Lecturer

Lukas Lewark email

Coordinator

Semyon Abramyan email

Content

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.

Exam

The oral exam will start with one randomly chosen question from this list, and continue with questions that might not be on the list.

Prerequisites

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.

Lectures

The lectures will take place

• Wednesday, 10:15-12:00, ML E12, and
• Friday, 14:15-16:00, HG G3.

All lectures are recorded. The recordings are available here.

Office hours

Office hours are held every week (except during Easter week) on Tuesdays 15:15-17:00 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 13:00-15:30 held by Lukas Lewark in his office HG G 66.3. You can send questions in advance by email, or just drop in.

Exercises

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.

Sheet 1	Solutions 1
Sheet 2 (typo in 4a) corrected)	Solutions 2 (typo in 3b) corrected)
Sheet 3	Solutions 3 (typos in 1) and 8c) corrected)
Sheet 4 (typos in 1 and 5a) corrected)	Solutions 4 (typo in 2a) corrected)
Sheet 5	Solutions 5 (added details in 5), typo in 9b) corrected)
Sheet 6	Solutions 6

Solutions written by students

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.

Lecture Contents

The hand-written 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	Overview 1. Tensor products of modules	Tensor products: Spanier (Sec. 4 in Intro, and Sec. 1 in Ch. 5) Hatcher (Sec. "A Künneth formula" in Ch. 3.2) Atiyah-Macdonald (Sec. "Tensor product of modules" in Ch. 2)	Notes, Clicker
23 February: Lecture 2	1. Tensor products of modules Category theory intermezzo 2. Homology with coefficients	Category Theory: Weibel (Sec. 1.1 and 1.2) gives many more details than the lecture Hatcher (Sec. "Categories and Functors" in Ch. 2.3) Homology with coefficients: Spanier (Sec. 1 in Ch. 5) Hatcher (Sec. "Homology with Coefficients" in Ch. 2.2)	Notes
28 February: Lecture 3	2. Homology with coefficients	Axioms for homology: Spanier (Sec. 8 in Ch. 4) Hatcher (Sec. "Axioms for Homology" in Ch. 2.3) uses slightly different axioms	Notes (version 2), Clicker
1 March: Lecture 4	2. Homology with coefficients 3. Calculations and the theorem of Borsuk-Ulam	3. Calculations and the theorem of Borsuk-Ulam: Bredon (Sec. 20 in Ch. IV) Spanier (Sec. 8 in Ch. 5) uses cohomology instead Hatcher (Sec. "The Borsuk-Ulam Theorem" in Ch. 2.B)	Notes (version 3)
6 March: Lecture 5	3. Calculations and the theorem of Borsuk-Ulam		Notes
8 March: Lecture 6	4. The Universal Coefficient Theorem for homology	4. The Universal Coefficient Theorem for homology: Spanier (Sec. 2 in Ch. 5) Hatcher (Ch. 3.A) Weibel (Ch. 2 and 3) treats resolutions, Tor, Ext, etc. in greater generality. Sec 3.6 is about Universal Coefficients.	Notes (version 4)
13 March: Lecture 7	4. The Universal Coefficient Theorem for homology		Notes, Clicker
15 March: Lecture 8	4. The Universal Coefficient Theorem for homology		Notes (version 3)
20 March: Lecture 9	5. Cohomology	Cohomology: Spanier (Sec. 4 and 5 in Ch. 5) Hatcher (Ch. 3.1) Bredon (Ch. V) focuses on de Rham cohomology instead	Notes (version 3)
22 March: Lecture 10	5. Cohomology		Notes (version 2)
27 March: Lecture 11	5. Cohomology		Notes (version 2)
29 March – 5 April	Easter
10 April: Lecture 12	6. The cup product	Cup product: Hatcher (Ch. 3.2) Spanier (Sec. 6 in Ch. 5) defines the cup product in a different way (using the cross product) Same goes for Bredon (Sec. 4 in Ch. VI)	Notes (version 2)
12 April: Lecture 13	6. The cup product		Notes
17 April: Lecture 14	6. The cup product		Notes (version 2)
19 April: Lecture 15	No lecture.
24 April: Lecture 16	7. Manifolds and orientations	Manifolds and orientations: Hatcher (Ch. 3.3) Spanier (Sec. 2, 3 in Ch. 6) Bredon (Sec. 7 in Ch. VI)	Notes (version 2)
26 April: Lecture 17	7. Manifolds and orientations		Notes
1 May	Labour Day
3 May: Lecture 18	7. Manifolds and orientations 8. Poincaré Duality	Poincaré Duality: Hatcher (Ch. 3.3) Spanier (Sec. 2, 3 in Ch. 6) Bredon (Sec. 5 (cap product), Sec. 8 (duality), Sec. 10 (applications) in Ch. VI)	Notes (version 2)
8 May: Lecture 19	8. Poincaré Duality		Notes (version 3)
10 May: Lecture 20	8. Poincaré Duality		Notes (version 2)
15 May: Lecture 21	8. Poincaré Duality 9. Cohomology with compact support and proof of PD Duality		Notes (version 2)
17 May: Lecture 22	9. Cohomology with compact support and proof of PD Duality		Notes (version 2)
22 May: Lecture 23	9. Cohomology with compact support and proof of PD Duality		Notes
24 May: Lecture 24	9. Cohomology with compact support and proof of PD Duality 10. Alexander Duality	Alexander Duality: Hatcher (Section Other forms of duality in Ch. 3.3) Spanier (Sec. 2 in Ch. 6) Bredon (Sec. 8 in Ch. VI)	Notes
29 May: Lecture 25	10. Alexander Duality		Notes
31 May: Lecture 26	11. Künneth Theorem (not in exam) 12. Homology with twisted coefficients (not in exam)	Künneth Theorem: Hatcher (Section A Künneth Formula in Ch. 3.2) Spanier (Sec. 3 in Ch. 5, for homology) Bredon (Sec. 1 in Ch. VI)	Notes

Literature

• Topology and Geometry (GTM 139, Springer, 1997) by G. Bredon
• Algebraic Topology (Cambridge University Press) by A. Hatcher
• Algebraic Topology (Springer) by E. H. Spanier
• Introduction to Commutative Algebra (Addison-Wesley) by M. Atiyah and I. G. Macdonald (for modules over rings, and in particular their tensor products)
• An introduction to homological algebra (Cambridge University Press) by C. A. Weibel (for category theory and homological algebra).

Last update: 6 August 2024.