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By I Juhasz

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Then, the surface is the connected sum of m projective planes and n tori, which by the lemma is homeomorphic to the connected sum of m + 2n projective planes. 1. 5. It is clear that we can also work the process described above backwards; whenever there are three pairs of the second kind, we can replace them by one pair of the second kind and two pairs of the first kind. 1 to any connected sum of which three or more of the summands are projective planes. 1, which may be preferable in some cases, results.

Tn, 8n, Tn+1 = T1, such that each e,- is an edge of T,- and Ti“, whereas each T,- has ej_1 and e,- as edges. An easy argument shows that there can only be one such cycle corresponding to each boundary component Bi. From the conditions imposed on the triangulation of M, it is clear that the union of the triangles T1, T2, . 31 illustrates how such a region might look when n = 17. - for each boundary component B;, l _S_ i g k. Let T1, . -, 1 _S_ i g k. 1 (as described in Section 7). 29(a)] results.

Note that in the above constructions we were careful to cut the holes in a straight line so that it was clear that we could make the cuts cl, . , ch in such a way that they would be disjoint except for one end point. Next, we consider triangulations of compact bordered surfaces. The definition is exactly the same as that given in Section 6 for the case of compact surfaces. 29 Orientable surface of genus 2 with four holes. surface, every edge is an edge of exactly two triangles. However, if a bordered surface is triangulated, some edges will be edges of only one triangle.

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