On Iwase's manifolds

TL;DR

This paper investigates specific 16-dimensional manifolds constructed by Iwase, providing counter-examples to conjectures about LS-category behavior, and establishes new lower bounds for the LS-category of their products.

Contribution

It demonstrates that certain Iwase manifolds counter Ganea's and the logarithmic LS-category conjectures, and constructs a degree-one map linking these manifolds to Rudyak's conjecture.

Findings

$M_3$ is a counter-example to the logarithmic law for LS-category.

Constructed a degree-one map from $N$ to $M_2 \times M_3$.

Established that $\operatorname{cat}_{LS}(M_2 \times M_3) \ge 4$.

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: ${\rm cat_{LS}} (M\times S^n)={\rm cat_{LS}} M+1$. 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: ${\rm cat_{LS}}(M\times M)=2{\rm cat_{LS}} M$. 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 ${\rm cat_{LS}}(M_2\times M_3)\ge 5$. We show that ${\rm cat_{LS}}(M_2\times M_3)\ge 4$.