In
mathematics, a piecewise linear (PL) manifold is a
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...
together with a piecewise linear structure on it. Such a structure can be defined by means of an
atlas
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth.
Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geograp ...
, such that one can pass from
chart to chart in it by
piecewise linear functions. This is slightly stronger than the topological notion of a
triangulation.
An
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
of PL manifolds is called a PL homeomorphism.
Relation to other categories of manifolds
![PDIFF](https://upload.wikimedia.org/wikipedia/commons/6/64/PDIFF.svg)
PL, or more precisely PDIFF, sits between DIFF (the category of
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF — for example, the
Generalized Poincaré conjecture is true in PL (with the possible exception of dimension 4, where it is equivalent to DIFF), but is false generally in DIFF — but is "worse behaved" than TOP, as elaborated in
surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while And ...
.
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 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.
Definition
A smooth structure on a manifold M is ...
s — they are not always ''smoothable.'' This relation can be elaborated by introducing the category
PDIFF
In geometric topology, PDIFF, for ''p''iecewise ''diff''erentiable, is the category of piecewise- smooth manifolds and piecewise-smooth maps between them. It properly contains DIFF (the category of smooth manifolds and smooth functions between ...
, 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
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines conn ...
in PL, but not in DIFF — the cone point is acceptable in PL.
A consequence is that the
Generalized Poincaré conjecture is true in PL for dimensions greater than four — the proof is to take a
homotopy sphere
In algebraic topology, a branch of mathematics, a ''homotopy sphere'' is an ''n''-manifold that is homotopy equivalent to the ''n''-sphere. It thus has the same homotopy groups and the same homology groups as the ''n''-sphere, and so every homotop ...
, remove two balls, apply the
''h''-cobordism 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 spheres.
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
The ''Hauptvermutung'' of geometric topology is a now refuted conjecture asking whether any two triangulations of a triangulable space have subdivisions that are combinatorially equivalent, i.e. the subdivided triangulations are built up in the s ...
.
The obstruction to placing a PL structure on a topological manifold is the
Kirby–Siebenmann class. To be precise, the Kirby-Siebenmann class is the
obstruction
Obstruction may refer to:
Places
* Obstruction Island, in Washington state
* Obstruction Islands, east of New Guinea
Medicine
* Obstructive jaundice
* Obstructive sleep apnea
* Airway obstruction, a respiratory problem
** Recurrent airway o ...
to placing a PL-structure on M x R and in dimensions n > 4, the KS class vanishes if and only if M has at least one PL-structure.
Real algebraic sets
An A-structure on a PL manifold is a structure which gives an inductive way of resolving the PL manifold to a smooth manifold. Compact PL manifolds admit A-structures. Compact PL manifolds are homeomorphic to
real-algebraic sets.
Put another way, A-category sits over the PL-category as a richer category with no obstruction to lifting, that is BA → BPL is a product fibration with BA = BPL × PL/A, and PL manifolds are real algebraic sets because A-manifolds are real algebraic sets.
Combinatorial manifolds and digital manifolds
* A
combinatorial manifold
Digital topology deals with properties and features of two-dimensional (2D) or three-dimensional (3D) digital images
that correspond to topological properties (e.g., connectedness) or topological features (e.g., boundaries) of objects.
Concepts ...
is a kind of manifold which is discretization of a manifold. It usually means a piecewise linear manifold made by
simplicial complexes
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial se ...
.
* A
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 piec ...
is a special kind of combinatorial manifold which is defined in digital space. See
digital topology
Digital topology deals with properties and features of two-dimensional (2D) or three-dimensional (3D) digital images
that correspond to topological properties (e.g., connectedness) or topological features (e.g., boundaries) of objects.
Concepts an ...
.
See also
*
Simplicial manifold
In physics, the term simplicial manifold commonly refers to one of several loosely defined objects, commonly appearing in the study of Regge calculus. These objects combine attributes of a simplex with those of a manifold. There is no standard ...
Notes
References
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{{refend
Structures on manifolds
Geometric topology
Manifolds