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  "body": "\n## Definition\n\nLet $M$ be a compact [[orientable|702SNFQG0W]] [[3-manifold|R20TB2KEQQ]].\nFor a compact surface $S$ with connected components $S_1, \\ldots, S_k$, set\n$$\\chi_-(S) = \\sum_{i=1}^{k} \\max(0, -\\chi(S_i)),$$\nwhere $\\chi$ is the [[Euler characteristic|RNDDWDAFJ5]].\nFor a class $a \\in H_2(M, \\partial M; \\Z)$ in [[homology|M0Q5CFJ7ZH]], the *Thurston norm* of $a$ is\n$$x(a) = \\min \\set{\\chi_-(S) : S \\text{ a properly embedded oriented surface in } M \\text{ with } [S] = a}.$$\nThe function $x$ satisfies $x(n a) = n\\, x(a)$ for integers $n \\ge 0$ and $x(a + b) \\le x(a) + x(b)$, and it extends uniquely to a continuous [[seminorm|FP5GVCN8GV]] on $H_2(M, \\partial M; \\R)$, also called the Thurston norm.\n\n## Remarks\n\nExample: let $F$ be a closed orientable surface of genus $g \\ge 2$ and $M = F \\times S^1$.\nThe fiber $F \\times \\set{1}$ represents a class $a$ with $\\chi_-(F) = 2g - 2$, and Thurston showed that fibers of a fibration over the circle minimise $\\chi_-$ in their class, so $x(a) = 2g - 2$.\n\nNon-example of definiteness: in $M = S^2 \\times S^1$ the class $a$ of $S^2 \\times \\set{1}$ is nonzero, but $\\chi_-(S^2) = 0$, so $x(a) = 0$.\nSo $x$ is in general only a seminorm, not a norm.\nIt is a norm when no nonzero class is represented by spheres, disks, tori and annuli, for example when $M$ is hyperbolic.\n",
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