{
  "id": "1403.4485",
  "version": "v2",
  "published": "2014-03-18T14:50:15.000Z",
  "updated": "2014-10-21T17:02:38.000Z",
  "title": "Big polygon spaces",
  "authors": [
    "Matthias Franz"
  ],
  "comment": "20 pages; new title, description of product structure in cohomology added, minor changes",
  "categories": [
    "math.AT"
  ],
  "abstract": "We study a new class of compact orientable manifolds, called big polygon spaces. They are intersections of real quadrics and related to polygon spaces, which appear as their fixed point set under a canonical torus action. What makes big polygon spaces interesting is that they exhibit remarkable new features in equivariant cohomology: The Chang-Skjelbred sequence can be exact for them and the equivariant Poincare pairing perfect although their equivariant cohomology is never free as a module over the cohomology ring of BT. More generally, big polygon spaces show that a certain bound on the syzygy order of the equivariant cohomology of compact orientable T-manifolds obtained by Allday, Puppe and the author is sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-18T14:50:15.000Z",
      "title": "Maximal syzygies in equivariant cohomology",
      "abstract": "Let T be a torus of rank r and X a compact orientable T-manifold. The equivariant cohomology H_T^*(X;Q) is a finitely generated module over the polynomial ring R = H^*(BT;Q). By a result of Allday, Puppe and the author, if H_T^*(X;Q) is a syzygy of order m >= r/2, then it is already free over R. In this note we show that this bound is sharp. Our examples, called 'big polygon spaces', are intersections of real quadrics and related to polygon spaces, which appear as their fixed point sets. They are also the first examples of compact orientable T-manifolds for which the equivariant Poincare pairing is perfect although H_T^*(X;Q) is not free over R.",
      "comment": "18 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-21T17:02:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55N91",
      "13D02",
      "57P10",
      "55R80"
    ],
    "keywords": [
      "equivariant cohomology",
      "maximal syzygies",
      "compact orientable t-manifold",
      "big polygon spaces",
      "real quadrics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.4485F"
    }
  }
}