### Background

It is well-known that the compact two-dimensional manifolds are completely classified (by their orientability and their Euler characteristic).

I'm also under the impression that there is also a classification for compact three-dimensional manifolds coming from the proof of the Geometrization Conjecture and related work.

Unfortunately for $n\ge4$ no similar classification is possible because it can be shown that it is at least as hard as the word problem for groups. Thus for higher-dimensional manifolds we instead focus on classifying all the _simply-connected_ compact manifolds.

### My question

Why in these "classification problems" are we only considering _compact_ manifolds? Is there an easy reason why we restrict ourselves to the classification of compact manifolds? Does a classification of general (not necessarily compact) manifolds follow easily from a classification of compact manifolds?