Hopf invariants and their application

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1. We obtained an affirmative answer to "the Ganea conjecture" on co-H-space of low dimensions, In the proof, we found an fundamental theorem on a co-Hopf space, However a counter-example was found in higher dimensions, The above fundamental theorem is used to show the example is a co-H-space.2. L-S category - a numrical invariant defined by Lusternik and Schnirelmann is studied in terms of "higher Hopf inuariuat" and "A_∞-structures on loop spaces". As a result, we found a counter exampled to Gamea's conjecture on L-S category as a simply connected closed manifold.3. when a space X is in "meta stable" range, End (ΣX) is not a ring but a "square" ring. We determine the structure of End (ΣX) completely in terms of a Hopf invariant, if X is a mapping cone of a sphere-mapping (in meta stable range).