesday, 10:15-12:00
• Friday, 14:15-16:00

Link to zoom for the lectures: https://ethz.zoom.us/j/92946091050?pwd=eUtmR2RRSGtTZ1JKbWt1eDYvYkkzQT09
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.

Content

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

• singular homology
• cell complexes and cellular homology
• the Eilenberg-Steenrod axioms
• cohomology

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

Prerequisites

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

Exercises

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.

Old Exams

Literature

• G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. (Through the ETH Zurich network members of ETH can legally download this and other ebooks.)
• A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. See also: A. Hatcher.
• E. Spanier, "Algebraic topology", Springer-Verlag