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, 34602). Essentially, no compactification can exist if the ends 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 and the book Ends of Complexes 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!