Ganea conjecture

rdf:langString Ganea conjecture

rdf:langString 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 . 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, , satisfied? It has been conjectured that these are precisely the spaces X for which equals a related invariant,