A function on the homology of 3–manifolds

Abstract

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