Balanced presentations of the trivial group and four-dimensional geometry
Boris Lishak, Alexander Nabutovsky

TL;DR
This paper explores complex geometric and topological properties of four-dimensional manifolds, demonstrating the existence of intricate structures and disconnections in metric spaces and triangulations.
Contribution
It establishes new results on the existence of non-trivial knots in higher dimensions and the disconnectedness of certain metric and triangulation spaces for 4-manifolds.
Findings
Existence of infinitely many non-trivial codimension one 'thick' knots in R^5
Disconnection of the space of Riemannian metrics with bounded injectivity radius on 4-manifolds
Existence of triangulations of 4-manifolds that cannot be connected by a bounded number of bistellar transformations
Abstract
We prove that 1) There exist infinitely many non-trivial codimension one "thick" knots in ; 2) For each closed four-dimensional smooth manifold and for each sufficiently small positive the set of isometry classes of Riemannian metrics with volume equal to and injectivity radius greater than is disconnected; 3) For each closed four-dimensional -manifold and any there exist arbitrarily large values of such that some two triangulations of with simplices cannot be connected by any sequence of bistellar transformations, where ( times).
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