Cohomological consequences of (almost) free torus actions

The long-standing Halperin--Carlsson conjecture (basically also known as the toral rank conjecture) states that the sum of all Betti numbers of a well-behaved space $X$ (with cohomology taken with coefficients in the cyclic group $\zz_p$ in characteristic $p>0$ respectively with rational coefficients for characteristic 0) is at least $2^n$ where $n$ is the rank of an $n$-torus $\zz_p \times \stackrel{(n)}...\times \zz_p$ (in characteristic $p>0$) respectively $\s^1 \times \stackrel{(n)}...\times \s^1$ (in characteristic 0) acting freely (characteristic $p>0$) respectively almost freely (characteristic 0) on $X$. This conjecture was addressed by a multitude of authors in several distinct contexts and special cases. However, having undergone various reformulations and reproofs over the decades, the best known general lower bound on the Betti numbers remains a very low linear one. This article investigates the conjecture in characteristics 0 and 2, which results in improving the lower bound in characteristic 0.