A classification of 3-thickenings of 2-polyhedra
The main result of the present paper guarantees the classification up to a homeomorphism of three-dimensional regular neighborhoods (3-thickenings) of a given 2-polyhedron \(P\). Two corollaries are derived for a special type of 2-polyhedra, containing fake surfaces. The first corollary proves that (i) 3-thickenings of \(P\) are classified by the restrictions of their first Stiefel-Whitney classes to \(P\) and (ii) for every two homotopic embeddings of \(P\) into the interior of a 3-manifold, the regular neighborhoods of their images are homeomorphic. The second corollary states that a fake surface is orientably 3-thickenable if and only if it does not contain a union of the Möbius band with an annulus (one of the boundary circles of the annulus attached to the middle circle of the Möbius band with a map of degree 1).
scientific article; zbMATH DE number 1317236