{
  "id": "1408.2010",
  "version": "v2",
  "published": "2014-08-09T02:52:09.000Z",
  "updated": "2014-09-10T04:38:24.000Z",
  "title": "Mapping spaces from projective spaces",
  "authors": [
    "Mitsunobu Tsutaya"
  ],
  "comment": "25 pages, 3 figures; made many corrections and changes, and added Figure 3",
  "categories": [
    "math.AT"
  ],
  "abstract": "We denote the $n$-th projective space of a topological monoid $G$ by $B_nG$ and the classifying space by $BG$. Let $G$ be a topological monoid of pointed homotopy type of a CW complex and $G'$ a grouplike topological monoid. We prove the weak equivalence between the pointed mapping space $\\mathrm{Map}_0(B_nG,BG)$ and the space of all $A_n$-maps from $G$ to $G'$. This fact has several applications. As the first application, we show that the connecting map $G\\rightarrow\\mathrm{Map}_0(B_nG,BG)$ of the evaluation fiber sequence $\\mathrm{Map}_0(B_nG,BG)\\rightarrow\\mathrm{Map}(B_nG,BG)\\rightarrow BG$ is delooped. As other applications, we consider higher homotopy commutativity, $A_n$-types of gauge groups and $T_k^f$-spaces by Iwase--Mimura--Oda--Yoon.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-09T02:52:09.000Z",
      "abstract": "We denote the $n$-th projective space of a topological monoid $G$ by $B_nG$ and the classifying space by $BG$. Let $G$ be a topological monoid of pointed homotopy type of a CW complex and $G'$ a grouplike topological monoid. We prove the weak equivalence between the pointed mapping space $\\mathrm{Map}_0(B_nG,BG)$ and the space of all $A_n$-maps from $G$ to $G'$. This fact has several applications. As the first application, we show that the evaluation fiber sequence $\\mathrm{Map}_0(B_nG,BG)\\rightarrow\\mathrm{Map}(B_nG,BG)\\rightarrow BG$ extends to the right. As other applications, we consider higher homotopy commutativity, $A_n$-types of gauge groups and $T_k^f$-spaces by Iwase--Mimura--Oda--Yoon.",
      "comment": "20 pages, 2 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-10T04:38:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54C35",
      "18D50",
      "55P45"
    ],
    "keywords": [
      "mapping space",
      "topological monoid",
      "evaluation fiber sequence",
      "higher homotopy commutativity",
      "cw complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}