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.

**Schedule**:

Week 1, *Feb. 21st, Review of some homological algebra I. * Note 1, Homework 1

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 4, Mar. 14th, Projectives, co-invariants and Definition of H_ast(G), Note 4, Homework 4

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 9, Apr. 18th, (Co)Homology of groups with Coefficients III, Note 9, Homework 9.

Week 10, (Co)Homology of groups with Coefficients IV. Note 10, Homework 10, [Br82], p.83 Exe 1.

Week 11, May 3rd, self reading.

Work 12, May 9th, low dimensional cohomology and group extension. Note 11, Homework 11.

Week 13, May 14th, self reading.

Week 14, May 16th, Spectral sequences I Note 12.

Week 15, May 23th, Spectral sequences II. Note 13.

**References:**

[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.