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Universal $L^2$-torsion and sutured decomposition for 3-manifolds

Jianru Duan

TL;DR

The paper develops a noncommutative invariant, the universal $L^2$-torsion, for admissible 3-manifolds and extends it to taut sutured manifolds. Central to the framework are the leading term map $L_oldsymbol{ ho}$ on Linnell’s skew field, the restriction and polytope maps, and a matrix-chain method for computation; together these yield a fiberedness criterion: a nonzero class is fibered iff the $oldsymbol{ ho}$-leading term of the torsion is trivial. A decomposition formula under taut sutured decompositions sharpens the parallel with sutured Floer theory and yields a product-detection result: a taut sutured manifold is a product iff its universal $L^2$-torsion is trivial. The theory is applied to group homomorphisms, with explicit Fox Jacobian formulas for free groups and concrete calculations for sutured handlebodies and chain-link complements, illustrating both the algebraic richness and geometric reach of the invariant.

Abstract

Given an admissible 3-manifold $M$ and a cohomology class $φ\in H^1(M;\mathbb R)$, we prove that the universal $L^2$-torsion of $M$ detects the fiberedness of $φ$, except when $M$ is a closed graph manifold that admits no non-positively curved metric. We further extend this invariant to sutured 3-manifolds and derive a decomposition formula for taut sutured decompositions. Moreover, we show that a taut sutured manifold is a product if and only if its universal $L^2$-torsion is trivial. Our methods are based on a detailed study of the leading term map over Linnell's skew field. As an application, we apply the theory to homomorphisms between finitely generated free groups, which enables explicit computations of the invariant for sutured handlebodies.

Universal $L^2$-torsion and sutured decomposition for 3-manifolds

TL;DR

The paper develops a noncommutative invariant, the universal -torsion, for admissible 3-manifolds and extends it to taut sutured manifolds. Central to the framework are the leading term map on Linnell’s skew field, the restriction and polytope maps, and a matrix-chain method for computation; together these yield a fiberedness criterion: a nonzero class is fibered iff the -leading term of the torsion is trivial. A decomposition formula under taut sutured decompositions sharpens the parallel with sutured Floer theory and yields a product-detection result: a taut sutured manifold is a product iff its universal -torsion is trivial. The theory is applied to group homomorphisms, with explicit Fox Jacobian formulas for free groups and concrete calculations for sutured handlebodies and chain-link complements, illustrating both the algebraic richness and geometric reach of the invariant.

Abstract

Given an admissible 3-manifold and a cohomology class , we prove that the universal -torsion of detects the fiberedness of , except when is a closed graph manifold that admits no non-positively curved metric. We further extend this invariant to sutured 3-manifolds and derive a decomposition formula for taut sutured decompositions. Moreover, we show that a taut sutured manifold is a product if and only if its universal -torsion is trivial. Our methods are based on a detailed study of the leading term map over Linnell's skew field. As an application, we apply the theory to homomorphisms between finitely generated free groups, which enables explicit computations of the invariant for sutured handlebodies.

Paper Structure

This paper contains 41 sections, 44 theorems, 180 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Motivations
  3. Why universal $L^2$-torsion
  4. Fiberedness and the leading term of torsions
  5. Analogy with sutured Floer homology
  6. Proof ingredients
  7. Algebraic preliminaries
  8. Universal $L^2$-torsion
  9. Leading term map, restriction map and polytope map
  10. Universal $L^2$-torsion for taut sutured manifolds
  11. Applications of the universal $L^2$-torsion
  12. Acknowledgments
  13. Algebraic preliminaries
  14. Hilbert modules
  15. Atiyah Conjecture and Linnell's skew field
  16. ...and 26 more sections

Key Result

Theorem 1.2

Suppose $M$ is an admissible 3-manifold that is not a closed graph manifold without an NPC metric. For any nonzero cohomology class $\phi\in H^1(M;\mathbb{R})$, the class $\phi$ is fibered if and only if $L_\phi\tau^{(2)}_u(M)=1\in \operatorname{Wh}^w(\pi_1(M))$.

Figures (7)

  • Figure 1: The $n$-chain link and the minimal-genus Seifert surface $\Sigma$, where $n=4$.
  • Figure 2: An illustration in one lower dimension. Left: A simplicial complex $N$ whose boundary is a union of a subcomplex $R_-$ (the arc $\wideparen{ABC}$) and a subcomplex $R_+$ (the arc $\wideparen{ADC}$). Middle: the dual cellular complex $N'$ (red). Right: the subcomplex $K\subset N'$ consisting of cells disjoint from $R_+$. The complex $N'$ deformation retracts to $K$ along a product neighborhood of $R_+$.
  • Figure 3: Decomposing $N$ along a separating decomposition surface $S$ (dotted region in left figure, with normal direction indicated by red arrows). The $R_+$-regions of the sutured manifolds are shown in green.
  • Figure 4: Consider $N$ as the union of $N\backslash\!\backslash S$ and $S\times I$.
  • Figure 5: Double of a sutured manifold $N$ with monodromy $f$
  • ...and 2 more figures

Theorems & Definitions (125)

  • Definition 1.1: Admissible 3-manifold
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1: Weak isomorphism
  • Definition 2.2: Atiyah Conjecture
  • Theorem 2.3: linnell1993division
  • Theorem 2.4: kielak2024group
  • Definition 2.5: Division closure
  • ...and 115 more