Lefschetz open book decompositions of 4 -manifolds

In this paper, we introduce the notion of Lefschetz open book decompositions for closed oriented smooth 4-manifolds. We define the notion of a Murasugi sum of Lefschetz open book decompositions. We show that the closed oriented 4-manifold associated to a Murasugi sum of Lefschetz open books of two closed oriented 4-manifolds X1 and X2 is the connected sum of X1 and X2. We also introduce the notion of a stabilization of a Lefschetz open book. We introduce the notion of a Lefschetz fibered link in a closed oriented 4-manifold and using the stabilization of Lefschetz open book, we show that in a simply connected closed oriented 4-manifold X, every oriented link B in X is Lefschetz fibered. We also discuss the existence of a folded Kähler structure on a closed oriented 4-manifold associated to a given Lefschetz open book which essentially follows by the work of Baykur.