We give a brief introduction to the Toral Rank Conjecture and some cohomological tools for the study of actions of compact Lie groups. The long-standing conjecture states that the sum of the betti numbers of a space with a free torus action is at least as big as the sum of the betti numbers of the acting torus.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/TopologySeminar