Ganea conjecture

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

\text{cat}(X \times S^n)=\text{cat}(X) +1, n>0 \,\!

where cat(X) is the Lusternik–Schnirelmann category of a topological space X, and Sn is the n dimensional sphere.

The inequality

\text{cat}(X \times Y) \le \text{cat}(X) +\text{cat}(Y)

holds for any pair of spaces, X and Y. Furthermore, cat(Sn)=1, for any sphere Sn, n>0. Thus, the conjecture amounts to cat(X × Sn) ≥ cat(X) + 1.

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

\text{cat}(M - {p})=\text{cat}(M) -1 ,

for a closed manifold M and p a point in M.

This work raises the question: For which spaces X is the Ganea condition, cat(X × Sn) = cat(X) + 1, satisfied? It has been conjectured that these are precisely the spaces X for which cat(X) equals a related invariant, Qcat(X).

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