A classification of 3-thickenings of 2-polyhedra

✍ Scribed by Dušan Repovš; Nikolaj B. Brodskij; Arkadij B. Skopenkov

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 93 KB
• Volume: 94
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(98)00036-4

✦ Synopsis

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).