{
  "id": "2310.14156",
  "version": "v1",
  "published": "2023-10-22T02:44:19.000Z",
  "updated": "2023-10-22T02:44:19.000Z",
  "title": "Configuration space integrals and formal smooth structures",
  "authors": [
    "Jianfeng Lin",
    "Yi Xie"
  ],
  "comment": "79 pages, comments welcome",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "Watanabe disproved the 4-dimensional Smale conjecture by constructing topologically trivial $D^{4}$-bundles over spheres and showing that they are smoothly nontrivial using configuration space integrals. In this paper, we define a new version of configuration space integrals that only relies on a formal smooth structure on the $D^{4}$-bundle (i.e., a vector bundle structure on the vertical tangent microbundle). It coincides with Watanabe's definition when the $D^{4}$-bundle is smooth. We obtain several applications. First, we give a lower bound (in terms of the graph homology) on the dimension of the rational homotopy and homology groups of $\\textrm{Top}(4)$ and $\\textrm{Homeo}(S^4)$ (the homeomorphism group of $\\mathbb{R}^4$ and $S^4$). In particular, this implies that $\\textrm{Top}(4)$ and $\\textrm{Homeo}(S^4)$ are not rationally equivalent to any finite-dimensional CW complexes. Second, we discover a generalized Miller-Morita-Mumford class $\\kappa_{\\theta}(\\pi)\\in H^{3}(B;\\mathbf{Q})$, which is defined for any topological 4-manifold bundle $X\\to E\\to B$. This class obstructs the existence of a formal smooth structure on the bundle. Third, we show that for any compact, orientable, smooth 4-manifold $X$ (possibly with boundary), the inclusion map from its diffeomorphism group to its homeomorphism group is not rationally $2$-connected (hence not a weak homotopy equivalence). This implies that the space of smooth structures on $X$ has a nontrivial rational homotopy group in dimension 2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-22T02:44:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "configuration space integrals",
      "formal smooth structure",
      "nontrivial rational homotopy group",
      "homeomorphism group",
      "vector bundle structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 79,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}