This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology including: homology with coefficients, cohomology of spaces, and Poincaré duality. See also Sara Kalisnik's previous course.
The oral exam will start with one randomly chosen question from this list, and continue with questions that might not be on the list.
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.
The lectures will take place
All lectures are recorded. The recordings are available here.Office hours are held every week (except during Easter week) on Tuesdays 15:1517:00 by Semyon Abramyan in his office HG FO 27.6. Please let Semyon know by email at what time you are going to come.
There will be another office hour on Tuesday, 23 July, from 13:0015:30 held by Lukas Lewark in his office HG G 66.3. You can send questions in advance by email, or just drop in.
Exercise sheets and solutions will be posted here, and will be discussed during office hours.
Here are all exercise sheets and solutions in a single file.
You may pick exercises from the sheets, write up their solutions cleanly in LaTeX, and email them to Lukas Lewark for feedback. After some polishing, your solution will then appear on the metaphor page, so everyone can profit from it. Only one student's solution per exercise: please refer to the following list, and check with Lukas Lewark beforehand that your exercise is really still free.
All solutions are now online.
The handwritten lecture notes will be posted here before every lecture.
Here are all lecture notes in a single file (version 8 from 6 August 2024).
All comments and corrections are highly welcome, even after the end of the lecture!Date  Topics  References  Material 

21 February: Lecture 1 

Tensor products:

Notes, Clicker 
23 February: Lecture 2 

Category Theory:

Notes 
28 February: Lecture 3 

Axioms for homology:

Notes (version 2), Clicker 
1 March: Lecture 4 

3. Calculations and the theorem of BorsukUlam:

Notes (version 3) 
6 March: Lecture 5 

Notes  
8 March: Lecture 6 

4. The Universal Coefficient Theorem for homology:

Notes (version 4) 
13 March: Lecture 7 

Notes, Clicker  
15 March: Lecture 8 

Notes (version 3)  
20 March: Lecture 9 

Cohomology:

Notes (version 3) 
22 March: Lecture 10 

Notes (version 2)  
27 March: Lecture 11 

Notes (version 2)  
29 March – 5 April  Easter  
10 April: Lecture 12 

Cup product:

Notes (version 2) 
12 April: Lecture 13 

Notes  
17 April: Lecture 14 

Notes (version 2)  
19 April: Lecture 15  No lecture.  
24 April: Lecture 16 

Manifolds and orientations:

Notes (version 2) 
26 April: Lecture 17 

Notes  
1 May  Labour Day  
3 May: Lecture 18 

Poincaré Duality:

Notes (version 2) 
8 May: Lecture 19 

Notes (version 3)  
10 May: Lecture 20 

Notes (version 2)  
15 May: Lecture 21 

Notes (version 2)  
17 May: Lecture 22 

Notes (version 2)  
22 May: Lecture 23 

Notes  
24 May: Lecture 24 

Alexander Duality:

Notes 
29 May: Lecture 25 

Notes  
31 May: Lecture 26 

Künneth Theorem:

Notes 
Last update: 6 August 2024.