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  "body": "\n## Definition\n\nLet $n \\ge 1$ and $0 \\le k \\le n$, and let $D^j$ be the closed unit ball of $\\R^j$.\nAn $n$-dimensional *$k$-handle* is $H = D^k \\times D^{n-k}$; its *attaching region* is $\\partial D^k \\times D^{n-k}$.\nLet $X$ be a smooth $n$-dimensional [[manifold with boundary|7DS6B2BFK8]], and let $\\varphi \\colon \\partial D^k \\times D^{n-k} \\to \\partial X$ be a smooth embedding onto a [[submanifold|VKFW27WXEZ]] of $\\partial X$.\nThe result of *attaching a $k$-handle* to $X$ along $\\varphi$ is $X \\cup_\\varphi H$.\nIt is the union of $X$ and $H$ with $y$ and $\\varphi(y)$ identified, with corners smoothed.\nLet $M$ be a compact smooth $n$-manifold, possibly with boundary.\nA *handle decomposition* of $M$ is a finite chain $\\emptyset = M_{-1} \\subseteq M_0 \\subseteq \\cdots \\subseteq M_r = M$ of smooth manifolds with boundary.\nEach $M_i$ must be obtained from $M_{i-1}$ by attaching one handle, up to diffeomorphism.\n\n## Remarks\n\nThe first step attaches a $0$-handle $D^0 \\times D^n = D^n$ to the empty set, whose attaching region is empty.\nA relative version starts from $\\partial_0 M \\times [0, 1]$ for a part $\\partial_0 M$ of the boundary instead of the empty set; this is the form used for cobordisms.\nExample: the sphere $S^n$ has a handle decomposition with one $0$-handle and one $n$-handle: the northern and southern hemispheres, glued along the equator $\\partial D^n \\times D^0 = S^{n-1}$.\nNon-example: gluing a second disc $D^2$ to $D^2$ along a single arc of its boundary circle is not a handle attachment.\nIn dimension $2$ the attaching region of a $1$-handle is $\\partial D^1 \\times D^1$, two disjoint arcs.\nThat of a $2$-handle is the whole circle $\\partial D^2 \\times D^0$, and that of a $0$-handle is empty; none of them is a single arc.\nIf $f \\colon M \\to [a, b]$ is a Morse function with critical values in $(a, b)$, then each critical point of index $k$ gives one $k$-handle in a handle decomposition of $M$ relative to $f^{-1}(a)$.\n",
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