By John G. Hocking

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**Example text**

We will also be able to choose vertices in the mesh within each Bi and join them to these paths in such a way as to obtain a map M which is a coarsening of T1n and which has one vertex within each Bi , one edge for each tube Ti , and one face for each face of T2 . It will then follow that the Euler characteristic is the same for M and T2 . Figure 11. A refinement of T1 Given an edge e of T2 running from vertex v1 to vertex v2 , let B1 and B2 denote the ε1 -balls around v1 and v2 , respectively, and let T denote the tube around e between B1 and B2 .

Now what about adding an inverted handle? Why has it been left off our list? It turns out that attaching an inverted handle is equivalent to attaching two M¨obius caps. Indeed, just as attaching a handle to two holes is equivalent to attaching a torus with a hole to a single hole, we can think of attaching an inverted handle as attaching a Klein bottle with a hole. A planar model of the Klein bottle on the 4-gon is given by the identifications aabb, and figure 27 suggests a proof that each of aa and bb is equivalent to attaching a M¨obius cap.

So far we have seen planar models for four different surfaces; two of these used the 2-gon and two the 4-gon. We can list these in terms of the identifications made between various sides as we complete a circuit around the boundary, as explained last time: edge identifications surface Euler characteristic aa−1 sphere 2 aa projective plane 1 −1 −1 aba b torus 0 abab−1 or aabb Klein bottle 0 To compute the Euler characteristic χ of a planar model on a 2m-gon, we may observe that F = 1 and E = m after passing to the quotient space, so the only variable is the number of vertices after all identifications have been made.