Lefschetz fibrations over the disc (Q2843974)

A Lefshetz fibration \(f:W\to{B}^2\) over the disc \(B^2\) determines a 2-handlebody structure for \(W\), with all handles of index \(\leq2\). Two such 2-handlebodies are \textit{2-equivalent} if they are related by isotopy, handle slides and the addition or deletion of cancelling pairs of handles of index \(\leq2\). The main result of this paper is the identification of three basic moves \(S,T\) and \(U\) such that two Lefshetz fibrations \(f\) and \(f'\) over the disc determine 2-equivalent 2-handlebodies if and only if they are related by fibred equivalence and applications of these moves. If \(f\) and \(f'\) are \textit{allowable}, meaning that the regular fibres have nonempty boundary and all vanishing cycles are homologically non-trivial, then fibred equivalence and the first two moves suffice.NEWLINENEWLINEOne of the key ideas used is a version of Rudolph's braiding procedure. The various notions -- ribbon surfaces; braided surfaces; the braiding procedure; 2-handlebodies; Lefshetz fibrations over the disc; and the moves \(S,T,U\) -- are described in detail before the proof of the main theorem, in \S8.NEWLINENEWLINEThe final two sections consider related results for \(\partial{W}\). Two further moves \(P\) and \(Q\) are defined. (These change the topology of \(W\).) Two allowable Lefshetz fibrations over the disc have diffeomorphic oriented boundaries if and only if they are related by fibred equivalence and the moves \(S,T,P\) and \(Q\). This leads to a version of Harer's equivalence theorem for 3-dimensional open books, which is more explicit in that the moves can be completely described in terms of the open book monodromy.