# 3-manifolds, cubes with handles

Somebody knows where I can find some proof of the following fact: If F is compact, connected 2-manifold with nonempty boundery why there exist n=1-X(F) pairwise disjoint properly embedded 1-cells {A1,...,An} in F which cut F to a 2-cell.

This claim is formulated in a proof of theorem 2.3, 3-manifolds, John Hempel

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This is better suited to math.stackexchange.com – Richard Kent Nov 1 '11 at 20:36
This is the classification theorem for compact 2-manifolds. You can find a proof in a variety of places. Singer and Thorpe's "Lecture Notes on Elementary Geometry and Topology" is a nice one. That's a bit old, though. I think this is also in many of the textbooks by Stillwell that involve surfaces. Proofs boil down to triangulability + Poincare duality + the simply connected case. – Ryan Budney Nov 1 '11 at 20:36

You can start from the classification theorem of compact surfaces (every compact connected 2-manifold is homeomorphic to either a connected sum of $g$ tori or a connected sum of $k$ projective planes, with a finite number of disks removed). To prove this you need the triangulation theorem, which is proved in Moise's Geometric Topology in Dimensions 2 and 3, and once you have a triangulation you can prove by hand that you will always get a connected sum of tori or projective planes.