Thurston norm

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.

Definition

Let be a differential manifold and . Then can be represented by a smooth embedding where is a (generally not connected) surface which is compact and without boundary. The Thurston norm of is then defined to be[1]

where the minimum is taken over all embedded surfaces (the being the connected components) representing as above and 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 extends to a function on which can then be extended by continuity to a seminorm on .[2] By Poincaré duality one can define the Thurston norm on .

When is compact with boundary the Thurston norm is defined in a similar manner on the relative homology group and its Poincaré dual .

It follows from further work of Gabai[3] that one can also defined the Thurston norm using only immersed surfaces. This implies that the Thurston norm is also equal to half the Gromov norm on homology.

Topological applications

The Thurston norm was introduced in view of its applications to fibrations 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 over the circle: if can be written as the mapping torus of a diffeomorphism of a surface then the embedding 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 .[3]

Notes

References

This article is issued from Wikipedia - version of the 7/11/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.