## Thin position and the recognition problem for $$S^ 3$$.(English)Zbl 0849.57009

In 1992 Rubinstein gave a series of lectures describing an algorithm to determine whether or not a triangulated 3-manifold is the 3-sphere. His proof uses the language of PL minimal surface theory. In the present paper a different proof is given that a simplified algorithm works using techniques from knot theory. Roughly, the algorithm is as follows. A normal surface in a triangulated 3-manifold $$M$$ is an embedded surface intersecting each 3-simplex in a certain simple pattern (nice triangles and quadrilaterals). Now a maximal collection of disjoint non-parallel normal 2-spheres (which can be constructed by a modification of an algorithm due to Haken) cuts $$M$$ into three possible types of components two of which are balls resp. punctured balls. Now $$M$$ is the 3-sphere if and only if also each component of the third type is a 3-ball which, by the main Lemma, is the case if and only if it contains an “almost normal” 2-sphere.

### MSC:

 57M40 Characterizations of the Euclidean $$3$$-space and the $$3$$-sphere (MSC2010)

### Keywords:

recognition problem for the 3-sphere; knot theory

Zbl 0849.57010
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