{
  "id": "2208.05143",
  "version": "v1",
  "published": "2022-08-10T04:40:44.000Z",
  "updated": "2022-08-10T04:40:44.000Z",
  "title": "Brieskorn spheres, cyclic group actions and the Milnor conjecture",
  "authors": [
    "David Baraglia",
    "Pedram Hekmati"
  ],
  "comment": "36 pages",
  "categories": [
    "math.GT",
    "math.DG"
  ],
  "abstract": "In this paper we further develop the theory of equivariant Seiberg-Witten-Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain the following applications. First, we show that the knot concordance invariants $\\theta^{(c)}$ defined by the first author satisfy $\\theta^{(c)}(T_{a,b}) = (a-1)(b-1)/2$ for torus knots, whenever $c$ is a prime not dividing $ab$. Since $\\theta^{(c)}$ is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture of a similar flavour to the proofs using the Ozsv\\'ath-Szab\\'o $\\tau$-invariant or Rasmussen $s$-invariant. Second, we prove that a free cyclic group action on a Brieskorn homology $3$-sphere $Y = \\Sigma(a_1 , \\dots , a_r)$ does not extend smoothly to any contractible smooth $4$-manifold bounding $Y$. This generalises to arbitrary $r$ the result of Anvari-Hambleton in the case $r=3$. Third, given a finite subgroup of the Seifert circle action on $Y = \\Sigma(a_1 , \\dots , a_r)$ of prime order $p$ acting non-freely on $Y$, we prove that if the rank of $HF_{red}^+(Y)$ is greater than $p$ times the rank of $HF_{red}^+(Y/\\mathbb{Z}_p)$, then the $\\mathbb{Z}_p$-action on $Y$ does not extend smoothly to any contractible smooth $4$-manifold bounding $Y$. We also prove a similar non-extension result for equivariant connected sums of Brieskorn homology spheres.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-10T04:40:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K10",
      "57K41"
    ],
    "keywords": [
      "milnor conjecture",
      "brieskorn spheres",
      "brieskorn homology spheres",
      "free cyclic group action",
      "knot concordance invariants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}