On the Toral Rank Conjecture and Variants of Equivariant Formality

We investigate the topological consequences of actions of compact connected Lie groups. Our focus lies on the \emph{Toral Rank Conjecture}, which states that a suitable space $X$ with an almost free $T^r$-action has to satisfy $\dim H^*(X;\mathbb{Q})\geq 2^r$. We investigate various refinements of formality in an equivariant setting and show that they imply the TRC in several cases. Furthermore, we study the properties of the newly developed terminology with regards to possible implications, inheritance under elementary topological constructions, and characterizations in terms of higher operations on the equivariant cohomology. We also attack the problem of finding bounds for $\dim H^*(X;\mathbb{Q})$ in the spirit of the TRC outside of the formal context. Different lower bounds are constructed and applied in particular to the case of cohomologically symplectic spaces