Ämnesdisposition

• Course information

Homological algebra is a versatile branch of algebra with applications in algebraic geometry, group theory, differential geometry, and, most notably, algebraic topology, which is why we offer a course combining these two topics. The central notion of homological algebra is the derived functor. Functors are an abstraction of the idea of a function between classes of mathematical objects that also transform maps: for instance, the assignment of the fundamental group to a topological space: any continuous map between spaces induces a group homomorphism between the fundamental groups. Homological algebra is mostly about functors between R-modules for various rings R. These functors can behave nicely (send injections to injections and surjections to surjections, among other conditions), but they do not have to. If they do not, they have one or more higher derived functors. Those derived functors are often computable and contain much information about the original functor.

Algebraic topology is a continuation of the idea of the fundamental group, i.e. of assigning a group (or other simple algebraic object) to a space in order to measure its properties. We will look at the homology groups of a space, which are invariants defined by using the machinery of homological algebra. These groups are easier to compute than the fundamental group but nevertheless powerful invariants. Here are some striking results that are easy to prove with the basic tools of algebraic topology:

• There are always two opposite points on the earth with the exact same temperature and humidity.
• There is always a place on the earth with no wind.
• However messily made, any sandwich with bread, cheese, and tomato can be cut by a straight cut into two halves with the exact same amount of bread, cheese, and tomato in each half.
Course syllabus

• Schedule

Tuesdays 15:15 - 17:00, see TimeEdit.

August 29 - October 10, KTH, V35 (Boij)

October 31 - December 12, SU, room 306, building 6 (Berglund)

Preliminary plan
Lecture Topic
1 Modules
2Complexes and homology
3 Hom-functors
4 Projective resolutions and Ext
5 Injective resolutions and Ext
6 Tensor products
7 Tor, flat modules and syzygies
8
Topological spaces (7.1-7.7)
9
Singular homology (8.1-8.4, 7.13)
10
Homotopy invariance of homology (7.12, 8.5)
11
Mayer-Vietoris sequence (8.6-8.7)
12
Homology of spheres and applications (8.8-8.10)
13
Cell attachments, cellular homology (7.9, 8.11)
14
Real projective spaces and their homology (7.10, 8.12-8.13)

• Course literature

Homological Algebra and Algebraic Topology by Wojciech Chachólski and Roy Skjelnes.

The assessment and grading will be based on weekly homework assignments. There will be fourteen assignments each giving a maximum of ten points. The twelve best results will be added and the minimum scores required for each grade are given by the following table:

ABCDEFXF
10090807060550

A score in the range 55 to 59 is a failing grade with the possibility to improve to an E grade by additional homework assignments.

• Homework assignments

The solutions to the homework assignments should be written in English and sent by email as pdf to Mattias Grey (mgrey@math.su.se).

Each homework is due before the beginning of the lecture in the following week. Late homework will not be accepted unless you have a compelling reason you cannot turn in the homework in time. Even in that case, you must get permission from the lecturer before the original deadline.

Discussing the homework problem with each other is admissible and even encouraged, but you have to formulate your solutions separately. This kind of cooperation should be clearly declared by all the participants in their homework. Identical or nearly identical solutions or solutions copied from sources on the internet are not acceptable.

The solutions should be based on material from the relevant chapter in the course literature or courses that are prerequisites to this course.

The homework assignments will be posted here one week before they are due.