2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/129968Given an abelian group $A$ and a Lie group $G$, we construct a bilinear pairing from $A\timesπ_1({\mathcal R})$ to $π_1(G)$, where $\mathcal R$ is a subvariety of the variety of representations $A\to G$. In the case where $A$ is the peripheral subgroup of a torus or two-bridge knot group, $G=S^1$ and $\mathcal R$ is a certain variety of representations arising from suitable SU(2)-representations of the knot group, we show that this pairing is not identically zero. We discuss the consequences of this result for the SU(2)-representations of fundamental groups of manifolds obtained by Dehn surgery on such knots.13 pages, 2 figuresGeometric TopologyGroup Theory57M25; 20F05, 20G05text