Is there a sequence of topological spaces $X_n$ (manifolds ideally), where the sum of the Betti numbers of $X_n$ remains bounded but the Lusternik–Schnirelmann category is unbounded, as $n \to \infty$? What about vice versa?

One might think of both of these numbers as very rough measures of the "complexity" of a space, and it is well known in particular that both quantities are lower bounds on the number of critical points of a Morse function. But it would be nice to hear about any facts governing the relations between them.