In spring 2022, I am teaching MATH820074 “Homological algebra and geometric applications”. The main topic of the course will be “Cohomology of groups“. We use [Br82] as the main resource of the lecture. [Bi81] and [Gr70] are also useful. Here is a quote from Robert Bieri on “How to Learn Mathematics” which could be equally useful.
Time: 9:55 – 12:30 every Monday at Tencent meeting, id: 818-5846-9116.
Office hours: By appointment.
Homework is due one week after it is assigned and please submit on elearning.
Final project: Please write down whatever you know about this problem.
Week 2, Feb. 28th, Review of some homological algebra II. Note 2, Homework 2: [Br82], p.12, Exe 1.
Week 3, Mar. 7th, Free resolution. Note 3, Homework 3: [Br82], p.17, Exe 3.
Week 5, Mar. 21st, Homology of a group with constant coefficients， Note 5, Homework 5: [Br82], p.36, Exe 1.
Week 6, Mar. 28th, Functoriality, homology of amalgamated free product, Note 6, Homework 6: [Br82], p.52, Exe 3.
Week 7, Apr. 2nd, (Co)Homology of groups with Coefficients I. Note 7 Homework 7: [Br82] p.56, Exe 1 and 2 (for exe 1, you need to first prove that a Z-module is flat if and only if it is torsionfree).
Week 8, Apr. 11th, (Co)Homology of groups with Coefficients II. Note 8, Homework 8: [Br82], p.67, Exe.
Week 10, (Co)Homology of groups with Coefficients IV. Note 10, Homework 10, [Br82], p.83 Exe 1.
Week 11, May 3rd, self reading.
Week 13, May 14th, self reading.
Week 14, May 16th, Spectral sequences I Note 12.
Week 15, May 23th, Spectral sequences II. Note 13.
[Bi81] Robert Bieri, Homological dimension of discrete groups. Second edition. Queen Mary College Mathematics Notes. Queen Mary College, Department of Pure Mathematics, London, 1981. iv+198 pp.
[Br82] Kenneth S. Brown, Cohomology of groups. Graduate Texts in Mathematics, 87. Springer-Verlag, New York-Berlin, 1982. x+306 pp.
[Gr70] Karl W. Gruenberg, Cohomological topics in group theory. Lecture Notes in Mathematics, Vol. 143 Springer-Verlag, Berlin-New York 1970 xiv+275 pp.