Covering the track of an isotopy

Abstract

We work in the Polyhedral Category as defined in [3], which consists of polyhedra and polymaps. A polyhedron is a space with a maximal nonempty family of p.l. related triangulations and a polymap is p.l. w.r.t. these triangulations. An ambient isotopy of a polyhedron X is a level-preserving homeo, H: XXI->XXI with HIXX1=1, Ht: X->X for tEI is the homeo induced by HIXXt. An isotopy of Y in X is a level-preserving embedding, F: YXI->XXI, Ft: Y-->X for tEI is the embedding induced by FJ YXt. We say that H covers F if HtFo = Ft for all t I, and we say that H covers the track of F if HtFo Y=Ft Y for all tEEI. Given an isotopy F of Y in X, it is not true that there is always an ambient isotopy of X which covers F or even just the track of F. For example, if we restrict attention to manifolds and proper embeddings, then classical knots of SI in S3 are isotopic embeddings which are not ambient isotopic. Ambient isotopy is rather more useful than isotopy, but isotopies are usually easier to construct. The following problem is therefore of interest. Problem A. Given an isotopy F of Y in X, under what conditions can we cover F by an ambient isotopy of X? For some purposes it is enough to cover the track of F, e.g. when considering knots as subspaces rather than embeddings, i.e. working set-wise rather than point-wise; so we have the weaker problem. Problem B. Under what conditions can we cover the track of an isotopy of Y in X? In [1 ] Zeeman and Hudson give a solution to A for proper embeddings of manifolds. Their condition for coverability is local-unknotting of Fl YXJ for any subinterval JCI, which is always true in codimension ? 3, see [1 ] for the precise definition. In this paper we will be concerned only with Problem B and our main result is that any locally collarable isotopy (definition below) can be track-covered. Local collarability is a necessary condition for track-covering and, in the case of proper embeddings of manifolds, is strictly weaker than local unknotting.