Ganea conjecture explained

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

\operatorname{cat}(X x S ^{n)=\operatorname{cat}(X)} +1 for all n> 0 , where \operatorname{cat}(X) is the Lusternik–Schnirelmann category of a topological space X, and Sⁿ is the n-dimensional sphere.

The inequality

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

\operatorname{cat}(M \setminus \{p\}) = \operatorname{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.

This work raises the question: For which spaces X is the Ganea condition,

\operatorname{cat}(X x S ^{n)} = \operatorname{cat}(X) + 1 , satisfied? It has been conjectured that these are precisely the spaces X for which \operatorname{cat}(X) equals a related invariant, \operatorname{Qcat}(X).

Furthermore, cat(X * S^n) = cat(X ⨇ S^n ⨧ Im Y + X Re X + Y) = 1 Im(X, Y), 1 Re(X, Y).

References

• Ganea . Tudor. Tudor Ganea . Some problems on numerical homotopy invariants . 10.1007/BFb0060892 . Berlin . 0339147 . 23–30 . Springer . . Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle Wash., 1971) . 249 . 1971.
• 10.1016/0040-9383(91)90006-P . Kathryn . Hess . Kathryn Hess. A proof of Ganea's conjecture for rational spaces . . 30 . 1991 . 2 . 205–214 . 1098914 .
• Norio . Iwase . Ganea's conjecture on Lusternik–Schnirelmann category . . 30 . 1998 . 6 . 623–634 . 10.1112/S0024609398004548 . 1642747 . 10.1.1.509.2343 .
• 10.1016/S0040-9383(00)00045-8 . Norio . Iwase . A_∞-method in Lusternik–Schnirelmann category . . 41 . 2002 . 4 . 695–723 . 1905835 . math/0202119 .
• Stanley. Donald. Rodríguez Ordóñez. Hugo. 2010. A minimum dimensional counterexample to Ganea's conjecture. Topology and Its Applications. 157. 14. 2304–2315. 10.1016/j.topol.2010.06.009. 2670507. free.
• 10.1016/S0040-9383(02)00007-1 . Lucile . Vandembroucq . Fibrewise suspension and Lusternik–Schnirelmann category . . 41 . 2002 . 6 . 1239–1258 . 1923222 .
• Zachary Marshall Gehring-Young.