A classification of 3-thickenings of 2-polyhedra

We classify 3-thickenings (i.e., 3-dimensional regular neighborhoods) of a given 2-polyhedron P up to a homeomorphism rel P. The partial case of our theorem is that for some class of 2-polyhedra, containing fake surfaces, 3-thickenings of P are classified by the restriction of their first Stiefel–Whitney class to P. The corollary is that for every two homotopic embeddings of a polyhedron P from our class into interior of a 3-manifold M, the regular neighborhoods of their images are homeomorphic. o prove that a fake surface is embeddable into some orientable 3-manifold 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).