Classes are held in ML E 12 on Wednesdays 10-12, and in HG G 3 on Fridays 13-15.
This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology including: cohomology of spaces, operations in homology and cohomology, duality.
General topology, linear algebra, singular homology of topological spaces (e.g. as taught in "Algebraic topology I").
Some knowledge of differential geometry and differential topology is useful but not absolutely necessary.
| Day | Pages | Summary |
|---|---|---|
| Week 1 (Feb. 19 and 21) | Hatcher, page 190 to corollary 3.3 (except the proof of Lemma 3.1) | Moral introduction to cohomology, cohomology of a chain-complex, universal coefficient theorem |
| Feb. 26 | Lemma 3.1, pages 197-198 | Ext doesn't depend on the resolution, definition of cohomology of a space |
| Feb. 28 | Pages 199-204 | Basic properties of cohomology, dual to those for homology |
| March 4 | Section 3.2 until Example 3.7 excluded, plus Proposition 3.10. (More discussion on pages 185-186.) |
Definition and basic properties of cup products |
| March 6 | Examples 3.7 and and 3.8 (both more general than discussed in class)
Note: We will develop tools to make those computations in different ways
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Examples of interesting cup products: projective plane and torus |
| March 11 | Pages 210-212 |
Cohomology ring, graded and commutative graded rings, the cohomology ring is commutative graded |
| March 13 | Pages 213-216 |
Examples of cohomology rings, cross product, tensor products, statement of the Kuenneth formula |
| March 18 | Pages 216-218 |
Outline of proof of the Kuenneth formula |
| March 20 | Pages 209-210, Theorem 3.18 (statement), and pages 220-222. |
Another version of the relative cup product, cohomology of projective spaces |
| March 25 | Exercise Class 1 | Notes on Exercise Class 1 and Recording of Exercise Class 1 on Zoom |
| March 27 | Example 3.20, Theorem 3.21 |
If R^n is a division algebra, then n is a power of 2 |
| April 1 |
The complex Hopf map (not needed for the exam) Notes on Lecture 12 and Recording of Lecture 12 on Zoom App for visualization of the Hopf map (by Samuel J. Li)
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| April 3 | Pages 224-225 |
James reduced product (not needed for the exam) |
| April 8 | Exercise Class 2 | Notes on Exercise Class 2 and Recording of Exercise Class 2 on Zoom |
| April 22 | Pages 231 and 233-234 |
Local orientations and orientations, orientation cover (statement only) Notes on Lecture 14 and Recording of Lecture 14 on Zoom: Part 1 and 3, Recording of Lecture 14 on Zoom: Part 2 |
| April 24 | Pages 234-236 |
Orientation cover (proof), R-orientability, structure of top dimensional homology (statement), fundamental class |
| April 29 | Pages 236-237 |
Structure of top dimensional homology (proof) |
| May 6 | Pages 239-245 |
Cap product, Poincare duality, cohomology with compact support |
| May 8 | Pages 246-249 |
Proof of Poincare' duality (not part of the exam) |
| May 13 | Pages 254-256, and Corollary 3.28+Proposition 3.29 |
Alexander duality and related results (not needed for the exam) |
| May 15 | Pages 525-527 and 255 |
Euclidean neighborhood retracts (not needed for the exam) |
| May 20 | Exercise Class 3 | Notes on Exercise Class 3 and Recording of Exercise Class 3 on Zoom |
| May 27 | Pages 87-91 |
Aspherical spaces, definition of group homology and cohomology (not needed for the exam) |
| May 29 |
Central extensions (not needed for the exam) |
The new exercises will be posted here.
| Exercise sheet | Solutions | Notes |
|---|---|---|
| Exercise sheet 1 | Solutions 1 | |
| Exercise sheet 2 | Solutions 2 | Small typo in the solution of exercise 4 corrected |
| Exercise sheet 3 | Solutions 3 | Typos in exercise 3&4 corrected |
| Exercise sheet 4 | Solutions 4 | Added a sentence in the solution of exercise 4 |
| Exercise sheet 5 | Solutions 5 | Added a paragraph in the solution of exercise 3 |
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