Removing local extrema of surfaces in open book decompositions (Intelligence of Low-dimensional Topology)
圭子 川室 · Kyoto University Research Information Repository (Kyoto University) · 2015
FOLIATIONSIn this section we recall some terminologies in the open book foliation package that are needed for this note.Open book foliations are defined by Ito and Kawamuro in [14].The idea of open book foliations came from Bennequin's work [1] and Birman and Menasco's braid foliations studied in the series of papers [2,3,4,5,6,7, 8,9,10].Many terminologies of open book foliations are taken from those of braid foliations, Let $S=S_{g,r}$ be a compact oriented genus $g$ surface with $r(>0)$ boundary components.Let $\phi$ : $Sarrow S$ be an orientation preserving diffeomorphism that fixes the boundary $\partial S$ point-wise.Let $M_{(S,\phi)}$ denote the 3-manifold given by the (abstract) open book $(S, \phi)$ .That is $M_{(S,\phi)}$ is obtained from the mapping torus $M_{\phi}$ of $\phi$ with solid tori glued trivially along all the boundary tori of $M_{\phi}$ .(See, for example, Etnyre's survey article [12] for more detail.)We denote the binding of the open book by $B$ and the pages by $S_{t}$ where $t\in[0$ , 1 $].$