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  "body": "\n## Definition\n\nLet $\\Gamma \\subset \\R^3$ be a connected finite graph with Euler characteristic $1 - g$.\nA *handlebody of genus $g$* is a $3$-manifold homeomorphic to a closed regular neighbourhood of $\\Gamma$ in $\\R^3$.\nIts boundary is a closed orientable surface of [[genus|NJ787GZS1N]] $g$.\n\nLet $M$ be a closed connected orientable [[3-manifold|R20TB2KEQQ]].\nA *Heegaard splitting* of $M$ is a pair $(V, W)$ of subsets of $M$ such that:\n\n1. $M = V \\cup W$;\n2. $V$ and $W$ are [[handlebodies|AP1KY4EHMA]] of the same genus $g$;\n3. $V \\cap W = \\partial V = \\partial W$.\n\nThe closed [[surface|VGJJVBZ1MQ]] $\\Sigma = V \\cap W$ is the *Heegaard surface*, and $g$ is the genus of the splitting.\n\n## Remarks\n\nEvery closed connected orientable $3$-manifold has a Heegaard splitting.\nTake a triangulation and let $V$ be a regular neighbourhood of its $1$-skeleton.\nThe closure $W$ of the complement is a regular neighbourhood of the dual $1$-skeleton.\nThe *Heegaard genus* of $M$ is the least genus of a Heegaard splitting.\n\nExample: $S^3 = \\set{(z, w) \\in \\C^2 : \\abs{z}^2 + \\abs{w}^2 = 1}$ is the union of the solid tori $V = \\set{\\abs{z} \\le \\abs{w}}$ and $W = \\set{\\abs{z} \\ge \\abs{w}}$, which meet in the torus $\\abs{z} = \\abs{w} = 1/\\sqrt{2}$; this is a splitting of genus $1$.\nLens spaces also have splittings of genus $1$.\n\nNon-example: let $N$ be a closed tubular neighbourhood of a trefoil knot in $S^3$ and $E$ the closure of its complement.\nThen $S^3 = N \\cup E$ with $N \\cap E = \\partial N = \\partial E$ a torus, and $N$ is a solid torus, but $E$ is not a handlebody.\nIndeed the fundamental group of $E$ is $\\langle x, y \\mid x^2 = y^3 \\rangle$, which is not free, while handlebodies have free fundamental groups.\nSo clause 2 fails.\n\n## Explanations\n\n### L1 human\n\nThe definition splits a closed $3$-dimensional space into two thickened graphs that share their whole boundary surface.\nAll the topology of the space is then in how the two boundary surfaces are glued.\n",
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