Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Circle-valued angle structures and obstruction theory

Craig D. Hodgson, Andrew J. Kricker, Rafał M. Siejakowski

TL;DR

The article develops a comprehensive obstruction-theoretic framework for circle-valued angle structures on ideal triangulations of 3-manifolds. It constructs maps from spaces of SAS to cohomology rings and proves that connected components are classified by $\mathbb{Z}_2$-cohomology obstructions to lifting PSL(2,$\mathbb{C}$) representations to SL(2,$\mathbb{C}$), including relative obstructions for boundary-parabolic representations. The authors connect these structures to Thurston’s gluing equations, Neumann’s combinatorial techniques, and a robust rectangle/fanning construction, yielding a bijection between component data and lifting obstructions that is triangulation-independent. They further relate the theory to the Garoufalidis–Kashaev meromorphic 3D-index, showing how the domain of integration for the index aligns with a peripherally trivial component and how obstruction classes determine contributing representations. An explicit hyperbolic example (the sister of the figure-eight knot complement) demonstrates the mechanism, computing obstruction classes for distinct boundary-parabolic representations and illustrating the component-based classification in practice.

Abstract

We study spaces of circle-valued angle structures, introduced by Feng Luo, on ideal triangulations of 3-manifolds. We prove that the connected components of these spaces are enumerated by certain cohomology groups of the 3-manifold with $\mathbb{Z}_2$-coefficients. Our main theorem shows that this establishes a geometrically natural bijection between the connected components of the spaces of circle-valued angle structures and the obstruction classes to lifting boundary-parabolic $PSL(2,\mathbb{C})$-representations of the fundamental group of the 3-manifold to boundary-unipotent representations into $SL(2,\mathbb{C})$. In particular, these connected components have topological and algebraic significance independent of the ideal triangulations chosen to construct them. The motivation and main application of this study is to understand the domain of the state-integral defining the meromorphic 3D-index of Garoufalidis and Kashaev, necessary in order to classify the boundary-parabolic representations contributing to its asymptotics.

Circle-valued angle structures and obstruction theory

TL;DR

The article develops a comprehensive obstruction-theoretic framework for circle-valued angle structures on ideal triangulations of 3-manifolds. It constructs maps from spaces of SAS to cohomology rings and proves that connected components are classified by -cohomology obstructions to lifting PSL(2,) representations to SL(2,), including relative obstructions for boundary-parabolic representations. The authors connect these structures to Thurston’s gluing equations, Neumann’s combinatorial techniques, and a robust rectangle/fanning construction, yielding a bijection between component data and lifting obstructions that is triangulation-independent. They further relate the theory to the Garoufalidis–Kashaev meromorphic 3D-index, showing how the domain of integration for the index aligns with a peripherally trivial component and how obstruction classes determine contributing representations. An explicit hyperbolic example (the sister of the figure-eight knot complement) demonstrates the mechanism, computing obstruction classes for distinct boundary-parabolic representations and illustrating the component-based classification in practice.

Abstract

We study spaces of circle-valued angle structures, introduced by Feng Luo, on ideal triangulations of 3-manifolds. We prove that the connected components of these spaces are enumerated by certain cohomology groups of the 3-manifold with -coefficients. Our main theorem shows that this establishes a geometrically natural bijection between the connected components of the spaces of circle-valued angle structures and the obstruction classes to lifting boundary-parabolic -representations of the fundamental group of the 3-manifold to boundary-unipotent representations into . In particular, these connected components have topological and algebraic significance independent of the ideal triangulations chosen to construct them. The motivation and main application of this study is to understand the domain of the state-integral defining the meromorphic 3D-index of Garoufalidis and Kashaev, necessary in order to classify the boundary-parabolic representations contributing to its asymptotics.
Paper Structure (23 sections, 19 theorems, 82 equations, 14 figures)

This paper contains 23 sections, 19 theorems, 82 equations, 14 figures.

Table of Contents

  1. Introduction
  2. General properties of the manifolds of circle-valued angle structures
  3. Circle-valued angle structures and obstruction theory
  4. Application to the Garoufalidis-Kashaev meromorphic 3D-index
  5. Preliminaries on ideal triangulations and angle structures
  6. Thurston's gluing equations
  7. Angle structures on ideal triangulations
  8. Tangential angle structures
  9. Spaces of circle-valued angle structures: obstruction classes and components
  10. Combinatorics of 3-manifold triangulations following Neumann
  11. Existence theorem for integer-valued pseudo-angle structures
  12. Truncated triangulations and rectangular cocycles
  13. The fanning construction and its inverse
  14. Definition of the obstruction maps
  15. Proof of Theorem \ref{['obstruction-theorem']}
  16. ...and 8 more sections

Key Result

Theorem 1.3

Consider an ideal triangulation $\mathcal{T}$, consisting of $N$ tetrahedra, of a connected orientable 3-manifold $M$ with $k$ toroidal ends.

Figures (14)

  • Figure 1.1: The cyclic ordering of normal quadrilaterals $\square\to\square'\to\square"\to\square$.
  • Figure 3.1: To a tetrahedral edge $e$ (bolded) we associate an $\mathbb{Z}_2$-cochain $\chi_{F_1}+\chi_{F_2}$ taking the value $1\in\mathbb{Z}_2$ on the bolded dual $1$-cells and the value $0\in\mathbb{Z}_2$ on the remaining $1$-cells.
  • Figure 3.2: Cross-sectional view of a neighbourhood of an edge $E$ of the triangulation $\mathcal{T}$ where two curves $\theta_1$ and $\theta_2$ deviate from one another. The tetrahedra $\Delta_1$ and $\Delta_2$ through which the curves enter and leave the neighbourhood of $E$ are shaded.
  • Figure 3.3: A doubly truncated tetrahedron in $\mathcal{T}_{00}$ and the three types of its edges.
  • Figure 3.4: Left: A large hexagonal face $F$ with $\sigma(F)=1$. Centre: The $1$-cochain $b_F$ is supported on the three long edges of the face $F$ (shown in bold). Right:$Y(\sigma) = \sigma + \delta^1(b_F)$ vanishes on $F$ but may have a non-zero value on rectangular $2$-cells of $\mathcal{T}_{00}$.
  • ...and 9 more figures

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 3.1
  • ...and 42 more