Hyperbolic rational homology spheres not admitting fillable contact structures (Q1651768)

An important open problem in low-dimensional topology is whether every hyperbolic \(3\)-manifold admits a tight contact structure. By \textit{Y. Eliashberg} and \textit{M. Gromov} [Proc. Symp. Pure Math. 52, 135--162 (1991; Zbl 0742.53010)], every weakly fillable contact structure is tight. Hence one may ask the refined question: Does every hyperbolic \(3\)-manifold admit a weakly fillable contact structure? By work of \textit{D. Gabai} [J. Differ. Geom. 18, 445--503 (1983; Zbl 0533.57013)] and \textit{Y. M. Eliashberg} and \textit{W. P. Thurston} [Confoliations. Providence, RI: American Mathematical Society (1998; Zbl 0893.53001)], if \(M\) is an irreducible \(3\)-manifold and \(b_1(M)>0\), then \(M\) admits a weakly fillable and hence tight contact structure. Thus one may restrict attention to the class of hyperbolic \(3\)-manifolds that are rational homology spheres. In this context, the authors provide a negative answer to the above question. Namely, the authors exhibit infinitely many hyperbolic rational homology \(3\)-spheres that do not admit any weakly fillable contact structure. Moreover, they note that most of these manifolds do admit a tight contact structure. The examples the authors give are \(3\)-manifolds that can be obtained by performing rational surgery on the pretzel knots \(P(-2,3,2m+1)\) for \(m\geq 3\) and the proof of nonexistence of weakly fillable contact structures that they give rests on theorems of \textit{B. Owens} and \textit{S. Strle} [Sel. Math., New Ser. 18, No. 4, 839--854 (2012; Zbl 1268.57006)] that provide an obstruction for a certain \(3\)-manifold to bound a negative definite \(4\)-manifold.