A Representation of Orientable Combinatorial 3-Manifolds

TL;DR: In this paper, it was shown that every closed, connected, orientable, 3-manifold is obtainable from S3 in the same way as shown in this paper.

Abstract: The following question has been posed by Bing [1]: "Which compact, connected 3-manifolds can be obtained from S3 as follows: Remove a finite collection of mutually exclusive (but perhaps knotted and linking) polyhedral tori T1, T2, * *, To from S3, and sew them back. " This paper answers that question by showing that every closed, connected, orientable, 3-manifold is obtainable from S3 in the above way. Whereas this fact can now be deduced from general theorems of differential topology, the combinatorial proof given here is direct and elementary; while, in the proof, a study is made of a certain type of homeomorphism of a two dimensional manifold that is of interest in itself. Having obtained the above mentioned result on 3-manifolds, it is then easy to deduce the well known result (Theorem 3) that the combinatorial cobordism group for orientable 3-manifolds is trivial.