Ganea conjecture

Ganea's conjecture is a now disproved claim in algebraic topology. It states that : \operatorname{cat}(X \times S^n)=\operatorname{cat}(X) +1 for all n0, where \operatorname{cat}(X) is the Lusternik–Schnirelmann category of a topological space X, and S**n is the n-dimensional sphere.

The inequality : \operatorname{cat}(X \times Y) \le \operatorname{cat}(X) +\operatorname{cat}(Y) holds for any pair of spaces, X and Y. Furthermore, \operatorname{cat}(S^n)=1, for any sphere S^n, n0. Thus, the conjecture amounts to \operatorname{cat}(X \times S^n)\ge\operatorname{cat}(X) +1.

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 : \operatorname{cat}(M \setminus {p})=\operatorname{cat}(M) -1 , for a closed manifold M and p a point in M.

A minimum dimensional counterexample to the conjecture was constructed by Don Stanley and Hugo Rodríguez Ordóñez in 2010. It has dimension 7 and \operatorname{cat}(X) = 2, and for sufficiently large n, \operatorname{cat}(X \times S^n) is also 2.

This work raises the question: For which spaces X is the Ganea condition, \operatorname{cat}(X\times S^n) = \operatorname{cat}(X) + 1, satisfied? It has been conjectured that these are precisely the spaces X for which \operatorname{cat}(X) equals a related invariant, \operatorname{Qcat}(X).

References

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