For an oriented closed 3-manifold $M$ and a group homomorphism  $\pi_1(M) \rightarrow SL_2(\mathbb{C})$, one can define the Chern–Simons invariant and the adjoint Reidemeister torsion. In recent years, several physicists and topologists have investigated reciprocity conjectures concerning these torsions. By analogy, I have formulated reciprocity conjectures for the Chern–Simons invariants of 3-manifolds and provided supporting evidence for them. In particular, I proved that the conjectures hold when the Galois descent of a certain algebraic $K_3$-group is satisfied. In this talk, I will present the background, explain the main results, and discuss possible future directions.

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