L-S category versus Betti numbers 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.
 A: The dumb answer is infinite projective space, which has zero Betti numbers as classically computed (over $\mathbb{Q}$).  But what you really want is very small 
cohomology and very large L-S category.
Let $X$ be a noncontractible 
acyclic space.  Then all Betti numbers are zero (no matter what coefficients you use), 
and the same is true for the 
$n$-fold product $X^n$.  But a theorem of Hilton says (as I recall) that an 
$n$-fold product of noncontractible spaces must have category at least $n$.
On the other hand, if $X$ is simply-connected and $X$ has category $n$, then $X$
must have cohomology in at least $n$ different dimensions.
A: 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.
