Algebraic Topology II Spring 2021

Paul Biran
Patricia Dietzsch


The lectures will take place

Link to zoom for the lectures:
Meeting ID: 929 4609 1050
Password: atop2FS21

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 a continuation course to Algebraic Topology I . The course will cover more advanced topics in algebraic topology including:


You should know the basics of point-set topology and the content of the course Algebraic Topology I .

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 Tensor products, homology with coefficients
Week 2 Borsuk-Ulam theorem and applications, cochain complexes and cohomology
Week 3 Cohomological universal coefficient theorem, Ext
Week 4 More on Ext, homological universal coefficient theorem, Tor
Week 5 More on Tor, Cohomology of spaces (homotopy equivalence, excision, cellular cohomology, Mayer-Vietoris)
Week 6 Homology cross product
Week 7 Eilenberg-Zilber theorem, Künneth formula, cohomological cross product
Week 8 Cohomological cross product continued, commutativity and associativity
Week 9 Cup product, Alexander-Whitney diagonal approximation
Week 10 Example of computation of cup product, cap product
Week 11 Application of cup product, manifolds, local homology
Week 12 Fundamental class, orientability
Week 13 Cohomology with compact support, Poincaré duality
Week 14 Proof of Poincaré duality, applications of Poincaré duality


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. If possible, please ask questions in advance, via email or in the forum. You may also ask questions spontaneously during the zoom session.

Exercise sheet Solutions Office hours
Exercise sheet 1
Exercise sheet on tensor products
Solutions 1
Solutions for the tensor product exercises
Exercise sheet 2 Solutions 2 Notes
Exercise sheet 3 Solutions 3 Notes 13.5.
Notes 20.5.
Exercise sheet 4 Solutions 4 Thursday, 17.6. at 16:15 via Zoom