Manifold phenomena in the theory of polyhedra
Ethan Akin · Transactions of the American Mathematical Society · 1969
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 (A', X0), denoted d(x; X, X0), to be max {/ : link pair of x in (X, X0) is a 7-fold suspension}, or equivalently, max {/ : there is a triangulation of (X, X0) with x in the interior of a /-simplex}. LetV(X, X0) = {xeX0: d(x; X, X0) á /}, called the intrinsic /-skeleton of (X, X0).For the absolute notion we associate the polyhedron A'with the pair (X, X), i.e. d(x; X) = d(x; X, X) and V(X) = V(X, X).The first fact about intrinsic dimension is that F(X, X0) is a subcomplex of any triangulation of (A', X0).