On perturbative invariants of combed three-manifolds

2025/10/22 by Yohan Mandin--Hublé, Mandin--Hublé, Yohan
Mathematics · Computer Science · #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2510.19633

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.

Citations

• A three-manifold invariant from graph configurations
• Higher order generalization of Fukaya's Morse homotopy invariant of 3-manifolds I. Invariants of homology 3-spheres
• Perturbative 3-manifold invariants by cut-and-paste topology
• Canonical framings for 3-manifolds
• Handlebody Construction of Stein Surfaces

Related

• A three-manifold invariant from graph configurations
• On a Universal Invariant of 3-Manifolds
• An introduction to finite type invariants of knots and 3-manifolds defined by counting graph configurations
• An Invariant of Integral Homology 3-Spheres Which Is Universal For All Finite Type Invariants
• Higher order generalization of Fukaya's Morse homotopy invariant of 3-manifolds I. Invariants of homology 3-spheres