On converting a side-pairing to a handle decomposition

TL;DR

This paper introduces a method to derive handle decompositions of n-manifolds from isometric side-pairings of polyhedra in Euclidean, spherical, or hyperbolic spaces, with applications in hyperbolic 3-manifolds and 4-sphere recognition.

Contribution

The paper presents a novel technique linking side-pairings of polyhedra to handle decompositions, facilitating manifold analysis and recognition tasks.

Findings

Method enables recognition of hyperbolic 3-manifold link complements.

Automatically produces link diagrams from side-pairings.

Shows a topological S^4 is diffeomorphic to the standard S^4.

Abstract

We give a method for obtaining a handle decomposition of an $n$-manifold if the manifold is given by isometric side-pairings of a polyhedron in $\en$, $\sn$ or $\hn$. Every cycle of $k$-faces on the polyhedron corresponds to an $(n-k)$-handle of the manifold. Two applications of the method are given. One helps recognize when a noncompact hyperbolic 3-manifold is a complement of a link in $S^3$ (and automatically produces the link diagram), the other shows that a topological $S^4$ described by the author in \cite{Ivansic3} is diffeomorphic to the standard differentiable $S^4$.