A∞-method in Lusternik–Schnirelmann category

To clarify the method behind (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623–634), 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 Ganeaʹs conjecture on LS category is obtained using the generalised higher Hopf invariants, which yields the main result of (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623–634) 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 Ganeaʹs conjecture on LS category and another closed manifold N is found to have the same LS category as its ‘punctured submanifold’ N−{P}, P∈N. But all examples obtained here support the conjecture in (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623–634).