On Iwase's manifolds

Abstract

In ~\cite{Iw2} Iwase has constructed two 16-dimensional manifolds $M_2$ and $M_3$ with LS-category 3 which are counter-examples to Ganea's conjecture: $\mathrm{ca}{\mathrm{t}}_{\mathrm{LS}}(M\times S^n)=\mathrm{ca}{\mathrm{t}}_{\mathrm{LS}}M+1$. 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: $\mathrm{ca}{\mathrm{t}}_{\mathrm{LS}}(M\times M)=2\mathrm{ca}{\mathrm{t}}_{\mathrm{LS}}M$. Also, we construct a map of degree one $$f:N\to M_2\times M_3$$ which reduces Rudyak's conjecture to the question whether $\mathrm{ca}{\mathrm{t}}_{\mathrm{LS}}(M_2\times M_3)\ge 5$. We show that $\mathrm{ca}{\mathrm{t}}_{\mathrm{LS}}(M_2\times M_3)\ge 4$.