Classes are held in ML E 12 on Wednesdays 1012, and in HG G 3 on Fridays 1315.
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 chaincomplex, universal coefficient theorem 
Feb. 26  Lemma 3.1, pages 197198  Ext doesn't depend on the resolution, definition of cohomology of a space 
Feb. 28  Pages 199204  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 185186.) 
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

Examples of interesting cup products: projective plane and torus 
March 11  Pages 210212 
Cohomology ring, graded and commutative graded rings, the cohomology ring is commutative graded 
March 13  Pages 213216 
Examples of cohomology rings, cross product, tensor products, statement of the Kuenneth formula 
March 18  Pages 216218 
Outline of proof of the Kuenneth formula 
March 20  Pages 209210, Theorem 3.18 (statement), and pages 220222. 
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)


April 3  Pages 224225 
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 233234 
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 234236 
Orientation cover (proof), Rorientability, structure of top dimensional homology (statement), fundamental class 
April 29  Pages 236237 
Structure of top dimensional homology (proof) 
May 6  Pages 239245 
Cap product, Poincare duality, cohomology with compact support 
May 8  Pages 246249 
Proof of Poincare' duality (not part of the exam) 
May 13  Pages 254256, and Corollary 3.28+Proposition 3.29 
Alexander duality and related results (not needed for the exam) 
May 15  Pages 525527 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 8791 
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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