A surface is simply a 2-manifold. The classification theorem for compact connected surfaces (with boundary) is commonly regarded in the categories TOP, DIFF and PL. Well known proofs (e.g. via triangulations, or Morse theory) yield the same classification because of results that connect these categories for surfaces. Informally speaking, here is what I know to be true for compact connected surfaces
[TOP & PL]. Topological surfaces always admit a triangulation, and any two triangulations of a surface are piecewise-linear equivalent (Hauptvermutung for surfaces)
[DIFF & PL (without using 1.)]. Every smooth surface admits a PL-structure, as every smooth manifold does (See the paper "On $C^{1}$ Complexes", by J.H.C. Whitehead).
Next is where I seek to be enlightened:
- [TOP & DIFF, (without using either 1. or 2.)]. Two smooth surfaces are diffeomorphic iff they are homeomorphic, and a topological surface always admits a smoothing.
Where can I find a formal statement, and a complete proof of 3.?
Finally, consider non-compact connceted surfaces (with boundary). There seems to be a complete classification of non-compact connected triangulable surfaces with boundary (See the paper "Classification of Noncompact Surfaces with Boundary", by A.O. Prishlyak and K.I. Mischenko).What about the TOP and DIFF categories? That is, do the results 1-3 above hold for non-compact surfaces?
NOTE: I want to mention the post Classification problem for non-compact manifolds for a related, yet different discussion. The paper: "On the Classification of Noncompact Surfaces", by Ian Richards is mentioned there in a comment. This paper considers the case of non-compact triangulable surfaces without boundary.
Thank you!