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Computing the twisted $L^2$-Euler characteristic

Jacopo G. Chen

TL;DR

This work develops an algorithm to compute Friedl and Lück’s twisted $L^2$-Euler characteristic for suitable $L^2$-acyclic CW complexes by leveraging Oki’s matrix expansion to estimate degrees of Dieudonné determinants within the Linnell skew field $\mathcal{D}(G)$. Grounded in the polytope interpretation of determinants and the universal $L^2$-torsion, the method expresses twists by a character $\phi$ as a single polytope-valued invariant whose degree recovers $\chi^{(2)}(\widetilde{M};\mathcal{N}(G),\phi)$; Lück’s approximation guides the finite-quotient computations, with error controlled by recent quantitative bounds. The paper shows that a truncated, human-assisted version yields accurate results in diverse settings, including hyperbolic link complements, closed census 3-manifolds, free-by-cyclic groups, and higher-dimensional manifolds such as the Ratcliffe–Tschantz fiber, illustrating a practical path to exploring a Thurston-like norm in higher dimensions. Empirically, the approach recovers Thurston and Alexander norm unit balls in several cases and suggests a deep link between twisted $L^2$-invariants and fibering phenomena, with broader implications for understanding geometry via noncommutative determinant theory. The work combines deep noncommutative algebra, $L^2$-invariants, and computational topology to produce an effective toolkit for probing norm-like invariants beyond dimension three.

Abstract

We present an algorithm that computes Friedl and Lück's twisted $L^2$-Euler characteristic for a suitable regular CW complex, employing Oki's matrix expansion algorithm to indirectly evaluate the Dieudonné determinant. The algorithm needs to run for an extremely long time to certify its outputs, but a truncated, human-assisted version produces very good results in many cases, such as hyperbolic link complements, closed census 3-manifolds, free-by-cyclic groups, and higher-dimensional examples, such as the fiber of the Ratcliffe-Tschantz manifold.

Computing the twisted $L^2$-Euler characteristic

TL;DR

This work develops an algorithm to compute Friedl and Lück’s twisted -Euler characteristic for suitable -acyclic CW complexes by leveraging Oki’s matrix expansion to estimate degrees of Dieudonné determinants within the Linnell skew field . Grounded in the polytope interpretation of determinants and the universal -torsion, the method expresses twists by a character as a single polytope-valued invariant whose degree recovers ; Lück’s approximation guides the finite-quotient computations, with error controlled by recent quantitative bounds. The paper shows that a truncated, human-assisted version yields accurate results in diverse settings, including hyperbolic link complements, closed census 3-manifolds, free-by-cyclic groups, and higher-dimensional manifolds such as the Ratcliffe–Tschantz fiber, illustrating a practical path to exploring a Thurston-like norm in higher dimensions. Empirically, the approach recovers Thurston and Alexander norm unit balls in several cases and suggests a deep link between twisted -invariants and fibering phenomena, with broader implications for understanding geometry via noncommutative determinant theory. The work combines deep noncommutative algebra, -invariants, and computational topology to produce an effective toolkit for probing norm-like invariants beyond dimension three.

Abstract

We present an algorithm that computes Friedl and Lück's twisted -Euler characteristic for a suitable regular CW complex, employing Oki's matrix expansion algorithm to indirectly evaluate the Dieudonné determinant. The algorithm needs to run for an extremely long time to certify its outputs, but a truncated, human-assisted version produces very good results in many cases, such as hyperbolic link complements, closed census 3-manifolds, free-by-cyclic groups, and higher-dimensional examples, such as the fiber of the Ratcliffe-Tschantz manifold.
Paper Structure (32 sections, 24 theorems, 82 equations, 5 figures, 10 tables)

This paper contains 32 sections, 24 theorems, 82 equations, 5 figures, 10 tables.

Table of Contents

  1. Preliminaries on L2̂-invariants
  2. The von Neumann algebra N(G)
  3. Von Neumann dimension for N(G)-modules
  4. L2̂-Betti numbers
  5. The Atiyah conjecture
  6. The polytope homomorphism
  7. The Dieudonné determinant
  8. Universal L2̂-torsion
  9. What about manifolds?
  10. Computing the twisted L2̂-Euler characteristic
  11. Degrees of determinants
  12. Lück's approximation theorem
  13. Estimating the error
  14. Implementation details
  15. Chain complex construction
  16. ...and 17 more sections

Key Result

Theorem 1

There exists an algorithm that, given a finite $L^2$-acyclic CW complex $M$, such that its fundamental group $G$ is residually finite and satisfies the Atiyah conjecture, and a character $\phi: G \to \mathbb Z$, computes the twisted $L^2$-Euler characteristic $\chi^{(2)}(\widetilde{M}; \phi)$.

Figures (5)

  • Figure 7.1: The Borromean rings.
  • Figure 7.2: The unit ball is determined by the $14$ marked points.
  • Figure 7.3: The two-component link L10n14 and the unit balls for its Thurston and Alexander norms.
  • Figure 7.4: Left: the Thurston norm unit ball for $\mathrm{v1539(5,1)}$, determined by the $10$ marked points. Right: the Newton polytope for the Alexander polynomial obtained from the group presentation (\ref{['eq:v1539-group']}). The two polytopes are dual to each other.
  • Figure 7.5: The unit balls of the seminorms $-\chi^{(2)}(G; -)$ for both free-by-cyclic examples, including the boundary points that determine them.

Theorems & Definitions (72)

  • Theorem
  • Corollary
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5: l2
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: compare l2
  • ...and 62 more