Ganea conjecture
Ganea's conjecture is a claim in algebraic topology, now disproved. It states that
for all, where 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, and. Furthermore,, for any sphere,. Thus, the conjecture amounts to.
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
for a closed manifold and a point in.
This work raises the question: For which spaces X is the Ganea condition,, satisfied? It has been conjectured that these are precisely the spaces X for which equals a related invariant,
