The Thurston norm is a mathematical concept in the field of low-dimensional topology, particularly in the study of 3-manifolds. It provides a way to assign a "norm" to elements of the second homology group \( H_2(M; \mathbb{R}) \) of a 3-manifold \( M \). This norm is associated with the concept of surface representations in the manifold and is used to measure the complexity of surfaces that can be embedded into the manifold.
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For a compact oriented three-manifold, the integral relative class has norm the minimum of over properly embedded oriented topological surfaces representing . Homogeneity and continuity extend it to real relative homology. It is a seminorm, since spheres, disks, annuli and tori have zero cost.