  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.
