The lectures will take place
Link to zoom for the lectures: https://ethz.zoom.us/j/92946091050?pwd=SXJuTlhwZFRzVFU4U0pPMzhqL2o4Zz09Please 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.
| 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 |
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 |
Notes |
| 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 |