Ganea conjecture
Ganea's conjecture is a claim in algebraic topology, now disproved. It states that
where cat(X) is the Lusternik–Schnirelmann category of a topological space X, and Sn is the n dimensional sphere.
The inequality
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
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". Lecture Notes in Mathematics. Berlin: Springer-Verlag. 249: 13–22. 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.