Implications of the Ganea condition

Abstract

Suppose the spaces X and X × A have the same LusternikSchnirelmann category: cat(X × A) = cat(X). Then there is a strict inequality cat(X × (A ⋊ B)) < cat(X) + cat(A ⋊ B) for every space B , provided the connectivity of A is large enough (depending only on X ). This is applied to give a partial verification of a conjecture of Iwase on the category of products of spaces with spheres.