I think it's because it's too hard when the manifolds are noncompact.

Closed orientable surfaces are classified nicely by their Euler characteristic, but it's not so clear how to classify noncompact surfaces. For instance, there are things like the sphere minus a Cantor set, or surfaces of genus g minus a Cantor set, or infinite genus things like the following.

Take the infinite graph obtained by looking at the union of the edges in the usual tiling of R^2 by squares. Take a 3-dimensional neighborhood of this and call the boundary S. Do the same thing with the graph obtained by taking the union of edges in the tiling of R^3 by cubes to get a surface T.

These aren't homeomorphic, but you need to do some work to differentiate them.

In higher dimensions the situation is even worse.

Some semi-related awesomeness: A large surface is a space that is a surface, except that you don't require it to be second countable. It's a theorem that there are precisely 2^{\aleph_1} connected large surfaces. I forget who proved this, but if anyone can find it in mathscinet, I would love to know the reference (I'd be your best friend, et cetera).

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I think it's because it's too hard when the manifolds are noncompact.

Closed orientable surfaces are classified nicely by their Euler characteristic, but it's not so clear how to classify noncompact surfaces. For instance, there are things like the sphere minus a Cantor set, or surfaces of genus g minus a Cantor set, or infinite genus things like the following.

Take the infinite graph obtained by looking at the union of the edges in the usual tiling of R^2 by squares. Take a 3-dimensional neighborhood of this and call the boundary S. Do the same thing with the graph obtained by taking the union of edges in the tiling of R^3 by cubes to get a surface T.

These aren't homeomorphic, but you need to do some work to differentiate them.

In higher dimensions the situation is even worse.