2017/08/26 by Lei Chen, Chen, Lei · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1708.07921
openalex publication_date 2017/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Section 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 Section 3.2) 3. If S=Sg a surface of genus g>1 and for n>1, we give an easy proof that the bundle fn(S) does not have a section.