Finite slope cyclic surgeries along toroidal Brunnian links and generalized Properties P and R

Abstract

Let $M_{\lambda}$ be the $\lambda$-component Milnor link. For $\lambda \ge 3$, we determine completely when a finite slope surgery along $M_{\lambda}$ yields a lens space including $S^3$ and $S^1\times S^2$, where {\it finite slope surgery} implies that a surgery coefficient of every component is not $\infty$. For $\lambda =3$ (i.e.\ the Borromean rings), there are three infinite sequences of finite slope surgeries yielding lens spaces. For $\lambda \ge 4$, any finite slope surgery does not yield a lens space. As a corollary, $M_{\lambda}$ for $\lambda \ge 3$ does not yield both $S^3$ and $S^1\times S^2$ by any finite slope surgery. We generalize the results for the cases of {\it Brunnian type links} and toroidal Brunnian type links (i.e.\ Brunnian type links including essential tori in the link complement). Our main tools are Alexander polynomials and Reidemeister torsions. Moreover we characterized toroidal Brunnian links and toroidal Brunnian type links in some senses.