I learned that "If $G$ is a finite group acting freely and continuously on $S^n$, the sphere then $G$ has periodic cohomology".


My question is: Are there any other similar theorems relating the free action of a finite group and its cohomology? 


maybe this question is too vague.



Fixed-point-free action and cohomology of a finite group
You may want to check out Alex Adem's paper:


*Adem, Alejandro*, [**Cohomological restrictions on finite group actions**](http://dx.doi.org/10.1016/0022-4049(88)90025-4), J. Pure Appl. Algebra 54, No.2-3, 117-139 (1988). [ZBL0686.57023](https://zbmath.org/?q=an:0686.57023).
If you look at the papers which cite this one on *Google Scholar* some wisdom may likewise ensue.


