On converting a side-pairing to a handle decomposition

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 $\text{en}$, $\text{sn}$ or $\text{hn}$. 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 $S^4$.