Thurston norm

Let ${\displaystyle M}$ be a differentiable 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 (not necessarily connected) surface that 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 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.