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Recognizing the 3‐Sphere

All manifolds in this paper are piecewise linear and are given in terms of a finite triangulation. In dimensions 5 3 this is completely general by virtue of the Moise triangulation theorem. By a closed manifold we mean a manifold that is connected, compact, and without boundary. The homeomorphism problem-to decide whether two closed manifolds are homeomorphic-is solved in dimensions 1 and 2 and, in general, unsolvable [9] in dimensions 24 . In three-dimensions, even the existence of an algorithm for deciding whether a manifold is a sphere, is open. In this paper the question for the 3-sphere is approached combinatorially, and an algorithm for recognizing the 3-sphere is conjectured. In 1904 PoincarC proved that the Betti and torsion numbers do not suffice to determine the 3-sphere among closed 3-manifolds and conjectured that the 3-sphere is characterized by being simply connected. An affirmative answer to the PoincarC conjecture would reduce the problem of recognizing the 3-sphere to the problem of determining algorithmically whether the fundamental group of a closed 3-manifold is trivial. This is also an open question. Let A be a potential algorithm for recognizing the 3-sphere. To confirm that A is valid, the following statements must be proved.