Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On perturbative invariants of combed three-manifolds

Yohan Mandin--Hublé

TL;DR

The paper introduces a new, combing-based universal finite type invariant $Z$ for oriented rational homology 3-spheres, leveraging propagators on the configuration space $C_2(M)$ to count graphs in $M$. It proves that degree-$n$ pieces $z_n(M,X)$ are well-defined and relate to the base invariant $z_n(M)$ via the combing’s Pontryagin-type invariant $p_1(X)$ and the beta anomaly: $z_n(M,X)=z_n(M)+\tfrac{1}{4}p_1(X)\beta_n$. The construction generalizes the parallelization-based approach and connects to Lescop’s, Kuperberg–Thurston’s, and Shimizu’s formalisms, while offering practical computation via Morse propagators and multisections. The work also provides a dual formulation and a multiframing generalization, tying the combing data to pseudo-parallelizations through a precise variation formula, and clarifies relations to the Theta invariant and existing invariants in the perturbative Chern–Simons framework. Overall, it advances concrete, combing-based methods for evaluating universal finite type invariants of rational homology spheres and broadens the toolkit for explicit computations in 3-manifold topology.

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.

On perturbative invariants of combed three-manifolds

TL;DR

The paper introduces a new, combing-based universal finite type invariant for oriented rational homology 3-spheres, leveraging propagators on the configuration space to count graphs in . It proves that degree- pieces are well-defined and relate to the base invariant via the combing’s Pontryagin-type invariant and the beta anomaly: . The construction generalizes the parallelization-based approach and connects to Lescop’s, Kuperberg–Thurston’s, and Shimizu’s formalisms, while offering practical computation via Morse propagators and multisections. The work also provides a dual formulation and a multiframing generalization, tying the combing data to pseudo-parallelizations through a precise variation formula, and clarifies relations to the Theta invariant and existing invariants in the perturbative Chern–Simons framework. Overall, it advances concrete, combing-based methods for evaluating universal finite type invariants of rational homology spheres and broadens the toolkit for explicit computations in 3-manifold topology.

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.
Paper Structure (25 sections, 37 theorems, 65 equations, 1 figure)

This paper contains 25 sections, 37 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Specifying the setting
  3. Combings and multisections
  4. Propagating forms associated with multisected combings
  5. Spaces of trivalent graphs
  6. Main result
  7. A generalisation of Theorem \ref{['thm_inv_combing']} using special multiframings
  8. A dual version of Theorem \ref{['thm_inv_combing']}
  9. Organisation of the paper
  10. Relation with other invariants
  11. The invariant z of a rational homology sphere
  12. The Theta invariant of a rational homology sphere equipped with a combing
  13. A definition of z via vector fields
  14. About multisections
  15. The Euler class of a principal circle bundle
  16. ...and 10 more sections

Key Result

Theorem 1.5

Let $g$ be a Riemannian metric on $\check{M}$. Let $[X]$ be a combing on $\check{M}$. Let $p\in\mathbb{N}\setminus\{0\}$ be a multiple of $\mathcal{O}(X)$. Let $\nu^p_{X}$ be a $p$-multisection of $UX^\perp$. Let $(\omega_i)_{i\in\{1,\dots,3n\}}$ be a family of $3n$ propagating forms of $(C_2(M),g,X of $\mathcal{A}_n^c(\emptyset)$ does not depend on the choice of the family $(\omega_i)_{i\in\{1,\d

Figures (1)

  • Figure 1: Our setting when $\hbox{slk}(K)=0$.

Theorems & Definitions (72)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 1.8
  • Theorem 1.9
  • Definition 1.11
  • ...and 62 more