  We study spaces of circle-valued angle structures, introduced by Feng Luo, on
ideal triangulations of 3-manifolds. We prove that the connected components of
these spaces are enumerated by certain cohomology groups of the 3-manifold with
$\mathbb{Z}_2$-coefficients. Our main theorem shows that this establishes a
geometrically natural bijection between the connected components of the spaces
of circle-valued angle structures and the obstruction classes to lifting
boundary-parabolic $PSL(2,\mathbb{C})$-representations of the fundamental group
of the 3-manifold to boundary-unipotent representations into
$SL(2,\mathbb{C})$. In particular, these connected components have topological
and algebraic significance independent of the ideal triangulations chosen to
construct them. The motivation and main application of this study is to
understand the domain of the state-integral defining the meromorphic 3D-index
of Garoufalidis and Kashaev, necessary in order to classify the
boundary-parabolic representations contributing to its asymptotics.
