Admissible transverse surgery does not preserve tightness (Q382319)

The paper under review discusses an operation called admissible transverse surgery along transverse knots in contact manifolds. (A substantial part of the paper is devoted to clarify the relationship of this operation to the better known Legendrian surgery operation.) The main result is the existence of (infinitely many) closed, universally tight contact manifolds for which an admissible transverse surgery on some transverse knot is overtwisted. The examples are supported by genus one open books with pseudo-Anosov monodromy and two binding components. The authors prove that in this case the admissible transverse 0-surgeries correspond to capping off a boundary component of the open book. The examples also provide new insights into the relationships between tightness, fillability and the Heegaard Floer contact invariants. The authors prove that there exist infinitely many atoroidal, tight contact manifolds with vanishing contact invariants. The examples are, in particular, hyperbolic and universally tight. Infinitely many of them are rational homology spheres and thus not weakly fillable.