Ganea's conjecture is a now disproved claim in algebraic topology.
is the Lusternik–Schnirelmann category of a topological space X, and Sn is the n-dimensional sphere.
The inequality holds for any pair of spaces,
Thus, the conjecture amounts to
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 for a closed manifold
A minimum dimensional counterexample to the conjecture was constructed by Don Stanley and Hugo Rodríguez Ordóñez in 2010.
, and for sufficiently large
This work raises the question: For which spaces X is the Ganea condition,
It has been conjectured that these are precisely the spaces X for which
equals a related invariant,
Qcat (
Furthermore, cat(X * S^n) = cat(X ⨇ S^n ⨧ Im Y + X Re X + Y) = 1 Im(X, Y), 1 Re(X, Y).