An application of Poénaru's ``zipping theory'' (Q307742)

The author presents an application of the ``zipping theory'' introduced by \textit{V. Poénaru} [Duke Math. J. 63, No. 2, 421--429 (1991; Zbl 0738.57008)], aimed to kill in a controlled way all the singularities of a nondegenerate simplicial map \(f:X\to M\) from a simplicial complex to a manifold. The main results are the following for the class of finite three-dimensional simplicial complexes:NEWLINENEWLINE 1. The set of complexes homeomorphic to the \(3\)-sphere \(S^3\) is recursively enumerable.NEWLINENEWLINE 2. The set of \(3\)-manifolds with a finite fundamental group is recursively enumerable.NEWLINENEWLINE 3. The set of simply connected complexes is recursively enumerable.NEWLINENEWLINE 4. The set of complexes with a finite fundamental group is recursively enumerable.NEWLINENEWLINE 5. If, for a \(3\)-manifold \(M\), deciding whether \(\pi_1M\) is finite or not is algorithmically solvable, then so is the problem whether \(M\) is homeomorphic to \(S^3\).