Ganea conjecture

Ganea's conjecture is a now disproved claim in algebraic topology. It states that
 * $$ \operatorname{cat}(X \times S^n)=\operatorname{cat}(X) +1$$

for all $$n>0$$, where $$\operatorname{cat}(X)$$ is the Lusternik–Schnirelmann category of a topological space X, and Sn is the n-dimensional sphere.

The inequality
 * $$ \operatorname{cat}(X \times Y) \le \operatorname{cat}(X) +\operatorname{cat}(Y) $$

holds for any pair of spaces, $$X$$ and $$Y$$. Furthermore, $$\operatorname{cat}(S^n)=1$$, for any sphere $$S^n$$, $$n>0$$. Thus, the conjecture amounts to $$ \operatorname{cat}(X \times S^n)\ge\operatorname{cat}(X) +1$$.

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
 * $$ \operatorname{cat}(M \setminus \{p\})=\operatorname{cat}(M) -1, $$

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, $$\operatorname{cat}(X\times S^n) = \operatorname{cat}(X) + 1$$, satisfied? It has been conjectured that these are precisely the spaces X for which $$\operatorname{cat}(X)$$ equals a related invariant, $$\operatorname{Qcat}(X).$$

Furthermore, cat(X * S^n) = cat(X ⨇ S^n ⨧ Im Y + X Re X + Y) = 1 Im(X, Y), 1 Re(X, Y).