In this paper, we give a close-to-sharp answer to the basic questions: When is there a continuous way to add a point to a configuration of n ordered points on a surface S of finite type so that all the points are still distinct? When this is possible, what are all the ways to do it? More precisely, let PConfn(S) be the space of ordered n-tuple of distinct points in S. Let fn(S) : PConfn+1(S) →PConfn(S) be the map given by fn(x0,x1,…,xn) := (x1,…,xn). We classify all continuous sections of fn up to homotopy by proving the following: (1)If S = ℝ2 and n > 3, any section of fn(S) is either “adding a point at infinity” or “adding a point near xk”. (We define these two terms in Sec. 2.1; whether we can define “adding a point near xk” or “adding a point at infinity” depends in a delicate way on properties of S.) (2)If S = S2 a 2-sphere and n > 4, any section of fn(S) is “adding a point near xk”; if S = S2 and n = 2, the bundle fn(S) does not have a section. (We define this term in Sec. 3.2). (3)If S = Sg a surface of genus g > 1 and for n > 1, we give an easy proof of [D. L. Gonçalves and J. Guaschi, On the structure of surface pure braid groups, J. Pure Appl. Algebra 182 (2003) 33–64, Theorem 2] that the bundle fn(S) does not have a section.

中文翻译：

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曲面构型空间的截面问题

在本文中，我们对基本问题给出了一个近乎尖锐的答案：什么时候有一种连续的方法可以将一个点添加到一个配置中n曲面上的有序点小号有限类型，以便所有点仍然不同？如果这是可能的，有什么方法可以做到这一点？更准确地说，让 PConfn(小号)是有序空间n- 不同点的元组小号. 让Fn(小号) ： 会议n+1(小号) →会议n(小号)是由给出的地图Fn(X0,X1,…,Xn) ：= (X1,…,Xn). 我们对所有连续部分进行分类Fn通过证明以下达到同伦： (1)如果小号 = ℝ2和n > 3, 的任何部分Fn(小号)要么是“在无穷远处加一个点”，要么是“在附近加一个点”Xķ”。（我们在 2.1 节中定义了这两个术语；我们是否可以定义“在附近添加一个点Xķ”或“在无穷远处添加一个点”以一种微妙的方式取决于小号.) (2)如果小号 = 小号2一种2-球体和n > 4, 的任何部分Fn(小号)是“在附近加一个点Xķ”；如果小号 = 小号2和n = 2, 捆绑Fn(小号)没有节。（我们在第 3.2 节中定义了这个术语）。 (3)如果小号 = 小号G属的表面G > 1并且对于n > 1，我们给出了 [DL Gonçalves 和 J. Guaschi, On the structure of surface pure braid groups,J.纯应用。代数 182(2003) 33–64, Theorem 2]Fn(小号)没有节。