So, yesterday I was trying to do Exercise 2.2.35 from Hatcher's Algebraic Topology, which goes like this:
35. Use the Mayer-Vietoris sequence to show that a nonorientable closed surface, or more generally a finite simplicial complex $X$ for which $H_1 (X)$ contains torsion, cannot be embedded as a subspace of $\mathbb{R}^3$ in such a way as to have a neighbourhood homeomorphic to the mapping cylinder of some map from a closed orientable surface to $X$. [This assumption on a neighbourhood is in fact not needed if one deduces the result from Alexander duality in §3.3.]
I found myself rather confused by this question. Sure, when you took the stuff in the square brackets into account, it started to make sense, what we were being asked to prove was simply a weaker version of the statement that closed non-orientable surfaces--like for instance, the Klein bottle--cannot be embedded as subspaces in three-dimensional space. Fair enough.
But this whole notion of a "neighbourhood homeomorphic to the mapping cylinder of some map from a closed orientable surface to $X$" just confused the living daylights out of me. Specifically, I could think of no surface $Y$, be it orientable, nonorientable, open, or closed, of any sort that could be embedded in $\mathbb{R}^3$ in such a way as to have a neighbourhood homeomorphic to the mapping cylinder of a closed orientable surface to $Y$.
In fact, the whole thing seemed a bit bizarre, since it appeared to me that a mapping cylinder from one surface to another ought to be homeomorphic to a volume, and thus trivially it ought to be something that you couldn't embed in a surface.
Nevertheless, looking around online, I found that this isn't some newfangled invention of Hatcher's purely for this question, and that there exists a notion of a "mapping cylinder neighbourhood" in topology.
Problem is, searching around, I cannot find any decent examples of a mapping cylinder neighbourhood that I can really visualize.
Thus, my question is, can somebody please give me an example of a surface $Y$, be it orientable, nonorientable, open, or closed, of any sort that can be embedded in $\mathbb{R}^3$ in such a way as to have a neighbourhood homeomorphic to the mapping cylinder of a closed orientable surface to $Y$.
I look forward to your answers.
Well, I think that I have sorted it out, and that, at the end of the day, I merely had misunderstood what Hatcher was asking for. When he says "embedded in $\mathbb{R}^3$ in such a way as to have a neighbourhood homeomorphic to the mapping cylinder of a closed orientable surface to $X$", he doesn't mean that there exists a neighbourhood contained in $X$ that is homeomorphic to the mapping cylinder of a closed orientable surface to $X$, rather, he means that there exists a neighbourhood in $\mathbb{R}^3$ "surrounding" $X$ that is homeomorphic to the mapping cylinder of a closed orientable surface to $X$.
Then all of a sudden the question makes sense, at the very least.