# a comparison between LS and cohomological dimension

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

• I very much doubt that the dimension of cohomology is greater – tarik Nov 1 '17 at 23:42
• also it is possible to using $nil ker\mu_A$ where $A$ is a CDGA of the rational homotopy type of the space $X$ with multiplication $\mu_A$, because $TC(X_\Bbb{Q})\leq~nil ker\mu_A$ – tarik Nov 3 '17 at 0:13

If $X$ has torsion in its cohomology, there’s nothing you can say. If $X$ is simply-connected with torsion free (co)homology then it has a homology decomposition, showing that its cone length, hence category, is at most the number of dimensions with nonzero (co)homology. Thus we can say $$\mathrm{cat}(X) \leq \dim ( H^*(X;\mathbb{Q}).$$ More can be said with information about the distribution of the nonzero groups (if there's clumping).

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.

• The Hilali conjecture holds for any coformal space X whose rational homotopy Lie algebra L is of nilpotency 1 or 2. so coformal under condtion – tarik Nov 4 '17 at 20:20