Piecewise linear manifold

# Piecewise linear manifold

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a piecewise linear (PL) manifold is a topological manifold
Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...

together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas
Atlas (topology)
In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...

, such that one can pass from chart to chart in it by piecewise linear functions.

An isomorphism of PL manifolds is called a PL homeomorphism.

## Relation to other categories of manifolds

PL, or more precisely PDIFF, sits between DIFF (the category of smooth manifolds) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF – for example, the Generalized Poincaré conjecture
Generalized Poincaré conjecture
In the mathematical area of topology, the term Generalized Poincaré conjecture refers to a statement that a manifold which is a homotopy sphere 'is' a sphere. More precisely, one fixes a...

is true in PL (with the possible exception of dimension 4, where it is equivilent to DIFF), but is false generally in DIFF – but is "worse behaved" than TOP, as elaborated in surgery theory
Surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...

.

### Smooth manifolds

Smooth manifolds have canonical PL structures – they are uniquely triangulizable, by Whitehead's theorem on triangulation – but PL manifolds do not always have smooth structure
Smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold....

s – they are not always smoothable. This relation can be elaborated by introducing the category PDIFF
PDIFF
In geometric topology, PDIFF, for piecewise differentiable, is the category of piecewise-smooth manifolds and piecewise-smooth maps between them...

, which contains both DIFF and PL, and is equivalent to PL.

One way in which PL is better behaved than DIFF is that one can take cones
Cone (topology)
In topology, especially algebraic topology, the cone CX of a topological space X is the quotient space:CX = /\,of the product of X with the unit interval I = [0, 1]....

in PL, but not in DIFF – the cone point is acceptable in PL.
A consequence is that the Generalized Poincaré conjecture
Generalized Poincaré conjecture
In the mathematical area of topology, the term Generalized Poincaré conjecture refers to a statement that a manifold which is a homotopy sphere 'is' a sphere. More precisely, one fixes a...

is true in PL for dimensions greater than 4 – the proof is to take a homotopy sphere, remove two balls, apply the h-cobordism
H-cobordism
A cobordism W between M and N is an h-cobordism if the inclusion mapsare homotopy equivalences...

theorem to conclude that this is a cylinder, and then attach cones to recover a sphere. This last step works in PL but not in DIFF, giving rise to exotic sphere
Exotic sphere
In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...

s.

### Topological manifolds

Not every topological manifold admits a PL structure, and of those that do, the PL structure need not be unique – it can have infinitely many. This is elaborated at Hauptvermutung
Hauptvermutung
The Hauptvermutung of geometric topology is the conjecture that any two triangulations of a triangulable space have a common refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by Steinitz and Tietze.This conjecture is now known to be false...

.

The obstruction to placing a PL structure on a topological manifold is the Kirby–Siebenmann class. To be precise, the Kirby-Sibenmann class is the obstruction to placing a PL-structure on M x R and in dimensions n > 4 this ensures that M has a PL-structure.

### Combinatorial manifolds and digital manifolds

• A combinatorial manifold is a kind of manifold which is discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes.
• A digital manifold
Digital manifold
In mathematics, a digital manifold is a special kind of combinatorial manifold which is defined in digital space i.e. grid cell space. A combinatorial manifold is a kind of manifold which is a discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes.-...

is a special kind of combinatorial manifold which is defined in digital space. See digital topology
Digital topology
Digital topology deals with properties and features of two-dimensional or three-dimensional digital imagesthat correspond to topological properties or topological features of objects....