Polygons in the boundary of a combinatorial solid torus

Abstract

For the definition of a combinatorial solid torus (CST), see the discussion just before Theorem 24.9. There we show that S is a CST if and only if S is the image of a product σ² × [0, 1] under a PL identification mapping ф which identifies σ² × {0} and σ² × {1} in such a way as to give an orientable 3-manifold with boundary. In fact, every polyhedral solid torus is a CST, but we are not yet in a position to prove it; it is a special case of the Hauptυermutung for 3-manifolds with boundary, and it is not easy to see how the special hypothesis can be used. Meanwhile we have the following.