  We prove that 1) There exist infinitely many non-trivial codimension one
"thick" knots in $\mathbb{R}^5$; 2) For each closed four-dimensional smooth
manifold $M$ and for each sufficiently small positive $\epsilon$ the set of
isometry classes of Riemannian metrics with volume equal to $1$ and injectivity
radius greater than $\epsilon$ is disconnected; 3) For each closed
four-dimensional $PL$-manifold $M$ and any $m$ there exist arbitrarily large
values of $N$ such that some two triangulations of $M$ with $<N$ simplices
cannot be connected by any sequence of $<M_m(N)$ bistellar transformations,
where $M_m(N)=\exp(\exp(\ldots \exp (N)))$ ($m$ times).
