Mathematics — Graph Theory · mayer-vietoris-sequence
Problem
Use the Mayer--Vietoris sequence to show that a nonorientable closed surface, or more generally a finite simplicial complex X for which H1(X) contains torsion, cannot be embedded as a subspace of R3 in such a way as to have a neighborhood homeomorphic to the mapping cylinder of some map from a closed orientable surface to X. [This assumption on a neighborhood is in fact not needed if one deduces the result from Alexander duality in §3.3.]