Balanced presentations of the trivial group and four-dimensional geometry

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 $\mathbb{R}^5$; 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).