On the LS-category of product of Iwase's manifolds

Iwase [Topology 42 (2003), pp. 701–713] has constructed two 16-dimensional manifolds M_2 and M_3 with LS-category 3 which are counter-examples to Ganea’s conjecture: \operatorname {cat} (M\times S^n)=\operatorname {cat} 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: \operatorname {cat}(M\times M)=2\operatorname {cat} M. Also we construct a map of degree one \begin{equation*} f:2(M_3\times S^2\times S^{14})\#-(M_2\times S^2\times S^{14})\to M_2\times M_3 \end{equation*} which reduces Rudyak’s conjecture to the question whether \operatorname {cat}(M_2\times M_3)\ge 5 and show that \operatorname {cat}(M_2\times M_3)\ge 4.