Lusternik-Schnirelmann category of a sphere-bundle over a sphere
  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. 
Norio Iwase