On contact type hypersurfaces in 4-space
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2020
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| _version_ | 1866917489051435008 |
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| author | Mark, Thomas E. Tosun, Bülent |
| author_facet | Mark, Thomas E. Tosun, Bülent |
| contents | We consider constraints on the topology of closed 3-manifolds that can arise as hypersurfaces of contact type in standard symplectic $R^4$. Using an obstruction derived from Heegaard Floer homology we prove that no Brieskorn homology sphere admits a contact type embedding in $R^4$, a result that has bearing on conjectures of Gompf and Kollár. This implies in particular that no rationally convex domain in $C^2$ has boundary diffeomorphic to a Brieskorn sphere. We also give infinitely many examples of contact 3-manifolds that bound Stein domains but not symplectically convex ones; in particular we find Stein domains in $C^2$ that cannot be made Weinstein with respect to the ambient symplectic structure while preserving the contact structure on their boundaries. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2008_02755 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On contact type hypersurfaces in 4-space Mark, Thomas E. Tosun, Bülent Geometric Topology Complex Variables Symplectic Geometry 57K33, 57K43, 32E20 We consider constraints on the topology of closed 3-manifolds that can arise as hypersurfaces of contact type in standard symplectic $R^4$. Using an obstruction derived from Heegaard Floer homology we prove that no Brieskorn homology sphere admits a contact type embedding in $R^4$, a result that has bearing on conjectures of Gompf and Kollár. This implies in particular that no rationally convex domain in $C^2$ has boundary diffeomorphic to a Brieskorn sphere. We also give infinitely many examples of contact 3-manifolds that bound Stein domains but not symplectically convex ones; in particular we find Stein domains in $C^2$ that cannot be made Weinstein with respect to the ambient symplectic structure while preserving the contact structure on their boundaries. |
| title | On contact type hypersurfaces in 4-space |
| topic | Geometric Topology Complex Variables Symplectic Geometry 57K33, 57K43, 32E20 |
| url | https://arxiv.org/abs/2008.02755 |