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  "body": "\n## Definition\n\nLet $g \\ge 0$ be an integer.\nA *handlebody of genus $g$* is a compact, connected, orientable $3$-dimensional smooth [[manifold with boundary|7DS6B2BFK8]] $V$ with the following property.\nIt has a [[handle decomposition|9V11N8VVTP]] with one $0$-handle, $g$ one-handles, and no other handles.\n\n## Remarks\n\nThe [[orientability|702SNFQG0W]] condition fixes the way each $1$-handle is attached, and then a handlebody of genus $g$ is unique up to diffeomorphism.\nIts boundary $\\partial V$ is the closed orientable surface of [[genus|NJ787GZS1N]] $g$, and $V$ is homotopy equivalent to a wedge of $g$ circles.\nEquivalently, for $g \\ge 1$, $V$ is the boundary connected sum of $g$ solid tori $S^1 \\times D^2$.\nAlso, $V$ is a closed regular neighbourhood of a connected finite graph in $\\R^3$ with Euler characteristic $1 - g$.\nExample: the solid torus $S^1 \\times D^2$ is a handlebody of genus $1$.\nIt is the $3$-ball $D^3$ with a $1$-handle $D^1 \\times D^2$ attached along two disjoint discs of $\\partial D^3$, in an orientation-compatible way.\nNon-example: the solid Klein bottle is obtained from $D^3$ by attaching one $1$-handle so that the result is non-orientable.\nIt has the right handles but is not orientable, so it is not a handlebody.\nIn higher-dimensional topology, *handlebody* often means any manifold with a handle decomposition, with no restriction on dimension or on the indices of the handles.\n",
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