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

Alberto S Cattaneo,Tatsuro Shimizu

doi:10.4171/qt/146

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.

Highlights

The ‚-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
Bott and the first author defined topological invariants of rational homology spheres with acyclic orthogonal local systems in [3] and [4]
These invariants were inspired by the Chern–Simons perturbation theory developed by M

Summary

Introduction

Bott and the first author defined topological invariants of rational homology spheres with acyclic orthogonal local systems in [3] and [4]. These invariants were inspired by the Chern–Simons perturbation theory developed by M. We show that a linear combination of Z‚ and another term ZO O is, a topological invariant of closed 3-manifolds with orthogonal acyclic local systems, A. We note that the second author proved that when G D SU.2/, Z‚ itself is an invariant of closed 3-manifolds with orthogonal local systems in [9]. The interval Œ0; 1 R is oriented from 0 to 1

The invariant

A framing correction

D Sign W