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Any point in the interior of an edge of F has a neighbourhood consisting of the union of the interiors of the edge and the two incident faces, which is clearly homeomorphic to an open disc. Given a vertex v of F , consider an open ball neighbourhood of small radius about its preimage in ∐T : this consists of a disjoint union of “open corners” of triangles of T , since the preimage of the vertex is just some subset of the 3f vertices of ∐T . The image of this open set in F consists of the open corners, glued together along their edges (two open corners meeting at each edge) and with common vertex v.

Here are two gluing patterns forming the torus, one combinatorial and one non-combinatorial. 15. 5) there is an analogous notion of combinatoriality. In this case a cone is simply two edges glued at a single common vertex. Thus, a circle made by gluing intervals is combinatorial if and only if it uses at least three edges. A “two-sided polygon” and a single interval with its ends glued together are ruled out. Finally, the following fact justifies the definitions we have just made: it allows us to consider only combinatorial surfaces, rather than having to work with arbitrary 2-manifolds.

10. Write down proofs of the two easy exercises just stated! 7. Classification of surfaces. The proof of the homeomorphism classification of closed connected combinatorial surfaces is actually based on a very simple idea: one simply looks for nonseparating curves in a surface and does surgery on them, repeating until there are none left. A simple lemma shows that a surface with no non-separating curves is a sphere. Rebuilding the original surface by reversing the surgeries (just as we reverse the pruning in the above lemma) makes it easily identifiable.

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