Mathematics – Geometric Topology
Scientific paper
2009-09-20
Mathematics
Geometric Topology
22 pages updated references, in particular we now learned that Theorem 17 had been proved independently by Abdelghani, Heusene
Scientific paper
Given a knot in an integral homology 3-sphere, there is a natural action of the cyclic group Z/n on the space of SL(n,C) representations of the knot group, and this induces an action on the SL(n,C) character variety. We identify the fixed points of the action on irreducible characters with characters of irreducible metabelian representations. We then show that for any irreducible metabelian representation, the first cohomology group with twisted coefficients in the adjoint bundle has dimension at least n-1. If equality holds, then we prove that the character is a smooth point in the character variety and that there exists a smooth (n-1)-dimensional family of characters of irreducible SL(n,C) representations of the knot group near the metabelian character. We relate the condition on the twisted first cohomology group to the vanishing of the untwisted first cohomology group of the 3-manifold obtained as the associated metabelian branched cover of the homology 3-sphere branched along the knot.
Boden Hans
Friedl Stefan
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