Super triangulations

Abstract

This paper concerns itself with continuous families of linear embeddings of triangulated complexes into E 2 . In [2] Cairns showed that if / and g are two linear embeddings of a triangulated complex (C, T) into E 2 so that there is an orientation preserving homeomorphism k of E 2 with kof=g, then there is a continuous family of linear embeddings h t : (C, T) -> E z (t e [[0, 1]) so that h 0 = / and h, = g. In this paper we prove various relative versions of this result when C is an arc, a 0-curve, or a disk.