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.

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).

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 it’s cone length, hence category, is at most the number of dimensions with nonzero (co)homology.

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.