  A criterion to determine the L-S category of a total space of a sphere-bundle
over a sphere is given in terms of homotopy invariants of its characteristic
map, and thus providing a complete answer to Ganea's Problem 4. As a result, we
obtain a necessary and sufficient condition for such a total space $N$ to have
the same L-S category as its `once punctured submanifold'
$N\smallsetminus\{P\}$, $P \in N$. Also a necessary condition for such a total
space $M$ to satisfy Ganea's conjecture is obtained.
