We use Menke’s JSJ-type decomposition theorem for symplectic fillings to
reduce the classification of strong and exact symplectic fillings of virtually
overtwisted contact structures on torus bundles to the same problem for
tight lens spaces. For virtually overtwisted structures on elliptic or parabolic
torus bundles, this gives a complete classification. For virtually overtwisted
structures on hyperbolic torus bundles, we show that every strong or
exact filling arises from a filling of a tight lens space via round symplectic
–handle
attachment, and we give a condition under which distinct tight lens space fillings
yield the same torus bundle filling.
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