Let $M$ be an $n$-dimensional manifold (smooth or topological). I call $\bar{M}$ a *compactification* of $M$ if it is an $n$-dimensional compact manifold with boundary $\partial \bar{M}$, an $(n-1)$-dimensional manifold, such that $M$ is the interior of $\bar{M}$. I understand that not every manifold $M$ has such a compactification. The obstructions have been discussed in some previous MO questions ([22441](https://mathoverflow.net/questions/22441), [34602](https://mathoverflow.net/questions/34602)). Essentially, no compactification can exist if the [ends](http://en.wikipedia.org/wiki/End_%2528topology%2529) of the manifold are too "wild".


However, I am more interested in how two compactifications are related to each other, provided they exist. For simplicity, let me assume that $M$ has only one end and a compactification exists. If $\bar{M}\_1$ and $\bar{M}\_2$ are both compactifications, what can be said about the relationship between them and between $\partial\bar{M}\_1$ and $\partial\bar{M}\_2$? Does there exist some kind of "universal" compactification $\bar{M}\_U$ from which both $\bar{M}\_1$ and $\bar{M}\_2$ could be obtained? If these questions are easier to answer in the topological category, then I would be quite happy with just that information.


Here's an example. Consider $M=\mathbb{R}^n$ ($n\ge 2$). It has one end. An obvious compactification is to consider it as the interior of a closed ball $\bar{M}\_1=\bar{B}^n$, so the boundary is a sphere $\partial\bar{M}\_1=S^{(n-1)}$. However, I can apply a homotopy to $\bar{B}^n$ which maps the boundary into itself and restricts to a homoemorphism on the interior. Composing this operation with the embedding of $M$ in $\bar{M}\_1$ I get a different compactification $\bar{M}\_2$. By construction, there is a map $\bar{M}\_1 \to \bar{M}\_2$, which is a homeomorphism between the interiors but only a homotopy between the boundaries. In particular, the homotopy could blow up a point on $\partial\bar{M}\_1$ to a closed set with open interior. Then two curves that had the same end point on the boundary of $\bar{M}\_1$ could be mapped to curves with distinct end points on the boundary of $\bar{M}\_2$. Based on this example, I would naively guess that $\mathbb{R}^n$ can be compactified by adding an $S^{(n-1)}$ boundary and all other compactifications can be obtained by applying a homotopies to any one element of this class of compactifications. So this compactification could be considered universal. How far is this intuition from reality?


I'm aware of resources like [L. Siebenmann's thesis](http://www.maths.ed.ac.uk/~aar/surgery/thesis.pdf) and the book [Ends of Complexes](http://www.maths.ed.ac.uk/~aar/books/ends.pdf) by Ranicki and Hughes. Unfortunately, I don't have enough algebraic and topological background to immediately see whether they contain the answer or how to dig it out if they do. So precise suggestions for where to look would also be helpful!



Uniqueness of compactification of an end of a manifold
Suppose $\overline{M}\_i$, $i=0,1$, are compact smooth manifolds with boundary whose interiors are diffeomorphic: let $\psi$ be such a diffeomorphism, and $M$ for either interior (identified via $\psi$). For both manifolds one can find a smooth collar $c\_i : \partial \overline{M}\_i \times [0, 1) \hookrightarrow \overline{M}\_i$, as they are compact. By shrinking $c\_0$ (and re-parametrising) we can suppose it has image inside that of $c\_1$ (restricted to the interior, compared via $\psi$), and then we obtain an embedding
$$\partial \overline{M}\_0 \times [\tfrac{1}{2}, 1) \hookrightarrow \partial \overline{M}\_1 \times (0, 1) \subset \overline{M}\_1 \times [0, 1).$$
The complement of this embedding has two components, precisely one of which is compact and is a cobordism $W : \partial \overline{M}\_0 \leadsto \partial \overline{M}\_1$. 


This is an $h$-cobordism, as each of the $\partial \overline{M}\_i \times [0, 1)$ and $W$ are homotopy equivalent to the space
$$\mathcal{E}(M) := \mathrm{holim}\_{K \subset M \text{compact}} M \setminus K$$
as there are evident cofinal subdiagrams indexed by $K = M \setminus c\_i(\partial \overline{M}\_i \times [0, \epsilon))$ for either $i$, which are homotopically constant.


Thus, the two boundaries $\partial \overline{M}\_i$ are $h$-cobordant (so diffeomorphic if they are simply-connected). If the boundaries are not simply-connected, then I imagine that this $h$-cobordism can be non-trivial, as in Benoît Kloeckner's discussion.


