Ganea's conjecture is a claim in algebraic topology, now disproved. It states that
\( {\displaystyle \operatorname {cat} (X\times S^{n})=\operatorname {cat} (X)+1} \)
for all n>0, where \( {\displaystyle \operatorname {cat} (X)} \) is the Lusternik–Schnirelmann category of a topological space X, and Sn is the n-dimensional sphere.
The inequality
\( {\displaystyle \operatorname {cat} (X\times Y)\leq \operatorname {cat} (X)+\operatorname {cat} (Y)} \)
holds for any pair of spaces, X and Y. Furthermore, \( {\displaystyle \operatorname {cat} (S^{n})=1} \), for any sphere \( S^{n} \), n>0. Thus, the conjecture amounts to \( {\displaystyle \operatorname {cat} (X\times S^{n})\geq \operatorname {cat} (X)+1} \).
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
\( {\displaystyle \operatorname {cat} (M\setminus \{p\})=\operatorname {cat} (M)-1,} \)
for a closed manifold M and p a point in M.
This work raises the question: For which spaces X is the Ganea condition, \( {\displaystyle \operatorname {cat} (X\times S^{n})=\operatorname {cat} (X)+1} \), satisfied? It has been conjectured that these are precisely the spaces X for which \( {\displaystyle \operatorname {cat} (X)} \) equals a related invariant, \( {\displaystyle \operatorname {Qcat} (X).} \)
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. CiteSeerX 10.1.1.509.2343. 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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Disproved conjectures
Borsuk's Chinese hypothesis Euler's sum of powers Ganea Hedetniemi's Hauptvermutung Hirsch Mertens Ono's inequality Pólya Ragsdale Schoen–Yau Seifert Tait's Von Neumann Weyl–Berry
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