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 M {\displaystyle M} be a differentiable manifold and c ∈ H 2 ( M ) {\displaystyle c\in H_{2}(M)} . Then c {\displaystyle c} can be represented by a smooth embedding S → M {\displaystyle S\to M} , where S {\displaystyle S} is a (not necessarily connected) surface that is compact and without boundary. The Thurston norm of c {\displaystyle c} is then defined to be
‖ c ‖ T = min S ∑ i = 1 n χ − ( S i ) {\displaystyle \|c\|_{T}=\min _{S}\sum _{i=1}^{n}\chi _{-}(S_{i})} , where the minimum is taken over all embedded surfaces S = ⋃ i S i {\displaystyle S=\bigcup _{i}S_{i}} (the S i {\displaystyle S_{i}} being the connected components) representing c {\displaystyle c} as above, and χ − ( F ) = max ( 0 , − χ ( F ) ) {\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:
‖ k c ‖ T = | k | ⋅ ‖ c ‖ T {\displaystyle \|kc\|_{T}=|k|\cdot \|c\|_{T}} for c ∈ H 2 ( M ) , k ∈ Z {\displaystyle c\in H_{2}(M),k\in \mathbb {Z} } ;
‖ c 1 + c 2 ‖ T ≤ ‖ c 1 ‖ T + ‖ c 2 ‖ T {\displaystyle \|c_{1}+c_{2}\|_{T}\leq \|c_{1}\|_{T}+\|c_{2}\|_{T}} for c 1 , c 2 ∈ H 2 ( M ) {\displaystyle c_{1},c_{2}\in H_{2}(M)} . These properties imply that ‖ ⋅ ‖ {\displaystyle \|\cdot \|} extends to a function on H 2 ( M , Q ) {\displaystyle H_{2}(M,\mathbb {Q} )} which can then be extended by continuity to a seminorm ‖ ⋅ ‖ T {\displaystyle \|\cdot \|_{T}} on H 2 ( M , R ) {\displaystyle H_{2}(M,\mathbb {R} )} . By Poincaré duality, one can define the Thurston norm on H 1 ( M , R ) {\displaystyle H^{1}(M,\mathbb {R} )} . When M {\displaystyle M} is compact with boundary, the Thurston norm is defined in a similar manner on the relative homology group H 2 ( M , ∂ M , R ) {\displaystyle H_{2}(M,\partial M,\mathbb {R} )} and its Poincaré dual H 1 ( M , R ) {\displaystyle H^{1}(M,\mathbb {R} )} . It follows from further work of David Gabai 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.
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