{
  "id": "2306.02139",
  "version": "v1",
  "published": "2023-06-03T15:40:11.000Z",
  "updated": "2023-06-03T15:40:11.000Z",
  "title": "Computing Topological Invariants Using Fixed Points",
  "authors": [
    "Loring W. Tu"
  ],
  "comment": "13 pages. This article is based on a talk given at the Sixth International Congress of Chinese Mathematicians, Taipei, Taiwan, in 2013",
  "journal": "Proceedings of the Sixth International Congress of Chinese Mathematicians, Vol. II, 285--298, Adv. Lect. Math. (ALM), 37, Int. Press, Somerville, MA, 2017",
  "categories": [
    "math.AT"
  ],
  "abstract": "When a torus acts on a compact oriented manifold with isolated fixed points, the equivariant localization formula of Atiyah--Bott and Berline--Vergne converts the integral of an equivariantly closed form into a finite sum over the fixed points of the action, thus providing a powerful tool for computing integrals on a manifold. An integral can also be viewed as a pushforward map from a manifold to a point, and in this guise it is intimately related to the Gysin homomorphism. This article highlights two applications of the equivariant localization formula. We show how to use it to compute characteristic numbers of a homogeneous space and to derive a formula for the Gysin map of a fiber bundle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-03T15:40:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55R10",
      "55N25",
      "14C17",
      "14M17"
    ],
    "keywords": [
      "fixed points",
      "computing topological invariants",
      "equivariant localization formula",
      "berline-vergne converts",
      "compact oriented manifold"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}