Ganea conjecture

Ganea’s conjecture asserted that multiplying any space by a sphere increases its Lusternik–Schnirelmann category by exactly one. Noiro Iwase found a counterexample in June 1997; later published work records and develops this result.

Known results

• Iwase’s closed-manifold counterexample was reported in 2002, disproving the general conjecture.
• Jessup and Hess established the rational version for $n\ge 2$.
• Singhof and Rudyak proved the equality for various classes of manifolds.
• Iwase’s $16$-dimensional manifolds $M_2$ and $M_3$ satisfy $\operatorname{cat}_{\mathrm{LS}}(M_i\times S^n)=\operatorname{cat}_{\mathrm{LS}}(M_i)=3$ for sufficiently large $n$; the equality holds for all $n$ for $M_3$ and $M_3\times S^1$.

Post-counterexample developments

Subsequent work studied implications of failures of the original equality. A large-$r$ theorem is known under a dimension–connectivity bound, but the associated small-$r$ question remains open; this does not revive Ganea’s original conjecture.

Current status (as of August 2026): Ganea’s original conjecture is settled false by Iwase’s counterexample; rational and several restricted versions are true, while related small-$r$ questions remain open.