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 $n\ge4$ 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?

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?