The \(\pi\)-orbifold group of a link
type: Item
instance of: scholarly article
title: The \(\pi\)-orbifold group of a link
zbMATH Open document ID: 0663.57006
DOI: 10.1007/BF01230281
author: Michel Boileau
author: Bruno Zimmermann
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174001
review text: 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)].
reviewed by: Francis Bonahon
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 4084575
zbMATH Keywords: realizing isomorphisms of the fundamental group by orbifold isomorphisms
zbMATH Keywords: link L in the 3-sphere
zbMATH Keywords: orbifold group
zbMATH Keywords: 2-fold covering
zbMATH Keywords: prime links
MaRDI profile type: Publication
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