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>=4 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?

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?