# A note on the $\Theta$-invariant of 3-manifolds

## TL;DR

This paper revisits the $ heta$-invariant of rational homology 3-spheres, providing a modified definition that applies even when certain cohomology groups do not vanish, extending its applicability.

## Contribution

It introduces a modified $ heta$-invariant definition that works without the vanishing cohomology condition, broadening the invariant's scope.

## Key findings

- Extended the $ heta$-invariant to non-vanishing cohomology cases
- Provided a modified construction of the invariant
- Maintained the invariant's relation to Chern-Simons theory

## Abstract

In this note, we revisit the $\Theta$-invariant as defined by R. Bott and the first author. 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.