Braids and branched coverings of dimension three (Intelligence of Low-dimensional Topology)
J. Scott Carter, Seiichi Kamada · Institutional Repositories DataBase (IRDB) · 2012
IntroductionThis is on a part of our work in progress, which was introduced at the conference "Intel- ligence of Low-dimensional Topology" held in RIMS in May, 2012.The purpose of our research is to understand branched coverings and $m$-dimensional braids which are gen- eralizations of classical braids.Here we discuss chart descriptions of branched coverings and braids in dimension $m=2$ first, and then those for which $m=3.$We work in the $PL$ category ([9, 20]).Let $S^{m}$ denote the $m$-sphere, and let $M^{m}$ denote a closed oriented $m$-manifold. PreliminariesWe start by giving some definitions and theorems on branched coverings.Definition 2.1 A PL map $f$ : $M^{m}arrow S^{m}$ is a branched covering (map) if there exists an $(m-2)$-subcomplex $L$ of $S^{m}$ such that the restriction $\underline{f}$ : $M^{m}\backslash f^{-1}(L)arrow S^{m}\backslash L$ is a covering map.We denote the covering degree by $d$ .We call $f$ a $d$ -fold branched covering.We assume that $L$ is minimum, i.e., $\forall y\in L,$ $\#(f^{-1}(y))