mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 mathe ...

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 geographic ...

, such that one can pass from

chart A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tabu ...

to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a

triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...

. 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 is ...

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 An ...

Smooth manifolds

Smooth manifolds have canonical PL structures — they are uniquely ''triangulizable,'' by Whitehead's theorem on

— 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 In mathematics, more specifically in topology, geometric topology, the Kirby–Siebenmann class is an obstruction for topological manifolds to allow a ''PL''-structure. The KS-class For a topological manifold ''M'', the Kirby–Siebenmann class \ka ...

. To be precise, the Kirby-Siebenmann class is the

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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