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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.

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This seems to be the same question as mathoverflow.net/questions/34602/compactification-of-a-manifold (which did not get any uplifting answers). –  Igor Rivin Dec 13 '11 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. –  Igor Khavkine Dec 14 '11 at 0:10

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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?

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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. –  Vitali Kapovitch Dec 14 '11 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? –  Igor Khavkine Dec 14 '11 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. –  Igor Khavkine Dec 14 '11 at 0:24
    
@Igor topology of cut locus can be very complicated (btw it need not be a manifold) and not much is known about it beyond dimension 2. It also strongly depends on the metric and can really be quite wild in high dimensions. This does show that there is no kind of uniqueness of compactification. In general it seems pretty clear to me (and should not be hard to prove) that an open manifold can densely openly embed into another one if and only if it can just openly embed. The latter can have topological obstructions as Igor Belegradek's example shows but this is really a different question. –  Vitali Kapovitch Dec 14 '11 at 0:47

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