Algebraic Topology II Spring 2024

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.

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

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.

Exercises

Exercise sheets and solutions will be posted here, and will be discussed during office hours.

Sheet 1 Solutions 1
Sheet 2 (typo in 4a) corrected)
Sheet 3
Sheet 4 (typos in 1 and 5a) corrected)

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.

Lecture Contents

The hand-written lecture notes will be posted here before every lecture. Here are all lecture notes in a single file. All comments and corrections are highly welcome!

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, 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
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 2)
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 2)
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 2)
22 March: Lecture 10
  • 5. Cohomology
 Notes
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
12 April: Lecture 13
  • 6. The cup product
 Notes
17 April: Lecture 14
  • 6. The cup product
 Notes
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
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)
8 May: Lecture 19
10 May: Lecture 20
15 May: Lecture 21
17 May: Lecture 22
22 May: Lecture 23
24 May: Lecture 24
29 May: Lecture 25
31 May: Lecture 26

Literature

Last update: 1 May 2024.