---
res:
  bibo_abstract:
  - "Let $M$ be a symplectic manifold carrying a Hamiltonian $S^1$-action with\r\nmomentum
    map $J:M \\rightarrow \\mathbb{R}$ and consider the corresponding\r\nsymplectic
    quotient $\\mathcal{M}_0:=J^{-1}(0)/S^1$. We extend Sjamaar's complex\r\nof differential
    forms on $\\mathcal{M}_0$, whose cohomology is isomorphic to the\r\nsingular cohomology
    $H(\\mathcal{M}_0;\\mathbb{R})$ of $\\mathcal{M}_0$ with real\r\ncoefficients,
    to a complex of differential forms on $\\mathcal{M}_0$ associated\r\nwith a partial
    desingularization $\\widetilde{\\mathcal{M}}_0$, which we call\r\nresolution differential
    forms. The cohomology of that complex turns out to be\r\nisomorphic to the de
    Rham cohomology $H(\\widetilde{ \\mathcal{M}}_0)$ of\r\n$\\widetilde{\\mathcal{M}}_0$.
    Based on this, we derive a long exact sequence\r\ninvolving both $H(\\mathcal{M}_0;\\mathbb{R})$
    and $H(\\widetilde{\r\n\\mathcal{M}}_0)$ and give conditions for its splitting.
    We then define a Kirwan\r\nmap $\\mathcal{K}:H_{S^1}(M) \\rightarrow H(\\widetilde{\\mathcal{M}}_0)$
    from the\r\nequivariant cohomology $H_{S^1}(M)$ of $M$ to $H(\\widetilde{\\mathcal{M}}_0)$\r\nand
    show that its image contains the image of $H(\\mathcal{M}_0;\\mathbb{R})$ in\r\n$H(\\widetilde{\\mathcal{M}}_0)$
    under the natural inclusion. Combining both\r\nresults in the case that all fixed
    point components of $M$ have vanishing odd\r\ncohomology we obtain a surjection
    $\\check \\kappa:H^\\textrm{ev}_{S^1}(M)\r\n\\rightarrow H^\\textrm{ev}(\\mathcal{M}_0;\\mathbb{R})$
    in even degrees, while\r\nalready simple examples show that a similar surjection
    in odd degrees does not\r\nexist in general. As an interesting class of examples
    we study abelian polygon\r\nspaces.@eng"
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Benjamin
      foaf_name: Delarue, Benjamin
      foaf_surname: Delarue
      foaf_workInfoHomepage: http://www.librecat.org/personId=70575
  - foaf_Person:
      foaf_givenName: Pablo
      foaf_name: Ramacher, Pablo
      foaf_surname: Ramacher
  - foaf_Person:
      foaf_givenName: Maximilian
      foaf_name: Schmitt, Maximilian
      foaf_surname: Schmitt
  bibo_doi: 10.1007/s00031-025-09924-0
  dct_date: 2025^xs_gYear
  dct_language: eng
  dct_title: Singular cohomology of symplectic quotients by circle actions and Kirwan  surjectivity@
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