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 ${\displaystyle M}$ be a differential manifold and ${\displaystyle c\in H_{2}(M)}$. Then ${\displaystyle c}$ can be represented by a smooth embedding ${\displaystyle S\to M}$ where ${\displaystyle S}$ is a (generally not connected) surface which is compact and without boundary. The Thurston norm of ${\displaystyle c}$ is then defined to be[1]

${\displaystyle \|c\|_{T}=\min _{S}\sum _{i=1}^{n}\chi _{-}(S_{i})}$

where the minimum is taken over all embedded surfaces ${\displaystyle S=\bigcup _{i}S_{i}}$ (the ${\displaystyle S_{i}}$ being the connected components) representing ${\displaystyle c}$ as above and ${\displaystyle \chi _{-}(F)=\max(0,-\chi (F))}$ is the absolute value of the Euler characteristic for surfaces which are not spheres (and 0 for spheres).

This function satisfies the following properties:

• ${\displaystyle \|kc\|_{T}=|k|\cdot \|c\|_{T}}$ for ${\displaystyle c\in H_{2}(M),k\in \mathbb {Z} }$;
• ${\displaystyle \|c_{1}+c_{2}\|_{T}\leq \|c_{1}\|_{T}+\|c_{2}\|_{T}}$ for ${\displaystyle c_{1},c_{2}\in H_{2}(M)}$.

These properties imply that ${\displaystyle \|\cdot \|}$ extends to a function on ${\displaystyle H_{2}(M,\mathbb {Q} )}$ which can then be extended by continuity to a seminorm ${\displaystyle \|\cdot \|_{T}}$ on ${\displaystyle H_{2}(M,\mathbb {R} )}$.[2] By Poincaré duality one can define the Thurston norm on ${\displaystyle H^{1}(M,\mathbb {R} )}$.

When ${\displaystyle M}$ is compact with boundary the Thurston norm is defined in a similar manner on the relative homology group ${\displaystyle H_{2}(M,\partial M,\mathbb {R} )}$ and its Poincaré dual ${\displaystyle H^{1}(M,\mathbb {R} )}$.

It follows from further work of 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.

Topological applications

The Thurston norm was introduced in view of its applications to fiberings and foliations of 3-manifolds.

The unit ball ${\displaystyle B}$ of the Thurston norm of a 3-manifold ${\displaystyle M}$ is a polytope with integer vertices. It can be used to describe the structure of the set of fiberings of ${\displaystyle M}$ over the circle: if ${\displaystyle M}$ can be written as the mapping torus of a diffeomorphism ${\displaystyle f}$ of a surface ${\displaystyle S}$ then the embedding ${\displaystyle S\hookrightarrow M}$ represents a class in a top-dimensional (or open) face of ${\displaystyle B}$: 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 ${\displaystyle M}$.[3]

Notes

1. Thurston 1986.
2. Thurston 1986, Theorem 1.
3. Gabai 1983.
4. Thurston 1986, Theorem 5.

References

• Gabai, David (1983). "Foliations and the topology of 3-manifolds". J. Differential Geom. 18: 445–503. MR 0723813.CS1 maint: ref=harv (link)
• Thurston, William (1986). "A norm for the homology of 3-manifolds". Mem. Amer. Math. Soc. 59 (33): i–vi and 99–130. MR 0823443.CS1 maint: ref=harv (link)