More on completion/compactification of open manifolds

This is a follow up to one of my previous questions (81714). Suppose that $M$ is an open manifold, say with a single end. Previously, I was concerned with realizing $M$ as the interior of a compact manifold with boundary $\bar{M}$, essentially adding a boundary to $M$. The responses there were quite helpful, but even after some more reading I hesitate to say that I have a satisfactory understanding of the situation.

Now I would like to consider a similar question in a slightly different direction. I wonder how to realize $M$ as a dense open subset of a compact manifold $\bar{M}$, now without boundary. That is, the extra points $C=\bar{M}\setminus M$ are all interior points of $\bar{M}$. My question is two-fold.

a. Given $M$, what are the possibilities for $C$ and $M$ as topological spaces/manifolds? Is there an analog equivalence relation between possible choices for $C$ like $h$-cobordism for potential boundaries (as in my previous question)?

b. If $C$ and $M$ are somehow fixed in advance (one example of how, though not the exact one I'm most interested in, is to consider $C$ to be a set of points supplied by Cauchy completion with respect to some metric on $M$), What do I need to give $\bar{M}=M\cup C$ manifold structure? In the previous case of adding $C$ as a boundary, I would need a collar neighborhood of the end of $M$ of the form $C\times[0,1)$. But it is not clear to me what I would need to give $C$ the structure of a set of interior points of $\bar{M}$.

Again, I would be interested in answers both in the smooth and topological categories. Specific pointers to the literature are also appreciated.

• This seems to be the same question as mathoverflow.net/questions/34602/compactification-of-a-manifold (which did not get any uplifting answers). Dec 13, 2011 at 18:39
• That question is definitely related. I was in a bit of a rush when posting, otherwise I would have included a link to it. However, I think my question is more focused, hence I hoped it would get some more attention from experts. Dec 14, 2011 at 0:10

If $M$, connected and without boundary but not compact, can be embedded as the interior of a smooth compact manifold $M'$ with boundary, then I believe it can be embedded as a dense set in a smooth closed manifold. Can't you just glue two copies of $M'$ together along the boundary and then let $C$ be a "spine" in the second copy of $M'$?
And $\mathbb R^n$ can be embedded densely in any connected closed $n$-manifold, can't it?
• Yes, $\mathbb R^n$ can be embedded densely into any smooth closed manifold $M$. Put any Riemannian metric on $M$ take any point $p\in M$ and look at the exponential map. The part of $T_pM$ before the cut locus embeds smoothly and the image is dense (with cut locus of $p$ being the complement). Of course the same works with any noncompact manifold as any manifold admits a complete Riemnannian metric. Dec 14, 2011 at 0:11
• Tom, thanks for your answer. Unfortunately, being a bit of a novice in differential topology, I find it a hard to interpret. I don't follow the "spine" part of your construction. How does $M$ become the dense interior of two glued copies of $M'$? Also, the factoid about $\mathbb{R}^n$ is definitely, interesting. Do you know of a reference for it? Dec 14, 2011 at 0:18
• Vitali, thanks! That's helpful. Now, what could be said about the topology of $C$? I would imagine that the possibilities should be somehow deducible from the topology of the end space of $M$ (the space of all proper maps of $[0,\infty)$ into $M$), as are the topologies of potential boundaries $\partial M$ that could be attached to $M$. The question about potential boundaries is quite well studied, but I've not yet identified the literature/terminology relevant for compactification without boundary. Dec 14, 2011 at 0:24