{
  "id": "1606.01828",
  "version": "v1",
  "published": "2016-06-06T17:02:32.000Z",
  "updated": "2016-06-06T17:02:32.000Z",
  "title": "Diffeomorphism Stability and Codimension Four",
  "authors": [
    "Curtis Pro",
    "Frederick Wilhelm"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "Given $k\\in \\mathbb{R},$ $v,$ $D>0,$ and $n\\in \\mathbb{N},$ let $\\left\\{ M_{\\alpha }\\right\\} _{\\alpha =1}^{\\infty }$ be a Gromov-Hausdorff convergent sequence of Riemannian $n$--manifolds with sectional curvature $\\geq k,$ volume $>v,$ and diameter $\\leq D.$ Perelman's Stability Theorem implies that all but finitely many of the $M_{\\alpha }$s are homeomorphic. The Diffeomorphism Stability Question asks whether all but finitely many of the $ M_{\\alpha }$s are diffeomorphic. We answer this question affirmatively in the special case when all of the singularities of the limit space occur along smoothly and isometrically embedded Riemannian manifolds of codimension $\\leq 4$. We then describe several applications. For instance, if the limit space is an orbit space whose singular strata are of codimension at $\\leq 4,$ then all but finitely many of the $M_{\\alpha }$s are diffeomorphic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-06T17:02:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20"
    ],
    "keywords": [
      "codimension",
      "diffeomorphism stability question asks",
      "perelmans stability theorem implies",
      "limit space occur",
      "gromov-hausdorff convergent sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}