Thurston norm

Summary

The Thurston norm is a seminorm on the second homology of a compact orientable 3-manifold. It measures the least complexity, in terms of Euler characteristic, of an embedded surface representing a given class. Thurston showed that its unit ball is a polyhedron. The classes represented by fibers of fibrations over the circle fill the cones over certain top-dimensional faces. It connects the topology of surfaces in $3$-manifolds with fibrations over the circle.