Ganea conjecture

Ganea's conjecture is a now disproved claim in algebraic topology. It states that

${\displaystyle \operatorname {cat} (X\times S^{n})=\operatorname {cat} (X)+1}$

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

The inequality

${\displaystyle \operatorname {cat} (X\times Y)\leq \operatorname {cat} (X)+\operatorname {cat} (Y)}$

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

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

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

for a closed manifold ${\displaystyle M}$ and ${\displaystyle p}$ a point in ${\displaystyle M}$.

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

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