Computing the twisted $L^2$-Euler characteristic

TL;DR

This paper introduces an algorithm to compute the twisted $L^2$-Euler characteristic for certain CW complexes, combining matrix expansion and determinant evaluation, with practical results on complex 3-manifolds and higher-dimensional examples.

Contribution

It presents a novel algorithm that computes the twisted $L^2$-Euler characteristic using matrix expansion and determinant evaluation techniques.

Findings

Algorithm successfully computes twisted $L^2$-Euler characteristic for various complex examples.

Truncated, human-assisted version yields accurate results in practical cases.

Applicable to hyperbolic link complements, 3-manifolds, and higher-dimensional manifolds.

Abstract

We present an algorithm that computes Friedl and L\"uck's twisted $L^2$-Euler characteristic for a suitable regular CW complex, employing Oki's matrix expansion algorithm to indirectly evaluate the Dieudonn\'e 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.