L2‐Euler characteristics and the Thurston norm

We assign to a finite CW-complex and an element in its first cohomology group a twisted version of the L2-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 twisted L2-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