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

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 ...

Notes

References

* * {{refend Structures on manifolds Geometric topology Manifolds