A note on the \(\Theta\)-invariant of 3-manifolds (Q2043517)

Summary: In this note, we revisit the \(\Theta\)-invariant as defined by \textit{R. Bott} and the first author in [J. Differ. Geom. 53, No. 1, 1--13 (1999; Zbl 1036.57500)]. The \(\Theta\)-invariant is an invariant of rational homology 3-spheres with acyclic orthogonal local systems, which is a generalization of the 2-loop term of the Chern-Simons perturbation theory. The \(\Theta\)-invariant can be defined when a cohomology group is vanishing. In this note, we give a slightly modified version of the \(\Theta\)-invariant that we can define even if the cohomology group is not vanishing.