Ganea's conjecture is a now disproved claim in algebraic topology. It states that for all n>0, where is the Lusternik–Schnirelmann category of a topological space X, and Sn is the n-dimensional sphere. The inequality holds for any pair of spaces, X and Y. Furthermore,, for any sphere S^n, n>0. Thus, the conjecture amounts to. The conjecture was formulated by Tudor Ganea in 1971. Many particular cases of this conjecture were proved, and Norio Iwase gave a counterexample to the general case in 1998. In a follow-up paper from 2002, Iwase gave an even stronger counterexample, with X a closed smooth manifold. This counterexample also disproved a related conjecture, which stated that for a closed manifold M and p a point in M. A minimum dimensional counterexample to the conjecture was constructed by Don Stanley and Hugo Rodríguez Ordóñez in 2010. This work raises the question: For which spaces X is the Ganea condition,, satisfied? It has been conjectured that these are precisely the spaces X for which equals a related invariant, Furthermore, cat(X * S^n) = cat(X ⨇ S^n ⨧ Im Y + X Re X + Y) = 1 Im(X, Y), 1 Re(X, Y).