Fibrewise suspension and Lusternik-Schnirelmann category
The Ganea conjecture \(\text{(cat} (X\times S^n)= \text{cat} X+1)\) has been the origin of a lot of interesting recent works on the Lyusternik-Shnirel'man category. Since Iwase disproved the conjecture the next question was to find a characterization of the spaces \(X\) that satisfy the conjecture. In [Fibrewise construction applied to Lusternik-Schnirelmann category, Isr. J. Math. 131, 333-359 (2002; Zbl 1011.55003)], \textit{H. Scheerer}, \textit{D. Stanley} and \textit{D. Tanré} introduced a new invariant \(\text{Qcat} (X)\) and conjectured that \(\text{cat}(X\times S^n)=\text{cat} X+1\) if and only if \(\text{cat}(X)= \text{Qcat}(X)\). In this paper the author proves that \(\text{Qcat}(X \times S^n)= \text{Qcat} X+1\). She deduces that the condition \(\text{cat} X =\text{Qcat}(X)\) implies the Ganea conjecture. She also proves that the Ganea conjecture is true for \((r-1)\)-connected CW complexes \(X\) with \(r.\text{cat} (X) \geq 3\) and \(\dim(X)\leq 2r.\text{cat}(X)-3\). This is in particular the case of Sp(3).
scientific article; zbMATH DE number 1838830