Ganea conjecture

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

${\displaystyle \operatorname {cat} (X\times S^{n})=\operatorname {cat} (X)+1}$

for all ${\displaystyle n>0}$, 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, ${\displaystyle X}$ and ${\displaystyle Y}$. Furthermore, ${\displaystyle \operatorname {cat} (S^{n})=1}$, for any sphere ${\displaystyle S^{n}}$, ${\displaystyle n>0}$. 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, 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

${\displaystyle \operatorname {cat} (M\setminus \{p\})=\operatorname {cat} (M)-1,}$

for a closed manifold ${\displaystyle M}$ and ${\displaystyle p}$ a point in ${\displaystyle M}$.

A minimum dimensional counterexample to the conjecture was constructed by Don Stanley and Hugo Rodríguez Ordóñez in 2010. It has dimension 7 and ${\displaystyle \operatorname {cat} (X)=2}$, and for sufficiently large ${\displaystyle n}$, ${\displaystyle \operatorname {cat} (X\times S^{n})}$ is also 2.

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, ${\displaystyle \operatorname {Qcat} (X).}$^{[by whom?]}

• Ganea, Tudor (1971). "Some problems on numerical homotopy invariants". Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle Wash., 1971). Lecture Notes in Mathematics. Vol. 249. Berlin: Springer. pp. 23–30. doi:10.1007/BFb0060892. ISBN 978-3-540-05715-4. 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.
• 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. MR 2670507.
• Vandembroucq, Lucile (2002). "Fibrewise suspension and Lusternik–Schnirelmann category". Topology. 41 (6): 1239–1258. doi:10.1016/S0040-9383(02)00007-1. MR 1923222.