Some Topological Properties of Surfaces in Z3

A basic property of a simple closed surface is the Jordan property: the complement of the surface has two connected components. We call back-component any such component, and the union of a back-component and the surface is called the closure of this back-component. We introduce the notion of strong surface as a surface which satisfies a strong homotopy property: the closure of a back-component is strongly homotopic to that back-component. This means that we can homotopically remove any subset of a strong surface from the closure of a back-component. On the basis of some results on homotopy, and strong homotopy, we have proved that the simple closed 26-surfaces defined by Morgenthaler and Rosenfeld, and the simple closed 18-surfaces defined by one of the authors are both strong surfaces. Thus, strong surfaces appear as an interesting generalization of these two notions of a surface.