---
tag: 15MKGRNM9K
kind: concept
name: Piecewise linear manifold
msc: [57Q99, 57Q25]
specializes: [QQQ9B1DK4P]
sources:
  - url: https://en.wikipedia.org/wiki/Piecewise_linear_manifold
same_as:
  wikidata: Q7191425
  wikipedia: Piecewise linear manifold
---

## Summary

A piecewise linear manifold, or PL manifold, is a topological manifold together with an atlas whose changes of chart are piecewise linear.
Equivalently, it is a manifold given with a triangulation in which the neighbourhood of every vertex is a standard piecewise linear ball.
PL manifolds sit between topological and smooth manifolds.
They are the natural setting for combinatorial methods such as handle moves, regular neighbourhoods and simple homotopy theory.

## Explanations

### L1 human

A PL manifold is a manifold built from flat pieces, such as triangles or tetrahedra, glued along faces.
The surface of a cube is a PL $2$-sphere: it is made of flat squares, and its corners are no problem.
Maps between PL manifolds are allowed to bend, but only along flat creases, never in a smooth curve.
Every smooth manifold can be given such a flat-pieces structure, but some topological manifolds cannot.