Ganea conjecture

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

[math]\displaystyle{ \operatorname{cat}(X \times S^n)=\operatorname{cat}(X) +1 }[/math]

for all [math]\displaystyle{ n\gt 0 }[/math], where [math]\displaystyle{ \operatorname{cat}(X) }[/math] is the Lusternik–Schnirelmann category of a topological space X, and Sⁿ is the n-dimensional sphere.

The inequality

[math]\displaystyle{ \operatorname{cat}(X \times Y) \le \operatorname{cat}(X) +\operatorname{cat}(Y) }[/math]

holds for any pair of spaces, [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math]. Furthermore, [math]\displaystyle{ \operatorname{cat}(S^n)=1 }[/math], for any sphere [math]\displaystyle{ S^n }[/math], [math]\displaystyle{ n\gt 0 }[/math]. Thus, the conjecture amounts to [math]\displaystyle{ \operatorname{cat}(X \times S^n)\ge\operatorname{cat}(X) +1 }[/math].

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

[math]\displaystyle{ \operatorname{cat}(M \setminus \{p\})=\operatorname{cat}(M) -1 , }[/math]

for a closed manifold [math]\displaystyle{ M }[/math] and [math]\displaystyle{ p }[/math] a point in [math]\displaystyle{ M }[/math].

A minimum dimensional counterexample to the 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, [math]\displaystyle{ \operatorname{cat}(X\times S^n) = \operatorname{cat}(X) + 1 }[/math], satisfied? It has been conjectured that these are precisely the spaces X for which [math]\displaystyle{ \operatorname{cat}(X) }[/math] equals a related invariant, [math]\displaystyle{ \operatorname{Qcat}(X). }[/math]

References

• "Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle Wash., 1971)". 249. Berlin: Springer. 1971. pp. 23–30. doi:10.1007/BFb0060892. 
• Hess, Kathryn (1991). "A proof of Ganea's conjecture for rational spaces". Topology 30 (2): 205–214. doi:10.1016/0040-9383(91)90006-P. 
• Iwase, Norio (1998). "Ganea's conjecture on Lusternik–Schnirelmann category". Bulletin of the London Mathematical Society 30 (6): 623–634. doi:10.1112/S0024609398004548. 
• Iwase, Norio (2002). "A_∞-method in Lusternik–Schnirelmann category". Topology 41 (4): 695–723. doi:10.1016/S0040-9383(00)00045-8. 
• Stanley, Donald; Rodríguez Ordóñez, Hugo (2010). "A minimum dimensional counterexample to Ganea's conjecture". Topology and Its Applications 157 (14): 2304–2315. doi:10.1016/j.topol.2010.06.009. 
• Vandembroucq, Lucile (2002). "Fibrewise suspension and Lusternik–Schnirelmann category". Topology 41 (6): 1239–1258. doi:10.1016/S0040-9383(02)00007-1.