The equivalence of close piecewise linear embeddings

Abstract

The main theorem (2.1) says that if N is an abstract regular neighborhood of a polyhedron X with collapsible retraction r: N → X, dimX < dimN − 3, and if g: N → intN is an embedding such that rg is close to the inclusion map X↪ N, then g is isotopic to the inclusion X↪ N by an ambient isotopy which is limited by r. As a corollary two close PL embeddings of a polyhedron X into a PL manifold Q are equivalent by small PL ambient isotopy if dim X < dimQ − 3 and if the embeddings are sufficiently close to a given topological embedding of X into Q. Some related results using a slightly weaker dimension restriction are also discussed, and some other corollaries are presented.