Theta-invariants of $\mathbb{Z}\pi$-homology equivalences to spherical 3-manifolds

We study Bott and Cattaneo’s $\mathrm{\Theta }$-invariant of 3-manifolds applied to $\mathbb{Z}\pi$-homology equivalences from 3-manifolds to a fixed spherical 3-manifold. The $\mathrm{\Theta }$-invariants are defined by integrals over configuration spaces of two points with local systems and by choosing some invariant tensors. We compute upper bounds of the dimensions of the space spanned by the Bott–Cattaneo $\mathrm{\Theta }$-invariants and of that spanned by Garoufalidis and Levine’s finite type invariants of type 2. The computation is based on representation theory of finite groups.