TITLE: L-S category versus Betti numbers
QUESTION [7 upvotes]: 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.

REPLY [4 votes]: 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.