Geometric proof of Neuwirth's theorem on the construction of 3-manifolds from 2-dimensional polyhedra
\textit{J. R. Stallings} [Fundam. Math. 51, 191-194 (1962; Zbl 0121.40006)] proved that there is no algorithm for checking if a given group is the fundamental group of some 3-dimensional manifold. \textit{L. Neuwirth} [Proc. Camb. Philos. Soc. 64, 603-613 (1968; Zbl 0162.27603)] described an algorithm for determining if a 2-dimensional polyhedron, given as a CW-complex with one vertex, can be thickened to a 3-dimensional manifold. We describe an algorithm for determining if any given 2-dimensional polyhedron can be thickened to some (or some orientable) 3-dimensional manifold. An algorithm for arbitrary CW-complexes is constructed similarly. This problem cannot be reduced to a problem for a complex with one vertex by a contraction of the maximal tree. Indeed, a contraction of an edge of a non-thickenable polyhedron can produce a thickenable polyhedron (for example, if \(N\) is the Möbius band with a disk glued to it along its middle line and \(I \subset N\) is the projection of the Möbius band onto the middle line, then \(N/I\) is thickenable, whereas \(N\) is not). In this article we use a simpler and more geometric method of proof than the one used by Neuwirth. Our method is based on reducing the property of being thickenable to embeddings of graphs into a sphere. Corollaries of our approach are \textit{P. Wright}'s theorem on the thickenability of fake surfaces [Topology 16, 435-439 (1977; Zbl 0378.57008)] and verification of the approximability of a mapping of the graph into the plane by embeddings. This solves a problem set by E. V. Shchepin. See [\textit{K. Sieklucki}, Fundam. Math. 65, 325-343 (1969; Zbl 0197.49201)] for related questions on embeddings of mappings.
scientific article; zbMATH DE number 830710