Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Guts determine the leading coefficients of $L^2$-Alexander torsions

Jianru Duan

TL;DR

The paper proves that for a connected, orientable, irreducible 3-manifold $N$, the leading coefficient $C(N,\phi)$ of the $L^2$-Alexander torsion is controlled by topological data: it equals the relative $L^2$-torsion obtained by cutting along a Thurston-norm-minimizing dual surface and also equals the relative torsion of the guts $\Gamma(\phi)$. A new convergence criterion for Fuglede--Kadison determinants under Alexander twists underpins this identification, enabling a direct link between the analytic invariant and a topological decomposition. The authors then prove that $C(N,\phi)$ is constant on any open Thurston cone, using the Agol–Zhang guts-invariance and a convexity argument for the logarithm of the leading coefficient, which yields finiteness of possible values and a subordination inequality. Collectively, these results provide a canonical, topological interpretation of the leading coefficient and integrate sutured-manifold theory, guts, and $L^2$-torsion into a coherent framework with potential implications for Thurston norms and 3-manifold topology.

Abstract

For 3-manifolds, the leading coefficient of the $L^2$-Alexander torsion is a numerical invariant of a real first cohomology class. We show that the leading coefficient equals the relative $L^2$-torsion of the manifold cut up along a norm-minimizing surface dual to the cohomology class. Furthermore, the leading coefficient equals the relative $L^2$-torsion of the guts associated to the cohomology class. Finally, we prove that the leading coefficient is constant on any open Thurston cone. The main ingredients are a new criterion for the convergence of Fuglede-Kadison determinants and the work of Agol and Zhang on guts of 3-manifolds.

Guts determine the leading coefficients of $L^2$-Alexander torsions

TL;DR

The paper proves that for a connected, orientable, irreducible 3-manifold , the leading coefficient of the -Alexander torsion is controlled by topological data: it equals the relative -torsion obtained by cutting along a Thurston-norm-minimizing dual surface and also equals the relative torsion of the guts . A new convergence criterion for Fuglede--Kadison determinants under Alexander twists underpins this identification, enabling a direct link between the analytic invariant and a topological decomposition. The authors then prove that is constant on any open Thurston cone, using the Agol–Zhang guts-invariance and a convexity argument for the logarithm of the leading coefficient, which yields finiteness of possible values and a subordination inequality. Collectively, these results provide a canonical, topological interpretation of the leading coefficient and integrate sutured-manifold theory, guts, and -torsion into a coherent framework with potential implications for Thurston norms and 3-manifold topology.

Abstract

For 3-manifolds, the leading coefficient of the -Alexander torsion is a numerical invariant of a real first cohomology class. We show that the leading coefficient equals the relative -torsion of the manifold cut up along a norm-minimizing surface dual to the cohomology class. Furthermore, the leading coefficient equals the relative -torsion of the guts associated to the cohomology class. Finally, we prove that the leading coefficient is constant on any open Thurston cone. The main ingredients are a new criterion for the convergence of Fuglede-Kadison determinants and the work of Agol and Zhang on guts of 3-manifolds.
Paper Structure (11 sections, 20 theorems, 76 equations, 1 figure)

This paper contains 11 sections, 20 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgment
  3. Preliminaries
  4. Hilbert modules and Fuglede--Kadison determinants
  5. $L^2$-Alexander torsions of CW-pairs
  6. $L^2$-Alexander torsion of 3-manifolds
  7. The leading coefficient and the relative $L^2$-torsion
  8. The guts and the leading coefficient
  9. Preliminaries to sutured manifolds
  10. The leading coefficient is determined by the guts
  11. The leading coefficient of an open Thurston cone

Key Result

Theorem 1.1

Let $N$ be a connected, orientable, irreducible, compact 3-manifold with empty or toral boundary. Let $\phi\in H^1(N;\mathbb{Z})$ and let $S$ be a Thurston norm-minimizing surface dual to $\phi$, then

Figures (1)

  • Figure 1: Decomposition along a product annulus (the above part) and a product disk (the lower part). By definition $R_+'$ and $R_-'$ each contains a copy of $C$ respectively, but one can isotope (by absorbing the copies of $C$ into the suture) so that $R_\pm'=R_\pm\setminus(\partial C\times (-1,1))$ as in the figure on the right. The two copies of $C$ in $\partial M'$ then become subsets of $\gamma'$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Theorem 2.7
  • ...and 28 more