{
  "id": "2311.06779",
  "version": "v1",
  "published": "2023-11-12T08:55:21.000Z",
  "updated": "2023-11-12T08:55:21.000Z",
  "title": "Non-existence of circle actions on oriented manifolds with three fixed points except in dimensions 4, 8 and 16",
  "authors": [
    "Hao Dong",
    "Jianbo Wang"
  ],
  "comment": "12 pages,1 table",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "Let $M$ be a smooth $4k$-dimensional orientable closed manifold, and assume that $M$ has at most two non-zero Pontrjagin numbers which are associated to the top dimensional Pontrjagin class and the square of the middle dimensional Pontrjagin class. We prove that the signature of $M$ being equal to $1$ and the $\\hat{A}$-genus of $M$ vanishing cannot hold at the same time except $\\dim M=8, 16$. As an application, we claim that the dimensions of oriented $S^1$-manifolds with exactly three fixed points are only $4, 8$ and $16$, and the rational projective plane whose dimension is greater than $16$ has no smooth non-trivial $S^1$-action.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-12T08:55:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57S15",
      "57S25",
      "57R20",
      "37B05"
    ],
    "keywords": [
      "fixed points",
      "circle actions",
      "oriented manifolds",
      "non-existence",
      "middle dimensional pontrjagin class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}