Ganea's conjecture is a now disproved claim in algebraic topology. It states that

is the Lusternik–Schnirelmann category of a topological space X, and Sn is the n-dimensional sphere.

The inequality

.

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

.

is also 2.

This work raises the question: For which spaces X, satisfied? It has been conjectured that these are precisely the spaces X

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