  In a recent paper, {\it Algorithms for Deforming and Contracting Simply
Connected Discrete Closed Manifolds (II)}, we discussed two algorithms for
deforming and contracting a simply connected discrete closed manifold into a
discrete sphere. The first algorithm was a continuation of work that began in
{\it Algorithms for Deforming and Contracting Simply Connected Discrete Closed
Manifolds (I)}, the second algorithm contained a more direct treatment of
contraction for discrete manifolds. In this paper, we clarify that we can use
this same method on standard piecewise linear (PL) complexes on the
triangulation of general smooth manifolds. Our discussion is based on
triangulation techniques invented by Cairns, Whitehead, and Whitney more than
half of a century ago. In this paper, we use PL or simplicial complexes to
replace certain concepts of discrete manifolds in previous papers. Note that
some details in the original papers related to discrete manifolds may also need
to be slightly modified for this purpose. In this paper, we use the algorithmic
procedure (a contraction process) to prove the following theorem: For a finite
triangulation of a simply-connected closed and orientable 3-manifold $M$ in
Euclidean space, if a (simply-connected closed) 2-cycle which was made by
2-cells of this triangulation separates $M$ into two connected components, then
each of the components will also be simply-connected. In addition, we can
algorithmically make $M$ to be homeomorphic to a 3-sphere. The relationship of
the theorem to a generalized Jordan separation problem, the general
Jordan-Schoenflies theorem, and other important problems are also discussed.
This revision adds more figures and explanations as well as two more special
cases. We will post the detailed algorithm/procedure for practical
triangulations in the following papers.
