  We provide a new topological interpretation of the symplectic properties of
gluing equations for triangulations of hyperbolic 3-manifolds, first discovered
by Neumann and Zagier. We also extend the symplectic properties to more general
gluings of PGL(2,C) flat connections on the boundaries of 3-manifolds with
topological ideal triangulations, proving that gluing is a K_2 symplectic
reduction of PGL(2,C) moduli spaces. Recently, such symplectic properties have
been central in constructing quantum PGL(2,C) invariants of 3-manifolds. Our
methods adapt the spectral network construction of Gaiotto-Moore-Neitzke to
relate framed flat PGL(2,C) connections on the boundary C of a 3-manifold to
flat GL(1,C) connections on a double branched cover S -> C of the boundary.
Then moduli spaces of both PGL(2,C) connections on C and GL(1,C) connections on
S gain coordinates labelled by the first homology of S, and inherit symplectic
properties from the intersection form on homology.
