Manifold phenomena in the theory of polyhedra

Abstract

Introduction. When can an isotopy be covered by an ambient isotopy? Let us restrict attention to the compact p.l. category. Hudson and Zeeman have shown that a locally unknotted isotopy of a manifold in a manifold can be covered by an ambient isotopy of the big manifold. By Zeeman's codimension > 3 unknotting theorem, an isotopy of manifolds is locally unknotted if the codimension is greater than or equal to 3. Hence, any isotopy of a manifold in a manifold of dimension at least 3 higher can be covered. Lickorish has generalized Zeeman's unknotting theorem to the case of a proper embedding of a cone in a ball of dimension at least three higher. From this, Hudson has shown any isotopy of a polyhedron in a manifold can be covered if the polyhedron has codimension at least 3. As a modest aim we would like a criterion of local unknottedness of a polyhedron in a manifold so that the original Hudson-Zeeman theorem would generalize. What we actually obtain is more general. We present a characterization of those isotopies of a polyhedron in a polyhedron which can be covered by ambient isotopies. Perhaps surprisingly, this question admits a rather elegant general solution. Intrinsic Dimension. Our major tool is the theory of intrinsic dimension developed by Armstrong. Given a polyhedral pair (X, X0) and a point x E X0, we define the intrinsic dimension of x in (X, X0), denoted d(x; X, X0), to be