Ganea conjecture

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

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

for all $n>0$, where $\mathrm{cat}(X)$ is the Lusternik–Schnirelmann category of a topological space X, and Sⁿ is the n-dimensional sphere.

The inequality

$\mathrm{cat}(X\times Y)\le \mathrm{cat}(X)+\mathrm{cat}(Y)$

holds for any pair of spaces, $X$ and $Y$. Furthermore, $\mathrm{cat}(S^n)=1$, for any sphere $S^n$, $n>0$. Thus, the conjecture amounts to $\mathrm{cat}(X\times S^n)\ge \mathrm{cat}(X)+1$.

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

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

for a closed manifold $M$ and $p$ a point in $M$.

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

This work raises the question: For which spaces X is the Ganea condition, $\mathrm{cat}(X\times S^n)=\mathrm{cat}(X)+1$, satisfied? It has been conjectured that these are precisely the spaces X for which $\mathrm{cat}(X)$ equals a related invariant, $\mathrm{Qcat}(X).$

References

• "Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle Wash., 1971)". 249. Berlin: Springer. 1971. pp. 23–30. doi:10.1007/BFb0060892. ISBN 978-3-540-05715-4. 
• 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. 
• Iwase, Norio (1998). "Ganea's conjecture on Lusternik–Schnirelmann category". Bulletin of the London Mathematical Society 30 (6): 623–634. doi:10.1112/S0024609398004548. 
• Iwase, Norio (2002). "A_∞-method in Lusternik–Schnirelmann category". Topology 41 (4): 695–723. doi:10.1016/S0040-9383(00)00045-8. 
• 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. 
• Vandembroucq, Lucile (2002). "Fibrewise suspension and Lusternik–Schnirelmann category". Topology 41 (6): 1239–1258. doi:10.1016/S0040-9383(02)00007-1.