{
  "id": "1011.0649",
  "version": "v2",
  "published": "2010-11-02T15:51:03.000Z",
  "updated": "2018-03-12T14:04:41.000Z",
  "title": "Quaternionic Grassmannians and Borel classes in algebraic geometry",
  "authors": [
    "Ivan Panin",
    "Charles Walter"
  ],
  "categories": [
    "math.AG",
    "math.KT"
  ],
  "abstract": "The quaternionic Grassmannian HGr(r,n) is the affine open subscheme of the ordinary Grassmannian parametrizing those 2r-dimensional subspaces of a 2n-dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular there is HP^{n} = HGr(1,n+1). For a symplectically oriented cohomology theory A, including oriented theories but also hermitian K-theory, Witt groups and symplectic and special linear algebraic cobordism, we have A(HP^{n}) = A(pt)[p]/(p^{n+1}). We define Borel classes for symplectic bundles. They satisfy a splitting principle and the Cartan sum formula, and we use them to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank 2 symplectic bundles determine Thom and Borel classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Borel classes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-11-02T15:51:03.000Z",
      "title": "Quaternionic Grassmannians and Pontryagin classes in algebraic geometry",
      "abstract": "The quaternionic Grassmannian HGr(r,n) is the affine open subscheme of the ordinary Grassmannian parametrizing those 2r-dimensional subspaces of a 2n-dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular there is HP^{n} = HGr(1,n+1). For a symplectically oriented cohomology theory A, including oriented theories but also hermitian K-theory, Witt groups and symplectic and special linear algebraic cobordism, we have A(HP^{n}) = A(pt)[p]/(p^{n+1}). We define Pontryagin classes for symplectic bundles. They satisfy a splitting principle and the Cartan sum formula, and we use them to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank 2 symplectic bundles determine Thom and Pontryagin classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Pontryagin classes.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2018-03-12T14:04:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F42",
      "14F99",
      "19E20",
      "19G38",
      "19G12"
    ],
    "keywords": [
      "algebraic geometry",
      "symplectic bundles determine thom",
      "thom classes",
      "special linear algebraic cobordism",
      "2n-dimensional symplectic vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.0649P"
    }
  }
}