Math 2.0

Manifolds and cell complexes Definition

Piecewise linear manifold

Builds on Topological manifoldAtlasOpen set

Definition

Let n≥0n \ge 0 and let UU and VV be open subsets of ℝn\R^n. A map f ⁣:U→Vf \colon U \to V is piecewise linear if UU is a union of a locally finite family of simplices in ℝn\R^n with ff affine on each. Any two simplices of the family must meet in a common face or not at all. Let MM be a topological manifold of dimension nn. A PL atlas on MM is an atlas of dimension nn all of whose transition maps are piecewise linear. Two PL atlases are equivalent if their union is a PL atlas. A piecewise linear manifold is a topological manifold together with an equivalence class of PL atlases, called a PL structure.

Remarks

The boundary QQ of the square [0,1]2[0,1]^2 is a PL manifold of dimension 11. For each vertex vv, let UvU_v be the union of vv and the two open edges at vv. Unfold UvU_v onto an open interval of ℝ\R by a map that sends vv to 00 and is affine on each of the two edges. These four charts cover QQ, and each transition map is affine on each of finitely many intervals, so it is piecewise linear.

The two stereographic charts x/(1−y)x/(1 - y) and x/(1+y)x/(1 + y) of the unit circle S1S^1 do not form a PL atlas. Their transition map t↦1/tt \mapsto 1/t on ℝ∖{0}\R \setminus \set{0} is not affine on any interval. The circle does have a PL structure, carried over from QQ by a homeomorphism Q→S1Q \to S^1.

Every smooth manifold has a PL structure, unique up to PL homeomorphism, by a theorem of J. H. C. Whitehead. Every topological manifold of dimension at most 33 has a unique PL structure, while in higher dimensions a manifold may have none or several. Equivalently, a PL manifold of dimension nn is the realization of a combinatorial manifold. This is a simplicial complex in which the link of each vertex is PL homeomorphic to the boundary of an nn-simplex.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Piecewise_linear_manifold
About this page
TagMX38D5Z5NY
URIm2://MX38D5Z5NY
Kinddefinition
WritingW0 stub
MSC57Q99, 57Q25
DefinesPiecewise linear manifold
UsesTopological manifold, Atlas, Open set
Changed2 November 2026

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