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Concept
Ganea conjecture
Summary
Ganea's conjecture is a now disproved claim in algebraic topology. It states that
for all , where is the 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, 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 and a point in .
A minimum dimensional counterexample to the conjecture was constructed by Don Stanley and Hugo Rodríguez Ordóñez in 2010.
Official source
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