Suspension of Ganea fibrations and a Hopf invariant

We introduce a sequence of numerical homotopy invariants σicat , i∈N , which are lower bounds for the Lusternik–Schnirelmann category of a topological space X . We characterize, with dimension restrictions, the behaviour of σicat with respect to a cell attachment by means of a Hopf invariant. Furthermore we establish for σicat a product formula and deduce a sufficient condition, in terms of the Hopf invariant, for a space X∪ep+1 to satisfy the Ganea conjecture, i.e., cat ((X∪ep+1)×Sm)=cat (X∪ep+1)+1 . This extends a recent result of Strom and a concrete example of this extension is given.