$L^2$-Euler characteristics and the Thurston norm

TL;DR

This paper introduces a twisted $L^2$-Euler characteristic for finite CW-complexes, linking it to the Thurston norm in 3-manifolds, and explores its implications for fundamental group epimorphisms.

Contribution

It defines a new twisted $L^2$-Euler characteristic and demonstrates its equivalence to the Thurston norm in certain 3-manifolds, providing insights into group epimorphisms.

Findings

The twisted $L^2$-Euler characteristic matches the Thurston norm for specific 3-manifolds.

It offers a new approach to compare Thurston norms via group epimorphisms.

The work establishes properties of the $L^2$-Euler characteristic in relation to topological invariants.

Abstract

We assign to a finite $CW$-complex and an element in its first cohomology group a twisted version of the $L^2$-Euler characteristic and study its main properties. In the case of an irreducible orientable $3$-manifold with empty or toroidal boundary and infinite fundamental group we identify it with the Thurston norm. We will use the $L^2$-Euler characteristic to address the problem whether the existence of a map inducing an epimorphism on fundamental groups implies an inequality of the Thurston norms.