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We introduce a new method to obstruct Liouville and weak fillability. Usingthis, we show that various rational homology 3-spheres admit strongly fillablecontact structures without Liouville fillings, which extends the result ofGhiggini on a family of Brieskorn spheres. We also make partial progress on aconjecture of Ghiggini and Van-Horn-Morris.

Strongly fillable contact structures without Liouville fillings