Fibrewise suspension and Lusternik–Schnirelmann category

Since Iwase disproved the Ganea conjecture the question became to find a characterization of the spaces X which satisfy the Ganea conjecture, i.e. for which the equality cat(X×Sk)=cat X+1 holds for any k⩾1. Recently Scheerer et al. (H. Scheerer, D. Stanley, D. Tanré, Fibrewise localization applied to Lusternik–Schnirelmann category, Israel J. Math. (2002) to appear.) have introduced an approximation of the category, denoted by Q cat, and have conjectured that, for a CW-complex X of finite dimension, we have cat(X×Sk)=cat X+1 for any k⩾1 if and only if Q cat X=cat X. In this paper, we establish the formula Q cat (X×Sk)=Q cat X+1 and deduce from this that if Q cat X=cat X then X satisfies the Ganea conjecture. In other words, a first direction of the conjecture of Scheerer et al. (H. Scheerer, D. Stanley, D. Tanré, Fibrewise localization applied to Lusternik–Schnirelmann category, Israel J. Math. (2002) to appear.) is proved. Using this new sufficient condition for a space to satisfy the Ganea conjecture, we prove that any (r−1)-connected CW-complex X with r cat(X)⩾3 and dim(X)⩽2r cat(X)−3 satisfies the Ganea conjecture. This shows for example that the Lie group Sp(3) satisfies the Ganea conjecture.