# Ganea conjecture

Ganea's conjecture is a claim in algebraic topology, now disproved. 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

$\operatorname {cat} (X\times Y)\leq \operatorname {cat} (X)+\operatorname {cat} (Y)$ 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 $\operatorname {cat} (X\times S^{n})\geq \operatorname {cat} (X)+1$ .

The conjecture was formulated by Tudor Ganea in 1971. Many particular cases of this conjecture were proved, till finally Norio Iwase gave a counterexample 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, stating that

$\operatorname {cat} (M\setminus \{p\})=\operatorname {cat} (M)-1,$ for a closed manifold $M$ and $p$ a point in $M$ .

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, $\operatorname {Qcat} (X).$ This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.