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

TL;DR

This paper investigates the Theta-invariant of 3-manifolds in relation to homology equivalences to spherical 3-manifolds, providing bounds on invariant spaces using representation theory.

Contribution

It introduces bounds on the dimensions of spaces spanned by Theta-invariants and finite type invariants for 3-manifolds, utilizing representation theory of finite groups.

Findings

Upper bounds on the space spanned by Theta-invariants

Upper bounds on the space spanned by finite type invariants of type 2

Application of representation theory to compute bounds

Abstract

We study Bott and Cattaneo's $\Theta$-invariant of 3-manifolds applied to $\mathbb{Z}\pi$-homology equivalences from 3-manifolds to a fixed spherical 3-manifold. The $\Theta$-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 $\Theta$-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.