EMBEDDINGS OF 2-COMPLEXES INTO 3-MANIFOLDS

Let f: K2 → M3 be an embedding of a compact, connected 2-complex into a compact, connected, orientable 3-manifold. We are interested to know, to which extent K2 determines (up to homeomorphism) its regular neighborhood N(f(K2)) ⊆ M3. We exhibit a list of four obstructions to uniqueness, and prove that in the absence of these, the regular neighborhoods are uniquely determined: Let f,K2,M3 be as above and assume M3 is prime and not a Poincare-counterexample. If N(f(K2)) does not contain essential annuli and has connected boundary, then N is determined by K2.