Approximating Embeddings of Polyhedra in Codimension Three ( J )

Let F be a p-dimensional polyhedron and let Ç be a PL qmanifold 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 polyhedron into Q (fl p > 3) can be approximated by PL embeddings. This generalizes 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. The "moreover" part of Theorem 1 is simply an application of an isotopy theorem proved by Connelly [8], and we shall make no further reference to it. Definitions and notations. By a polyhedron we mean the underlying space of a locally finite simplicial complex embedded as a closed subset of some euclidean space (or manifold). We use the standard definition of piecewise linear (PL) map and PL manifold. When we say that R is a subpolyhedron of a polyhedron P, we shall always assume that R is a closed subset of P. If /: P—» 0 is a PL map, Received by the editors November 18, 1970. AMS 1970 subject classifications. Primary 57C05, 57C55, 57C35.