# A road map through group cohomology

I am a PhD student in algebraic topology, and I would like to learn something about group cohomology.

The final goal would be to present one or two seminars on this topic, in order to give my mates a gently introduction to this subject and at the same time showing them some striking result/application of this theory. Ideally, my plan for the seminar is:

1. Introduce group cohomology, with a lot of motivations and examples
2. Explain what makes group cohomology awesome
3. Focus on a specific result, and showing some pretty applications of it (something that could be interesting to an algebraic topologist if possible) in order to strenghthen point 2

I am not looking for books, which are already given in these questions:

https://math.stackexchange.com/questions/2697778/reference-for-group-cohomology

https://math.stackexchange.com/questions/695613/reference-request-introduction-to-finite-group-cohomology?rq=1

So my questions are:

1. Does anyone know any introductory papers/lecture notes where I can find a concise introduction to group cohomology? I am looking for something which do not contains all the details but which gives me a general view of the main results and applications of the theory. Youtube videos/lecture series are also very welcome. Of course if you want to mention book that are not in the previous answers it is fine aswell.
2. Are there any suggestions about results/applications that I can put in points 2 and 3 of the seminar? As I said before the idea is to present this material to other students of algebraic topology, so I would prefer theorems/applications that will appeal to this kind of audience.

EDIT: an answer of this kind References and resources for (learning) chromatic homotopy theory and related areas is also very welcome and pertinent! Thank you in advance, Tommaso

• A nice application for topologists is the study of finite groups that can act freely on spheres, using some spectral sequence argument Mar 21 at 19:49
• Thank you a lot! Have you got some good reference for this topic? It seems perfect to my purpose! Mar 21 at 20:43
• I'm not sure about a reference, but Mike Miller explains here (math.stackexchange.com/questions/3106886/… - it can also be recovered from the spectral sequence my answer mentions) why the cohomology of such a group must be periodic; so that rules out lots of examples. Swan has a paper (namely Periodic Resolutions for Finite Groups) where he proves a sort of converse (that a group with periodic cohomology acts freely on a proxy-sphere) Mar 21 at 21:06
• Regarding 2, one nice feature of group cohomology is that it converts a nonabelian structure into a (graded) commutative one. As a result, you can apply the tools of commutative algebra to study nonabelian groups. This may be most helpful when using the mod $p$ cohomology of a group with order divisible by $p$. Mar 21 at 22:43
• Kevin Buzzard gives a Lean workshop right now and discusses group cohomology in this context, this might provide some useful intuitions from a somewhat unusual angle: xenaproject.wordpress.com/2021/03/15/… Mar 22 at 10:51