Balanced presentations of the trivial group and four-dimensional geometry

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 $ϵ$ the set of isometry classes of Riemannian metrics with volume equal to $1$ and injectivity radius greater than $ϵ$ 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 $M_m(N)=\exp (\exp (\dots \exp (N)))$ ($m$ times).