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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.