Implications of the Ganea condition

Abstract

Suppose the spaces $X$ and $X\times A$ have the same Lusternik–Schnirelmann category: $cat\left(X\times A\right)=cat\left(X\right)$. Then there is a strict inequality $cat\left(X\times \left(A⋊B\right)\right)<cat\left(X\right)+cat\left(A⋊B\right)$ 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.