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Floer Homology for Knots and $\SU(2)$-Representations for Knot Complements and Cyclic Branched Covers

In this article, using 3-orbifolds singular along a knot with underlying space a homology sphere $Y^3$, the question of existence of non-trivial and non-abelian $\SU(2)$-representations of the fundamental group of cyclic branched covers of $Y^3$ along a knot is studied. We first use Floer Homology for knots to derive an existence result of non-abelian $\SU(2)$-representations of the fundamental group of knot complements, for knots with a non-vanishing equivariant signature. This provides information on the existence of non-trivial and non-abelian $\SU(2)$-representations of the fundamental group of cyclic branched covers. We illustrate the method with some examples of knots in $S^3$.
 MSC Classifications: 57R57 - Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX] 57M12 - Special coverings, e.g. branched 57M25 - Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M05 - Fundamental group, presentations, free differential calculus