Tight, not semi-fillable contact circle bundles (Q1423616)

Let \(Y_{g,n}\) denote an oriented circle bundle over a closed, oriented surface of genus g with Euler number n. There are two virtually overtwisted (that is, the pullback to some finite cover of \(Y_{g,n}\) becomes overtwisted) contact structures \(\xi_0\), \(\xi_1\) for \(n\geq 2g>0\). The authors prove that \(\xi_0\) and \(\xi_1\) are not symplectically semi-fillable. A contact 3-manifold is symplectically fillable if it is the oriented boundary of some symplectic 4-manifold and the restriction of the symplectic form on the contact bundle does not vanish. A contact 3-manifold is symplectically semi-fillable if it is a part of some symplectically fillable contact 3-manifold. The idea of the proof is to compare the homotopy classes of these two tight and symplectically semi-fillable contact structures, and to show the difference.