中文摘要：

将紧致平环中的曲线复形的定义推广到带有三个洞的球面P上，称这个曲线复形为曲面P的弧复形，研究曲面P中的本质弧和P的弧复形中的距离，证明了以下的一个结果：若P是一个带有三个洞的球面，a，b为P的弧复形中的任意两个顶点，则dc（a，b）≤4．

英文摘要：

Firstly, the definition of the curve complex for a compact annulus, which is called arc complex of the surface P, is generalized to a there holed sphere P. Secondly, we study the essential arcs of the surface P and the distance in the acr complex of this surface. Finally, it is shown that if P is a there holed sphere, and a and b are distinct vertices of the acr complex, then dc （a, b）≤4.