The \(\pi\)-orbifold group of a link
The authors associate to a link L in the 3-sphere \(S^ 3\) what they call its \(\pi\)-orbifold group O(L), defined as follows. Consider the 2-fold covering V of \(S^ 3\) branched along the link L, and let W be the universal covering of V. Then O(L) is the group acting on W which is generated by the fundamental group of V and by any lift of the covering involution of V. The main result of the article is that prime links with infinite \(\pi\)-orbifold groups are classified by these groups. Namely, if L and \(L'\) are two prime links such that O(L) is infinite and is abstractly isomorphic to \(O(L')\), then there is a diffeomorphism of \(S^ 3\) sending L to \(L'\). They also prove that, for a prime link L with infinite O(L), the symmetry group \(\pi_ 0(Diff(S^ 3,L))\) of L is isomorphic to the outer automorphism group of O(L). Some related material can be found in a paper of \textit{Y. Takeuchi} [Trans. Am. Math. Soc. 307, 481-502 (1988; Zbl 0663.57009)].
realizing isomorphisms of the fundamental group by orbifold isomorphisms
scientific article; zbMATH DE number 4084575
scientific article