{
  "id": "1708.07921",
  "version": "v1",
  "published": "2017-08-26T02:50:02.000Z",
  "updated": "2017-08-26T02:50:02.000Z",
  "title": "Section problems for configuration spaces of surfaces",
  "authors": [
    "Lei Chen"
  ],
  "comment": "23 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "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 PConf$_n(S)$ be the space of ordered $n$-tuples of distinct points in $S$. Let $f_n(S): \\text{PConf}_{n+1}(S) \\to \\text{PConf}_n(S)$ be the map given by $f_n(x_0,x_1,\\ldots,x_n):=(x_1,\\ldots,x_n)$. We will classify all continuous sections of $f_n$ by proving: 1. If $S=\\mathbb{R}^2$ and $n>3$, any section of $f_{n}(S)$ is either \"adding a point at infinity\" or \"adding a point near $x_k$\". (We define these two terms in Section 2.1, whether we can define \"adding a point near $x_k$\" or \"adding a point at infinity\" depends in a delicate way on properties of $S$.) 2. If $S=S^2$ a $2$-sphere and $n>4$, any section of $f_{n}(S)$ is \"adding a point near $x_k$\", if $S=S^2$ and $n=2$, the bundle $f_n(S)$ does not have a section. (We define this term in Section 3.2) 3. If $S=S_g$ a surface of genus $g>1$ and for $n>1$, the bundle $f_{n}(S)$ does not have a section.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-26T02:50:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "section problems",
      "configuration spaces",
      "basic questions",
      "close-to-sharp answer",
      "distinct points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}