A function on the homology of 3-manifolds

In analogy with the Thurston norm, we define for an orientable 3-manifold $M$ a numerical function on $H_2(M;Q/Z)$. This function measures the minimal complexity of folded surfaces representing a given homology class. A similar function is defined on the torsion subgroup of $H_1(M)$. These functions are estimated from below in terms of abelian torsions of $M$.