On Iwases manifolds

In ~cite{Iw2} Iwase has constructed two 16-dimensional manifolds $M_2$ and $M_3$ with LS-category 3 which are counter-examples to Ganeas conjecture: ${rm cat_{LS}} (Mtimes S^n)={rm cat_{LS}} M+1$. We show that the manifold $M_3$ is a counter-example to the logarithmic law for the LS-category of the square of a manifold: ${rm cat_{LS}}(Mtimes M)=2{rm cat_{LS}} M$. Also, we construct a map of degree one $$f:Nto M_2times M_3$$ which reduces Rudyaks conjecture to the question whether ${rm cat_{LS}}(M_2times M_3)ge 5$. We show that ${rm cat_{LS}}(M_2times M_3)ge 4$.