On the Kontsevich-Kuperberg-Thurston construction of a configuration-space invariant for rational homology 3-spheres

TL;DR

This paper reviews the Kontsevich-Kuperberg-Thurston construction of a topological invariant for rational homology 3-spheres, providing detailed proofs of its invariance and laying groundwork for future splitting formulae.

Contribution

It offers a detailed review and elementary proofs of the invariant's invariance, serving as a foundation for subsequent work on splitting formulas.

Findings

Z is a universal finite type invariant for integral homology spheres

Elementary proofs of invariance are provided

Prepares for proving splitting formulas in future work

Abstract

M. Kontsevich proposed a topological construction for an invariant Z of rational homology 3-spheres using configuration space integrals. G. Kuperberg and D. Thurston proved that Z is a universal real finite type invariant for integral homology spheres in the sense of Ohtsuki, Habiro and Goussarov. We review the Kontsevich-Kuperberg-Thurston construction and we provide detailed and elementary proofs for the invariance of Z. This article is the preliminary part of a work that aims to prove splitting formulae for this powerful invariant of rational homology spheres. It contains the needed background for the proof that will appear in the second part.