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

for all $n>0$ , 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)\leq {\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 ${\text{cat}}(X\times S^{n})\geq {\text{cat}}(X)+1$ .

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\setminus \{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).

References

Ganea, Tudor (1971). "Some problems on numerical homotopy invariants". Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle Wash., 1971). Lecture Notes in Mathematics. 249. Berlin: Springer. pp. 23–30. doi:10.1007/BFb0060892. MR 0339147.
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. MR 1098914.
Iwase, Norio (1998). "Ganea's conjecture on Lusternik–Schnirelmann category". Bulletin of the London Mathematical Society. 30 (6): 623–634. doi:10.1112/S0024609398004548. MR 1642747.
Iwase, Norio (2002). "A_∞-method in Lusternik–Schnirelmann category". Topology. 41 (4): 695–723. doi:10.1016/S0040-9383(00)00045-8. MR 1905835.
Vandembroucq, Lucile (2002). "Fibrewise suspension and Lusternik–Schnirelmann category". Topology. 41 (6): 1239–1258. doi:10.1016/S0040-9383(02)00007-1. MR 1923222.

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