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  "body": "\n## Definition\n\nLet $n \\ge 0$ and let $U$ and $V$ be [[open subsets|02BH3MCBQB]] of $\\R^n$.\nA map $f \\colon U \\to V$ is *piecewise linear* if $U$ is a union of a locally finite family of simplices in $\\R^n$ with $f$ affine on each.\nAny two simplices of the family must meet in a common face or not at all.\nLet $M$ be a [[topological manifold|QQQ9B1DK4P]] of dimension $n$.\nA *PL atlas* on $M$ is an [[atlas|J7P32JQTVE]] of dimension $n$ all of whose transition maps are piecewise linear.\nTwo PL atlases are *equivalent* if their union is a PL atlas.\nA *piecewise linear manifold* is a topological manifold together with an equivalence class of PL atlases, called a *PL structure*.\n\n## Remarks\n\nThe boundary $Q$ of the square $[0,1]^2$ is a PL manifold of dimension $1$.\nFor each vertex $v$, let $U_v$ be the union of $v$ and the two open edges at $v$.\nUnfold $U_v$ onto an open interval of $\\R$ by a map that sends $v$ to $0$ and is affine on each of the two edges.\nThese four charts cover $Q$, and each transition map is affine on each of finitely many intervals, so it is piecewise linear.\n\nThe two stereographic charts $x/(1 - y)$ and $x/(1 + y)$ of the unit circle $S^1$ do not form a PL atlas.\nTheir transition map $t \\mapsto 1/t$ on $\\R \\setminus \\set{0}$ is not affine on any interval.\nThe circle does have a PL structure, carried over from $Q$ by a homeomorphism $Q \\to S^1$.\n\nEvery smooth manifold has a PL structure, unique up to PL homeomorphism, by a theorem of J. H. C. Whitehead.\nEvery topological manifold of dimension at most $3$ has a unique PL structure, while in higher dimensions a manifold may have none or several.\nEquivalently, a PL manifold of dimension $n$ is the realization of a combinatorial manifold.\nThis is a simplicial complex in which the link of each vertex is PL homeomorphic to the boundary of an $n$-simplex.\n",
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