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# 数学代写|拓扑学代写TOPOLOGY代考|MATH6280 Compact Surfaces Have Finite Triangulations

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## 数学代写|拓扑学代写TOPOLOGY代考|Compact Surfaces Have Finite Triangulations

In the previous section, we stated that one of the invariants used in the classification theorem is the Euler characteristic. However, so far we do not necessarily know how to compute Euler characteristics for all compact surfaces. The problem is that we have defined the Euler characteristic in terms of triangulations. Thus, in order to guarantee that Euler characteristic makes sense for all compact surfaces, we need to show that every compact surface admits a triangulation.

Theorem 4.18 Let $S$ be a compact surface. Then $S$ has a triangulation into finitely many triangles.

This theorem is not easy to prove, and we will skip the proof here. You can find a relatively elementary-but rather long and intricate-proof in Thomassen’s paper [Tho92].

It is worth pointing out that Theorem $4.18$ is not nearly as obvious as it seems. While it is true for surfaces, and also for 3-dimensional manifolds broken up into tetrahedra, it is false in higher dimensions. Freedman in [Fre82] provided an example of a 4-dimensional manifold that is not triangulable, and Manolescu in [Man16] proved that there are also examples in all dimensions $\geq 5$.

## 数学代写|拓扑学代写TOPOLOGY代考|Proof of the Classification Theorem

The proof of the classification theorem uses the ID space representation of a surface. Therefore, we have to begin by showing that every connected compact surface is homeomorphic to an ID space consisting of a polygon of some finite number $2 \mathrm{~N}$ of sides that are identified in pairs. We can argue, as follows, that this is true. First, apply Theorem $4.18$ to the compact surface $S$ to decompose it into a union of cells $\left{T_1, \ldots, T_N\right}$ that satisfy all the properties of a valid triangulation given in the last chapter. Label each edge on $S$ with a unique identifier $a_1, \ldots, a_M$, and transfer these labels to the appropriate edges of all the $T_i$. Thus, each identifier is used exactly twice as a label among the edges of all the $T_i$ ‘s. Also, for each $i$, there is a planar triangle $T_i^{\prime}$ that is homeomorphic to the cell $T_i$. Let us label the edges of the $T_i^{\prime}$ ‘s using the same labels as the $T_i$ ‘s.

At this point, we have already shown that $S$ is homeomorphic to an ID spacenamely the union of all triangles $T_1^{\prime}, \ldots, T_N^{\prime}$ whose edges are identified according to the assignment of labels we have made. However, this ID space isn’t a polygon! To get a polygon, first start by choosing any two triangles in $\left{T_1^{\prime}, \ldots, T_N^{\prime}\right}$ that have an edge with the same label, and gluing them together along the labeled edge, as shown in Figure 4.5. Now we have an ID space for $S$ consisting of $N-2$ triangles and one lozenge. Now keep repeating this process until there are no triangles left. Every time you add a triangle, a pair of commonly labeled edges disappears. Thus the process terminates in a polygon with $2 \mathrm{~N}$ boundary edges identified in pairs.

We will now prove the classification theorem by applying cut-and-paste operations to this ID space until we have an equivalent ID space that we recognize as one of the three different kinds of surfaces listed in the theorem. The following five steps allow us to reach this goal.

## 数学代写|拓扑学代写TOPOLOGY代考|紧凑曲面具有有限三角形

Missing or unrecognized delimiter for \left中的任意两个三角形，它们有一条具有相同标签的边，并将它们沿着标签的边粘在一起，如图4.5所示。现在我们有一个$S$的ID空间，由$N-2$三角形和一个菱形组成。现在继续重复这个过程，直到没有三角形了。每增加一个三角形，就会有一对共同标记的边消失。因此，这个过程的终点是一个具有成对标识的2美元边界边的多边形。

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