Ganea conjecture

Ganea conjecture

Ganea's conjecture is a claim in algebraic topology, now disproved. It states that: ext{cat}(X imes S^n)= ext{cat}(X) +1, n>0 ,!where cat("X") is the Lusternik–Schnirelmann category of a topological space "X", and "S""n" is the "n" dimensional sphere.

The inequality : ext{cat}(X imes Y) le ext{cat}(X) + ext{cat}(Y) holds for any pair of spaces, "X" and "Y". Furthermore, cat("S""n")=1, for any sphere "S""n", "n">0. Thus, the conjecture amounts to cat("X" × S"n") > cat("X").

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 : ext{cat}(M - {p})= ext{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, cat("X" × S"n") = cat("X") + 1, satisfied? It has been conjectured that these are precisely the spaces "X" for which cat("X") equals a related invariant, Qcat("X").

References

* Tudor Ganea, "Some problems on numerical homotopy invariants", Lecture Notes in Mathematics, vol. 249, Springer-Verlag, Berlin, 1971, 13--22. MathSciNet| id=0339147
* [http://dx.doi.org/10.1016/0040-9383(91)90006-P] Kathryn Hess, "A proof of Ganea's conjecture for rational spaces", Topology 30 (1991), no. 2, 205--214. MathSciNet| id=1098914
* [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=19333] Norio Iwase, "Ganea's conjecture on Lusternik-Schnirelmann category", Bulletin of the London Mathematical Society 30 (1998), no. 6, 623--634. MathSciNet| id=1642747
* [http://dx.doi.org/S0040-9383(00)00045-8] Norio Iwase, "A∞-method in Lusternik-Schnirelmann category", Topology 41 (2002), no. 4, 695--723. MathSciNet| id=1905835
* [http://dx.doi.org/10.1016/S0040-9383(02)00007-1] Lucile Vandembroucq, "Fibrewise suspension and Lusternik-Schnirelmann category", Topology 41 (2002), no. 6, 1239--1258. MathSciNet| id=1923222


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