Balanced presentations of the trivial group and four-dimensional geometry

Abstract

We prove that: (1) There exist infinitely many nontrivial codimension one “thick” knots in [Formula: see text]; (2) For each closed four-dimensional smooth manifold [Formula: see text] and for each sufficiently small positive [Formula: see text] the set of isometry classes of Riemannian metrics with volume equal to [Formula: see text] and injectivity radius greater than [Formula: see text] is disconnected; and (3) For each closed four-dimensional [Formula: see text]-manifold [Formula: see text] and any [Formula: see text] there exist arbitrarily large values of [Formula: see text] such that some two triangulations of [Formula: see text] with [Formula: see text] simplices cannot be connected by any sequence of [Formula: see text] bistellar transformations, where [Formula: see text] ([Formula: see text] times).