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.
We also 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).
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