On perturbative invariants of combed three-manifolds

TL;DR

This paper introduces a new, more flexible definition of a universal finite type invariant for rational homology spheres, based on combings instead of parallelizations, inspired by perturbative Chern-Simons theory.

Contribution

It replaces the traditional parallelization approach with combings to define the invariant, enhancing flexibility and convenience.

Findings

Provides a new combinatorial framework for invariants

Simplifies the construction of finite type invariants

Connects invariants to perturbative Chern-Simons theory

Abstract

We give a new definition of a universal finite type invariant of three-dimensional oriented rational homology spheres which counts configurations of trivalent graphs in such manifolds. Kontsevich introduced this invariant following Witten's study of the perturbative expansion of the Chern-Simons theory, using parallelizations of three-manifolds. In this article, we use combings instead of parallelizations to get a more flexible and convenient definition.