Operations on open book foliations
Let \((S,\varphi)\) be an open book decomposition of a closed oriented \(3\)-manifold \(M\), where \(S=S_{g,r}\) is a genus-\(g\) surface with \(r\) boundary components, and \(\varphi\in\text{Diff}^+(S,\partial S)\) is an orientation-preserving diffeomorphism of \(S\) that fixes the boundary pointwise. Let \(B\) denote the binding of the open book and \(\pi:M\setminus B\to S^1\) the fibration whose fiber \(S_t=\pi^{-1}(t)\) is a page. An oriented link \(L\) in \(M\) is called a closed braid with respect to the open book \((S,\varphi)\) if \(L\) is disjoint with the binding \(B\) and positively transverse to each page \(S_t\). Let \(F\) be an oriented, connected, compact surface smoothly embedded in \(M\) whose boundary \(\partial F\) is in braid position with respect to the open book \((S,\varphi)\), and let \({\mathcal F}={\mathcal F}(F)\) be the singular foliation on \(F\) induced by the pages \(\{S_t;\;t\in S^1\}\). The singular foliation \({\mathcal F}_{\text{ob}}(F)\) is an open book foliation if the following conditions are satisfied: (i)\,the binding \(B\) pierces the surface \(F\) transversely in finitely many points, and \(p\in B\cap F\) if and only if there exists a disc neighborhood \(N_p\subset\text{Int}(F)\) of \(p\) on which the foliation \({\mathcal F}(N_p)\) is radial with the node \(p\), (ii)\,the leaves of \({\mathcal F}\) along \(\partial\,F\) are transverse to \(\partial\,F\), (iii) all but finitely many fibers \(S_t\) intersect \(F\) transversely, (iv)\,all the tangencies of \(F\) and fibers are of saddle type.NEWLINENEWLINEIn this paper, the authors define and study three types of operations on open book foliations on surfaces that are realized by isotopies of the surfaces: (i)\,the \(b\)-arc foliation change, (ii)\,bypass move, and (iii)\,exchange move. A \(b\)-arc foliation change and an exchange move are generalizations of Birman-Menasco's foliation change and exchange move in braid foliation theory. A bypass move can be seen as an analogue of Honda's bypass attachment in convex surface theory. The authors study the effect of stabilizations of open books on open book foliations and show that the open book foliation of a surface changes in two ways after a stabilization of the open book. Also, they show that the resulting two open book foliations are related to each other by bypass moves and exchange moves. As applications of \(b\)-arc foliation changes and exchange moves, the authors prove the split/composite closed braid theorems of Birman and Menasco in the setting of general open books.
scientific article; zbMATH DE number 6369102
scientific article