On the Imbedding of Polyhedra in 3-Space
Ralph H. Fox · Annals of Mathematics · 1948
The object of our investigation is spherical 3-space S and its polyhedral subsets. The sets which come into consideration are all to be polyhedral (or polygonal) even when this is not explicitly stated; all homeomorphisms considered are to be semi-linear. Given in S any finite (positive) number of non-intersecting closed surfaces, the closure of any component of the residual space will be called a connected elementary figure. An elementary figure [1] is the union of a finite collection of non-intersecting connected elementary figures. It is known [2] that any polyhedron P in S which is not the whole space has a closed neighborhood, called a regular neighborhood, which is an elementary figure and of which P is a deformation retract. Thus in any investigation of properties of the homotopy types [3] of polyhedra in S one may restrict oneself to consideration of connected elementary figures. A further simplification is effected by theorem (1) below, whose proof is our main objective. We shall often have to deal with the closure of the complement of an elementary figure; this set, which is itself an elementary figure, will be designated as the closed complement, or more simply as the complement, of the given elementary figure. A regular neighborhood of a finite (but not necessarily connected) linear graph will be called a tubular figure. The combinatorial type, and hence the topological type, of such a figure is uniquely determined by the given graph: a connected graph of Euler characteristic 1 p gives rise to a connected tubular figure whose boundary is an orientable surface of genus p and which is of the topological type of the Cartesian product of a 1-cell with a closed simplyor multiply-connected region of the plane. An example of an elementary figure which is not tubular may be obtained by boring through a 3-cell a system of tunnels, which may be knotted or linked and need not have entrances or exits. That this is a description. of the general non-tubular elementary figure is the content of theorem (1). MAIN THEOREM (1) Every connected elementary figure in S is homeomorphic to the closed complement of some tubular figure in S. An immediate consequence of this theorem is that any connected polyhedron in S belongs to the homotopy type of the residual space of some finite graph in S. This shows that questions concerning the homotopy type of polyhedra that can be imbedded in Euclidean 3-space may be studied by the methods of knot theory. Following the proof of (1) several results of this nature are derived. We begin by applying a suitable modification of a method, used by Alexander [4], to obtain a proof of the following lemma: (2) Let Ui, 022 )... * Ur be a system of non-intersecting surfaces in S and suppose that these surfaces are not all of genus zero. Then there exists a 2-cell C whose interior C C is disjoint to a = ai + . . . + Urn and whose boundary curve C is on u (i.e. is a subset of a) but does not bound any 2-cell on a.