Compactifications of manifolds
In manifold topology, we often like embedding topological spaces into Euclidean space, in particular, as a compact subset because of the special and nice properties compact spaces have. The idea of compactification itself, which involves adding "points at infinity" to a given space in order to create a new, larger space with desirable properties, was first introduced by French mathematician Émile Borel in the early 1920s. Borel's work on compactification was further developed by other mathematicians in the decades that followed, including Alexandroff, Stone, and Čech, and the theory of compactifications became an important tool in many areas of mathematics, including analysis, algebraic geometry, and topology. Although there are various ways to compactify a given space, a compactification that preserves certain properties of the original space often piques interest. One desirable way to compactify a manifold is called completion, in which the final product is a compact manifold with boundary and the "compactification points" all lie in that boundary. For instance, $R^n$ can be compactified by addition of an $(n - 1)$-sphere. Manifolds are not always completable even contractible open manifolds. For example, the Whitehead manifold is not completable. In higher dimensions, the Davis contractible open manifolds are also not completable. Here the "fundamental group at infinity" plays a key role. Stallings gave a characterization of $R^n$ using that idea. Then Siebenmann, in his 1965 PhD thesis, characterized manifolds with compact boundary which are completable. As part of my dissertation, Craig Guilbault and I proved a characterization of completable $M^m (m \geq 6)$ which removes the requirement that $\partial M^m$ be compact. This resolved a problem that had persisted since the 60s, when geometric topology began to shift from a descriptive to a constructive approach. Nonetheless, our classification, while helpful, still have limitations. Some interesting manifolds, such as the Davis contractible open manifolds, are not completable...
The recognition of manifolds
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