Lefschetz open book decompositions of 4-manifolds

Abstract

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