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Register a new userA Smale-Barden manifold admitting K-contact but not Sasakian structure
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Authors
Vicente Mu~noz
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We give the first example of a simply connected compact 5-manifold (Smale-Barden manifold) which admits a K-contact structure but does not admit any Sasakian structure, settling a long standing question of Boyer and Galicki.
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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 are obtained mainly by two techniques. These are either links (Boyer and Galicki), or semi-regular Seifert fibrations over smooth orbifolds (Kollar). Recently, the second named author of this article started the systematic development of quasi-regular Seifert fibrations, that is, over orbifolds which are not necessarily smooth. The present work is devoted to several applications of this theory. First, we develop constructions of a Smale-Barden manifold admitting a quasi-regular Sasakian structure but not a semi-regular K-contact structure. Second, we determine all Smale-Barden manifolds that admit a null Sasakian structure. Finally, we show a counterexample in the realm of cyclic Kahler orbifolds to the algebro-geometric conjecture that claims that for an algebraic surface with $b_1=0$ and $b_2>1$ there cannot be $b_2$ smooth disjoint complex curves of genus g>0 spanning the (rational) homology.
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 connected 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 $M$ are classified by their second homology $H_2(M,{mathbb Z})$ and the Barden invariant $i(M)$. It is an important and dificult question to decide when $M$ admits a Sasakian structure in terms of these data. In this work we show methods of doing this. In particular we realize all $M$ with $H_2(M)={mathbb Z}^koplus(oplus_{i=1}^r{mathbb Z}_{m_i}^{2g_i})$ and $i=0,infty$, provided that $kgeq 1$, $m_igeq 2$, $g_igeq 1$, $m_i$ are pairwise coprime. Using our methods we also contribute to the problem of the existence of definite Sasakian structures on rational homology spheres. Also, we give a complete solution to the problem of the existence of Sasakian structures on rational homology spheres in the class of semi-regular Sasakian structures.
We develop the Gompf fiber connected sum operation for symplectic orbifolds. We use it to construct a symplectic 4-orbifold with $b_1=0$ and containing symplectic surfaces of genus 1 and 2 that are disjoint and span the rational homology. This is used in turn to construct a K-contact Smale-Barden manifold with specified 2-homology that satisfies the known topological constraints with sharper estimates than the examples constructed previously. The manifold can be chosen spin or non-spin.
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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