TITLE: a comparison between LS and cohomological dimension
QUESTION [6 upvotes]: Let $X$ a simply connected elliptic space. Assume $\pi_\star(X)\otimes\Bbb{Q}$ is concentrated in odd degrees. So, we have $dim~\pi_\star(X)\otimes\Bbb{Q}=TC(X_\Bbb{Q})=catX_\Bbb{Q}$ (ie) the topological complexity and the Lusternik-Schnirelmann category. I want to compare the LS category or the $TC$ with $dim~H^\star(X;\Bbb{Q}).$ I just want an idea if it's possible

REPLY [2 votes]: The Hilali conjecture is a well-known conjecture in rational homotopy theory, which asks whether the inequality 
$$
\dim \pi_*(X)\otimes\mathbb{Q}\leq H^*(X;\mathbb{Q})
$$
always holds for $1$-connected rationally elliptic spaces  $X$. You ask about the oddly generated case. I believe the Hilali conjecture is still unkown in this generality, but is known to hold if in addition the space is coformal, meaning the differential in the minimal model is purely quadratic.