On the geography of Stein fillings of certain 3-manifolds (Q1411310)

The author considers the problem of describing characteristic numbers of Stein fillings of a given (closed) contact \(3\)-manifold \((Y, \xi)\). A Stein filling of \((Y, \xi)\) is a \(4\)-manifold \(W\) that is a Stein domain such that \((Y, \xi)\) is contactomorhpic to \(\partial W\) with the contact structure induced by the complex tangencies. The characteristic numbers for \(W\) are the Betti number \(b_1(W)\), signature \(\sigma(W)\), and Euler characteristic \(\chi(W)\). The author conjectures that for a given contact \(3\)-manifold \((Y, \xi)\) the set consisting of these three invariants for all Stein fillings of \((Y, \xi)\) is finite. His main result is that for a given \((Y, \xi)\) the number \(3\sigma(W)\)+\(2\chi(W)\) is bounded below. As Corollaries, the conjecture is proved in particular cases: It is true if every Stein filling of \((Y, \xi)\) has vanishing \(b_2^+\)--invariant. It is also true if \(Y\) is a circle bundle over the Riemann surface of genus \(g\) with Euler number n satisfying \(| n| \) \(>\) \(2g-2\).