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.



L-S category versus Betti numbers
The 2d surfaces have unbounded Betti numbers, but bounded category.


A matrix in $SL\_n(\mathbb Z)$ describes a diffeomorphism of the $n$-torus. We may form the mapping torus, a bundle of tori over the circle, with monodromy the matrix. If the matrix is generic, so that none of its exterior powers have eigenvalues that are roots of unity, then the homology of the manifold will be the same as that of $S^1\times S^n$, hence total Betti number $4$. But its universal cover has cup length about $n$, hence large category.


