# Classification Theorem for Non-Compact 2-Manifolds? 2-Manifolds With Boundary?

I recently learned about the Classification Theorem for compact 2-manifolds. Is there a similar classification theorem for ALL 2-manifolds, not just the compact ones?

Moreover, is there a theorem which classifies the 2-manifolds with boundary?

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Any open subset of R^2 is a 2-manifold, so my impression is that this classification is hopeless in general. – Qiaochu Yuan Sep 27 '10 at 18:33
It's pretty fussy but not quite hopeless. That's one of the most interesting things about doing mathematics -- in that you get to see the blurry line where "hard" problems live evolve and take shape, some problems become "doable" and some become "hopeless", some stay mysteriously "hard" and never budge... – Ryan Budney Sep 27 '10 at 19:00
For the classification of compact connected 2-manifolds with boundary, you can look at Massey's "Algebraic Topology. An introduction". Notice that "the other" Massey, "Basic course in Algebraic Topology", though merging the former with his "Singular homology theory", does no longer contain the aforementioned classification. – a.r. Sep 27 '10 at 23:30

Yes, there's a classification theorem for non-compact 2-manifolds.

This paper gives the classification for triangulable 2-manifolds:

http://www.jstor.org/stable/1993768

That an arbitrary (2nd countable, Hausdorff) topological 2-manifold admits a triangulation is fairly classical. Ahlfors book "Riemann Surfaces" has a proof. There are others available, see for example this list:

http://mathoverflow.net/questions/17578/triangulating-surfaces

If all you're interested in is compact manifolds with boundary, you get that classification immediately from the closed manifold case. Because if you have a compact manifold with boundary, its boundary is a disjoint union of circles. So cap those circles off with discs to produce a closed manifold. So compact manifolds with boundary are classified by the closed manifold you get by "capping off" and the number of boundary circles you started with.

Non-compact manifolds have a more delicate classification -- think for example about the complement of a Cantor set in a compact surface.

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Should I interpret that complement of the Cantor set is taken account of by the classification theorem in the first paper that you have linked to? – Anirbit Dec 13 '10 at 20:16
Yes, that's included in this classification. – Ryan Budney Dec 13 '10 at 20:22

Amazingly, the complete classification of noncompact 2-manifolds with boundary was not completed until 2007. Here's the reference:

A. O. Prishlyak and K. I. Mischenko, Classification of noncompact surfaces with boundary, Methods Funct. Anal. Topology 13 (2007), no. 1, 62–66.

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