# 数学代写|黎曼曲面代写Riemann surface代考|The degree of a holomorphic map between compact Riemann surfaces

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## 数学代写|黎曼曲面代写Riemann surface代考|The degree of a holomorphic map between compact Riemann surfaces

5.3 Theorem/Definition: Let $f: R \rightarrow S$ be a non-constant holomorphic map between compact, connected Riemann surfaces. Then the number
$$\operatorname{deg}(f)=\sum_{r \in f^{-1}(s)} v_f(r)$$
is independent of the choice of point $s \in S$, and is called the degree of the map $f$.
5.4 Note: If $f$ is constant, we define $\operatorname{deg}(f)=0$. Note that $\operatorname{deg}(f)>0$ otherwise.
5.5 Proposition: For all but finitely many $s \in S, \operatorname{deg}(f)=\left|f^{-1}(s)\right|$, the number of solutions to $f(x)=s$. For any $s,\left|f^{-1}(s)\right| \leq \operatorname{deg}(f)$.

Proof: Clear from the theorem and the fact that the points $r$ with $v_f(r)>1$ are finite in number (see ‘good behaviour’ theorem).
For the proof of (5.3), we need the following lemma.
5.6 Lemma: Let $f: X \rightarrow Y$ be a continuous map of topological Hausdorff spaces, with $X$ compact. Let $y \in Y$ and $U$ be a neighbourhood of $f^{-1}(y)$. Then there exists some neighbourhood $V$ of $y$ with $f^{-1}(V) \subseteq U$.

Proof of the lemma: As $V$ varies over all neighbourhoods of $y \in Y, \cap \bar{V}={y}$, by the Hausdorff property. Then, $\bigcap f^{-1}(\bar{V})=f^{-1}(y)$. But then, $\cap f^{-1}(\bar{V}) \cap(X \backslash U)=\emptyset$. Now the $f^{-1}(\bar{V})$ and $(X \backslash U)$ are closed sets, and by compactness of $X$, some finite intersection is already empty. So $X \backslash U \cap f^{-1}\left(\bar{V}_1\right) \cap \cdots \cap f^{-1}\left(\bar{V}_n\right)=\emptyset$, or $f^{-1}\left(\bar{V}_1 \cap \cdots \cap \bar{V}_n\right) \subseteq U$, in particular $f^{-1}\left(V_1 \cap \cdots \cap V_n\right) \subseteq U$. But $V_1 \cap \cdots \cap V_n$ is a neighbourhood of $y$ in $Y$.

## 数学代写|黎曼曲面代写Riemann surface代考|The Riemann-Hurwitz formula

For a holomorphic map $f$ between compact connected Riemann surfaces $R$ and $S$, the theorem gives a formula relating

the degree of the map,

the topologies of $R$ and $S$,

the (local) valencies of the map.

As a preliminary, we need the following result from topology.
5.15 Classification of compact orientable surfaces: Any compact, connected, orientable surface without boundary is homeomorphic to one of the following: $g$ is the genus and counts the ‘holes’. There is another depiction of these surfaces as ‘spheres with handles’,and then $g$ counts the number of handles. The Euler characteristic of these surfaces is (provisionally) defined as $2-2 g$.
5.16 Definition: Let $f: R \rightarrow S$ be a non-constant holomorphic map between compact connected Riemann surfaces. The total branching index $b$ of $f$ is
$$\sum_{s \in S} \sum_{r \in f^{-1}(s)}\left(v_f(r)-1\right)=\sum_{s \in S}\left(\operatorname{deg}(f)-\left|f^{-1}(s)\right|\right) .$$
Note that this sum is finite. It counts the total number of ‘missing’ solutions to $f(x)=s$.
5.17 Theorem (Riemann-Hurwitz formula): With $f$ as above,
$$\chi(R)=\operatorname{deg}(f) \chi(S)-b$$
Equivalently, in terms of the genus,
$$g(R)-1=(\operatorname{deg} f)(g(S)-1)+\frac{1}{2} b,$$
where $g(X)$ denotes the genus of $X$.

# 黎曼曲面代写

## 数学代写|黎曼曲面代写Riemann surface代考|The degree of a holomorphic map between compact Riemann surfaces

5.3定理/定义:设$f: R \rightarrow S$为紧连黎曼曲面间的非常全纯映射。然后是数字
$$\operatorname{deg}(f)=\sum_{r \in f^{-1}(s)} v_f(r)$$

5.4注:如果$f$为常数，则定义$\operatorname{deg}(f)=0$。请注意$\operatorname{deg}(f)>0$否则。
5.5命题:对于除有限数$s \in S, \operatorname{deg}(f)=\left|f^{-1}(s)\right|$外的所有解$f(x)=s$的个数。对于任何$s,\left|f^{-1}(s)\right| \leq \operatorname{deg}(f)$。

5.6引理:设$f: X \rightarrow Y$为拓扑Hausdorff空间的连续映射，且$X$紧。让$y \in Y$和$U$成为$f^{-1}(y)$的邻居。那么$y$与$f^{-1}(V) \subseteq U$存在邻域$V$。

## 数学代写|黎曼曲面代写Riemann surface代考|The Riemann-Hurwitz formula

$R$和$S$的拓扑结构，

5.15紧致可定向曲面的分类:任何紧致的、连通的、无边界的可定向曲面都是同纯的:$g$是属并计算“孔”。还有另一种描述这些表面为“带手柄的球体”，然后$g$计算手柄的数量。这些表面的欧拉特性(暂时)定义为$2-2 g$。
5.16定义:设$f: R \rightarrow S$为紧连黎曼曲面间的非常全纯映射。$f$的总分支索引$b$为
$$\sum_{s \in S} \sum_{r \in f^{-1}(s)}\left(v_f(r)-1\right)=\sum_{s \in S}\left(\operatorname{deg}(f)-\left|f^{-1}(s)\right|\right) .$$

5.17定理(黎曼-赫维茨公式):有$f$，
$$\chi(R)=\operatorname{deg}(f) \chi(S)-b$$

$$g(R)-1=(\operatorname{deg} f)(g(S)-1)+\frac{1}{2} b,$$

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