Ganea conjecture

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

for all , where 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, and . Furthermore, , for any sphere , . Thus, the conjecture amounts to .

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 and a point in .

This work raises the question: For which spaces X is the Ganea condition, , satisfied? It has been conjectured that these are precisely the spaces X for which equals a related invariant,


  • 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. CiteSeerX doi:10.1112/S0024609398004548. MR 1642747.
  • Iwase, Norio (2002). "A-method in Lusternik–Schnirelmann category". Topology. 41 (4): 695–723. arXiv:math/0202119. 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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