  To clarify the method behind the paper "Ganea's conjecture on
Lusternik-Schnirelman category" by the author, a generalisation of
Berstein-Hilton Hopf invariants is defined as `higher Hopf invariants'. They
detect the higher homotopy associativity of Hopf spaces and are studied as
obstructions not to increase the LS category by one by attaching a cone. Under
a condition between dimension and LS category, a criterion for Ganea's
conjecture on LS category is obtained using the generalised higher Hopf
invariants, which yields the main result of "Ganea's ..." for all the cases
except the case when $p=2$. As an application, conditions in terms of homotopy
invariants of the characteristic maps are given to determine the LS category of
sphere-bundles-over-spheres. Consequently, a closed manifold $M$ is found not
to satisfy Ganea's conjecture on LS category and another closed manifold $N$ is
found to have the same LS category as its `punctured submanifold' $N-\{P\}$, $P
\in N$. But all examples obtained here support the conjecture in "Ganea's ...".
