Lusternik–Schnirelmann category of a sphere-bundle over a sphere

We determine the Lusternik–Schnirelmann (L–S) category of a total space of a sphere-bundle over a sphere in terms of primary homotopy invariants of its characteristic map, and thus providing a complete answer to Ganeaʹs Problem 4. As a result, we obtain a necessary and sufficient condition for a total space N to have the same L–S category as its ‘once punctured submanifold’ N\{P}, P∈N. Also, necessary and sufficient conditions for a total space M to satisfy Ganeaʹs conjecture are described.