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تسجيل مستخدم جديدNegative Sasakian structures on simply-connected 5-manifolds
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We study several questions on the existence of negative Sasakian structures on simply connected rational homology spheres and on Smale-Barden manifolds of the form $#_k(S^2times S^3)$. First, we prove that any simply connected rational homology sphere admitting positive Sasakian structures also admits a negative one. This result answers the question, posed by Boyer and Galicki in their book [BG], of determining which simply connected rational homology spheres admit both negative and positive Sasakian structures. Second, we prove that the connected sum $#_k(S^2times S^3)$ admits negative quasi-regular Sasakian structures for any $k$. This yields a complete answer to another question posed in [BG].
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We construct the first example of a 5-dimensional simply connected compact manifold that admits a K-contact structure but does not admit a semi-regular Sasakian structure. For this, we need two ingredients: (a) to construct a suitable simply connecte
d symplectic 4-manifold with disjoint symplectic surfaces spanning the homology, all of them but one of genus 1 and the other of genus g>1, (b) to prove a bound on the second Betti number $b_2$ of an algebraic surface with $b_1=0$ and having disjoint complex curves spanning the homology when all of them but one are of genus 1 and the other of genus g>1.
Smale-Barden manifolds are simply-connected closed 5-manifolds. It is an important and difficult question to decide when a Smale-Barden manifold admits a Sasakian or a K-contact structure. The known constructions of Sasakian and K-contact structures
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