Ganea Conjecture

Ganea's conjecture is a claim in algebraic topology, now disproved. 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, 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(M \setminus \)=\operatorname(M) -1 ,$
for a closed manifold $M$ and $p$ a point in $M$.

A minimum dimensional counterexample to Ganea’s 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, $\operatorname(X\times S^n) = \operatorname(X) + 1$, satisfied? It has been conjectured that these are precisely the spaces ''X'' for which $\operatorname(X)$ equals a related invariant, $\operatorname(X).$

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{{Disproved conjectures

Disproved conjectures
Algebraic topology