{
  "id": "2405.13257",
  "version": "v1",
  "published": "2024-05-22T00:09:03.000Z",
  "updated": "2024-05-22T00:09:03.000Z",
  "title": "On Topological Complexity of $(r,ρ(R))$-mild spaces",
  "authors": [
    "Smail Benzaki",
    "Youssef Rami"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AT"
  ],
  "abstract": "In this paper, we first prove the existence of relative free models of morphisms (resp. relative commutative models) in the category of $DGA(R)$ (resp. $CDGA(R)$), where $R$ is a principal ideal domain containing $\\frac{1}{2}$. Next, we restrict to the category of $(r,\\rho(R))$-H-mild algebras and we introduce, following Carrasquel's characterization, $secat(-, R)$, the sectional category for surjective morphisms. We then apply this to the $n$-fold product of the commutative model of an $(r,\\rho(R))$-mild CW-complex of finite type to introduce $TC_n(X,R)$, $mTC_n(X,R)$ and $HTC_n(X,R)$ which extend well known rational topological complexities. We do the same for $\\operatorname{sc(-, \\mathbb{Q})}$ to introduce analogous algebraic $\\operatorname{sc(-,R)}$ in terms of their commutative models over $R$ and prove that it is an upper bound for $secat(-, R)$. This also yields, for any $(r,\\rho(R))$-mild CW-complex, the algebraic $tc_n(X,R)$, $mtc_n(X,R)$ and $Htc_n(X,R)$ whose relation to the homology nilpotency is investigated. In the last section, in the same spirit, we introduce in $DGA(R)$, $secat(-, R)$, $\\operatorname{sc(-,R)}$ and their topological correspondents. We then prove, in particular, that $ATC_n(X,R)\\leq TC_n(X,R)$ and $Atc_n(X,R)\\leq tc_n(X,R)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-22T00:09:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P62",
      "55M30"
    ],
    "keywords": [
      "topological complexity",
      "mild spaces",
      "commutative model",
      "mild cw-complex",
      "relative free models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}