In mathematics, the Thurston norm is a function on the second homology group of an oriented 3-manifold introduced by William Thurston, which measures in a natural way the topological complexity of homology classes represented by surfaces.
can be represented by a smooth embedding
is a (not necessarily connected) surface that is compact and without boundary.
The Thurston norm of
is then defined to be[1] where the minimum is taken over all embedded surfaces
being the connected components) representing
is the absolute value of the Euler characteristic for surfaces which are not spheres (and 0 for spheres).
This function satisfies the following properties: These properties imply that
which can then be extended by continuity to a seminorm
[2] By Poincaré duality, one can define the Thurston norm on
is compact with boundary, the Thurston norm is defined in a similar manner on the relative homology group
It follows from further work of David Gabai[3] that one can also define the Thurston norm using only immersed surfaces.
This implies that the Thurston norm is also equal to half the Gromov norm on homology.
The Thurston norm was introduced in view of its applications to fiberings and foliations of 3-manifolds.
The unit ball
of the Thurston norm of a 3-manifold
is a polytope with integer vertices.
It can be used to describe the structure of the set of fiberings of
can be written as the mapping torus of a diffeomorphism
represents a class in a top-dimensional (or open) face of
: moreover all other integer points on the same face are also fibers in such a fibration.
[4] Embedded surfaces which minimise the Thurston norm in their homology class are exactly the closed leaves of foliations of