曲面的拓扑复杂性
The Topological Complexity of a Surface
摘要：
设$p$是黎曼曲面到黎曼球面$\mathbb{P}^1$的分支覆盖，具有分支集$B \subset \mathbb{P}^1$ .如果$\mathbb{P}^1 \setminus B$不具有双曲结构，则定义$p$的复杂度为无穷大，否则定义其次数与$\mathbb{P}^1 \setminus B$的双曲面积的乘积为无穷大.曲面$S$的拓扑复杂度定义为分支覆盖$M \to \mathbb{P}^1$的所有复杂度之集的下确界，其中$M$是与$S$同胚的黎曼曲面.我们证明了如果$S$是亏格$g$的连通、闭、可定向曲面，则其拓扑复杂度$C_{\text{top}}(S)$由下式给出：\[C_{\text{top}}(S)= \left\{ \begin{array}{cl} 2π(2g+1) & \mbox{if } g \geq 1, 6 π& \mbox{if } g=0. \end{array} \right.\]
英文摘要：
Let $p$ be a branched covering of a Riemann surface to the Riemann sphere $\mathbb{P}^1$, with branching set $B \subset \mathbb{P}^1$. We define the complexity of $p$ as infinity, if $\mathbb{P}^1 \setminus B$ does not admit a hyperbolic structure, or the product of its degree and the hyperbolic area of $\mathbb{P}^1 \setminus B$, otherwise. The topological complexity of a surface $S$ is defined as the infimum of the set of all complexities of branched coverings $M \to \mathbb{P}^1$, where $M$ is a Riemann surface homeomorphic to $S$. We prove that if $S$ is a connected, closed, orientable surface of genus $g$, then its topological complexity, $C_{\text{top}}(S)$, is given by: \[C_{\text{top}}(S)= \left\{ \begin{array}{cl} 2π(2g+1) & \mbox{if } g \geq 1, 6 π& \mbox{if } g=0. \end{array} \right.\]
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