Ganea Conjecture

Ganea's conjecture is a now disproved claim in algebraic topology. It states that
: $\operatorname(X \times S^n)=\operatorname(X) +1$
for all $n>0$, where $\operatorname(X)$ is the Lusternik–Schnirelmann category of a topological space ''X'', and ''S''^''n'' is the ''n''-dimensional sphere.

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

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(M \setminus \)=\operatorname(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. It has dimension 7 and $\operatorname(X) = 2$, and for sufficiently large $n$, $\operatorname(X \times S^n)$ is also 2.

This work raises the question: For which spaces ''X'' is the Ganea condition, $\operatorname(X\times S^n) = \operatorname(X) + 1$, satisfied? It has been conjectured that these are precisely the spaces ''X'' for which $\operatorname(X)$ equals a related invariant, $\operatorname(X).$

References

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*{{cite journal , doi=10.1016/S0040-9383(02)00007-1 , first=Lucile , last=Vandembroucq , title=Fibrewise suspension and Lusternik–Schnirelmann category , journal=Topology , volume=41 , year=2002 , issue=6 , pages=1239–1258 , mr=1923222 , doi-access=

Disproved conjectures
Algebraic topology