It is well known that a regular neighbourhood of a polyhedron in a piecewise linear manifold may be regarded as a simplicial mapping cylinder. The aim of this paper is to show that if the polyhedron is a locally unknotted submanifold of the interior then the class of maps giving rise to such regular neighbourhoods has a simple characterization. At the same time, it is possible to answer the question: Given a simplicial map f defined on a combinatorial manifold, when is the image of f also a combinatorial manifold? Marshall Cohen has answered this question when the image is required to be isomorphic to the domain; the methods used here are those developed in (1), to which the reader is referred for definitions and notation.