Approximating embeddings of polyhedra in codimension three

Abstract

Let F be a p-dimensional polyhedron and let C be a PL q- manifold without boundary. (Neither is necessarily compact.) The purpose of this paper is to prove that, if q — p > 3, then any topological embedding of P into Q can be pointwise approximated by PL embeddin-gs. The proof of this theorem uses the analogous result for embeddings of one PL manifold into another obtained by Cernavskii and Miller. Introduction. Recently Cernavskii and Miller have shown that any topological embedding of a PL zzz-manifold M into a PL o-manifold Q can be approximated by PL embeddings provided q - m > 3- (See (5), (6), and (10).) In this paper we use this fact to prove that a topological embedding of an arbitrary p-dimensional poly- hedron into Q (fl - p > 3) can be approximated by PL embeddings. This general- izes the results of Berkowitz (2) and Weber (13)- Our theorem can be stated as follows. Theorem 1. Suppose that P is a (not necessarily compact) p-dimensional polyhedron, that 0 is a PL q-manifold without boundary (q — p > 3), and that f: P —* Q is a topological embedding. Then for each continuous function c P—» (0, 00) there exists a PL embedding g; P—»0 such that d(f(x),g(x)) < e(x) for each x £ P. Moreover, if R is a subpolyhedron of P on which f\R is PL, then we may choose g so that g\R=f\R.