Triangulations with shellable 3-cells

Abstract

M. E. Rudin and R. H. Bing have given some of the better known examples of unshellable 3-cells. Nevertheless, L. B. Treybig has shown that every minimal triangulation of a compact 3-manifold with boundary has the free cell property (i.e. the property that every 3-cell which is the union of two or more 3-simplexes has two free 3-simplexes). Murray extended this result by showing that the above triangulation need only be minimal with respect to a predetermined triangulation of the boundary of the manifold. In this dissertation we investigate further the properties of minimal triangulations and expand the class of triangulated 3-manifolds which have the free cell property. Also, we present a new application of shelling 3-cells which involves certain algebraic representations of loops contained in 3-cells. It is shown herein that a certain class of triangulated, noncompact 3-manifolds, which include E³, has the free cell property. In fact, such a triangulation of E³ is actually constructed. These results are obtained by using a variation of the techniques of Milnor. That is, we show the operations of forming a certain type of connected sum of two triangulations and that of adding a handle in a certain manner to a triangulation, preserve the free cell property. We then apply these techniques to show the existence of noncompact 3-manifolds having triangulations with the free cell property. Several examples are included which further demonstrate these methods. Also, an algorithm is given for showing certain algebraic representations, similar to those of L.P. Neuwirth, of loops contained in a shellable triangulated 3-cell are trivial. This algorithm depends almost entirely on a shelling order for the 3-cell, but is generalized somewhat to an arbitrary triangulation of a 3-cell, provided there is a known bound on the number of 3-simplexes in a shellable subdivision..