  Suppose the spaces X and X cross A have the same Lusternik-Schnirelmann
category: cat(X cross A)= cat(X). Then there is a strict inequality cat(X cross
(A halfsmash B)) < cat (X) + cat(A halfsmash 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.
