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  "body": "\n## Definition\n\nLet $S$ be a closed [[orientable|0TD904MG06]] [[surface|VGJJVBZ1MQ]], that is, a compact connected orientable surface without boundary.\nCall simple closed curves $c_1, \\ldots, c_k$ in $S$ *non-separating* when they are pairwise disjoint and $S \\setminus (c_1 \\cup \\cdots \\cup c_k)$ is [[connected|0V032VWTHD]].\nThe *genus* $g(S)$ is the largest integer $k \\ge 0$ for which such curves $c_1, \\ldots, c_k$ exist.\n\n## Remarks\n\nThe genus is finite and satisfies $\\chi(S) = 2 - 2g(S)$, where $\\chi$ is the Euler characteristic.\nBy the classification of surfaces, $S$ is homeomorphic to the sphere if $g = 0$, and to the connected sum of $g$ tori if $g \\ge 1$.\n\nExample: for the torus $T^2 = S^1 \\times S^1$, the curve $c_1 = S^1 \\times \\set{1}$ has connected complement $S^1 \\times (S^1 \\setminus \\set{1})$, an open annulus.\nAny two disjoint simple closed curves in the torus disconnect it, so $g(T^2) = 1$, in accord with $\\chi(T^2) = 0$.\nNon-example of a non-separating curve: on the sphere $S^2$ every simple closed curve separates the sphere into two pieces, by the Jordan curve theorem, so $g(S^2) = 0$.\n\nFor a closed non-orientable surface, the *non-orientable genus* is the number $k$ of copies of $\\R P^2$ in its connected sum decomposition; then $\\chi = 2 - k$.\n\n## Explanations\n\n### L1 human\n\nThe genus counts how many separate cuts along closed loops a surface can survive while staying in one piece.\nEach handle allows exactly one such cut, around the handle.\nSo the genus is the number of handles.\n",
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