{"abstract":[{"text":"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.","lang":"eng"}],"publication":"arXiv:2312.03634","title":"Singular cohomology of symplectic quotients by circle actions and Kirwan  surjectivity","date_updated":"2024-04-11T12:38:25Z","language":[{"iso":"eng"}],"department":[{"_id":"548"}],"type":"preprint","citation":{"chicago":"Delarue, Benjamin, Pablo Ramacher, and Maximilian Schmitt. “Singular Cohomology of Symplectic Quotients by Circle Actions and Kirwan  Surjectivity.” <i>ArXiv:2312.03634</i>, 2023.","mla":"Delarue, Benjamin, et al. “Singular Cohomology of Symplectic Quotients by Circle Actions and Kirwan  Surjectivity.” <i>ArXiv:2312.03634</i>, 2023.","apa":"Delarue, B., Ramacher, P., &#38; Schmitt, M. (2023). Singular cohomology of symplectic quotients by circle actions and Kirwan  surjectivity. In <i>arXiv:2312.03634</i>.","short":"B. Delarue, P. Ramacher, M. Schmitt, ArXiv:2312.03634 (2023).","ieee":"B. Delarue, P. Ramacher, and M. Schmitt, “Singular cohomology of symplectic quotients by circle actions and Kirwan  surjectivity,” <i>arXiv:2312.03634</i>. 2023.","bibtex":"@article{Delarue_Ramacher_Schmitt_2023, title={Singular cohomology of symplectic quotients by circle actions and Kirwan  surjectivity}, journal={arXiv:2312.03634}, author={Delarue, Benjamin and Ramacher, Pablo and Schmitt, Maximilian}, year={2023} }","ama":"Delarue B, Ramacher P, Schmitt M. Singular cohomology of symplectic quotients by circle actions and Kirwan  surjectivity. <i>arXiv:231203634</i>. Published online 2023."},"user_id":"70575","year":"2023","author":[{"first_name":"Benjamin","id":"70575","full_name":"Delarue, Benjamin","last_name":"Delarue"},{"first_name":"Pablo","last_name":"Ramacher","full_name":"Ramacher, Pablo"},{"full_name":"Schmitt, Maximilian","last_name":"Schmitt","first_name":"Maximilian"}],"external_id":{"arxiv":["2312.03634"]},"date_created":"2024-04-11T12:30:59Z","status":"public","_id":"53412"}