On the local structure of \(SL(2,{\mathbb C})\)-character varieties at reducible characters
type: Item
instance of: scholarly article
title: On the local structure of \(SL(2,{\mathbb C})\)-character varieties at reducible characters
zbMATH Open document ID: 1021.57011
DOI: 10.1016/S0166-8641(01)00278-4
published in: Topology and its Applications
publication date: 2002-08-22T00:00:00Z
review text: Let \(M\) be a compact connected irreducible orientable 3-manifold whose boundary is an incompressible torus, and \(\Gamma\) be the fundamental group of \(M\). The \(SL(2,\mathbb{C})\)-representation variety of \(M\) is a complex affine algebraic variety \(R(M)\) whose points correspond to the representations of \(\Gamma\) in \(SL(2,\mathbb{C})\). The quotient of \(R(M)\) by the natural action of \(SL(2,\mathbb{C})\) is the character variety which is denoted by \(X(M)\). The author studies the local behavior of the projection \(R(M)\to X(M)\) at reducible representations, and describes a 1-cocyle condition which guarantees the smoothness of a reducible character in \(X(M)\). The author applies the results mentioned above and the results of [\textit{L. Ben Abdelghani} and \textit{S. Boyer}, Proc. Lond. Math. Soc. (3) 83, No. 1, 235-256 (2001; Zbl 1029.57001)] to exceptional Dehn fillings. In particular, he proves the following statement: Let \(K\) be a hyperbolic knot in \(S^3\) with exterior \(M\); then there is at most one slope \(r\) on the boundary \(\partial M\) of \(M\) such that the corresponding manifold \(M(r)\) obtained by Dehn filling is a Seifert fibered space whose base orbifold is Euclidean; if there is such a slope, it is integral, though not longitudinal.
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1787535
zbMATH Keywords: 3-manifold
zbMATH Keywords: character variety
zbMATH Keywords: reducible character
author: Steven Boyer
reviewed by: Ilia Itenberg
MaRDI profile type: Publication
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full work available at URL: s0166-8641(01)00278-4
OpenAlex ID: W2043834059
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description: scientific article; zbMATH DE number 1787535